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Theorem fsnex 6503
 Description: Relate a function with a singleton as domain and one variable. (Contributed by Thierry Arnoux, 12-Jul-2020.)
Hypothesis
Ref Expression
fsnex.1 (𝑥 = (𝑓𝐴) → (𝜓𝜑))
Assertion
Ref Expression
fsnex (𝐴𝑉 → (∃𝑓(𝑓:{𝐴}⟶𝐷𝜑) ↔ ∃𝑥𝐷 𝜓))
Distinct variable groups:   𝐴,𝑓,𝑥   𝐷,𝑓,𝑥   𝑓,𝑉,𝑥   𝜓,𝑓   𝜑,𝑥
Allowed substitution hints:   𝜑(𝑓)   𝜓(𝑥)

Proof of Theorem fsnex
StepHypRef Expression
1 fsn2g 6370 . . . . . . . 8 (𝐴𝑉 → (𝑓:{𝐴}⟶𝐷 ↔ ((𝑓𝐴) ∈ 𝐷𝑓 = {⟨𝐴, (𝑓𝐴)⟩})))
21simprbda 652 . . . . . . 7 ((𝐴𝑉𝑓:{𝐴}⟶𝐷) → (𝑓𝐴) ∈ 𝐷)
32adantrr 752 . . . . . 6 ((𝐴𝑉 ∧ (𝑓:{𝐴}⟶𝐷𝜑)) → (𝑓𝐴) ∈ 𝐷)
4 fsnex.1 . . . . . . 7 (𝑥 = (𝑓𝐴) → (𝜓𝜑))
54adantl 482 . . . . . 6 (((𝐴𝑉 ∧ (𝑓:{𝐴}⟶𝐷𝜑)) ∧ 𝑥 = (𝑓𝐴)) → (𝜓𝜑))
6 simprr 795 . . . . . 6 ((𝐴𝑉 ∧ (𝑓:{𝐴}⟶𝐷𝜑)) → 𝜑)
73, 5, 6rspcedvd 3306 . . . . 5 ((𝐴𝑉 ∧ (𝑓:{𝐴}⟶𝐷𝜑)) → ∃𝑥𝐷 𝜓)
87ex 450 . . . 4 (𝐴𝑉 → ((𝑓:{𝐴}⟶𝐷𝜑) → ∃𝑥𝐷 𝜓))
98exlimdv 1858 . . 3 (𝐴𝑉 → (∃𝑓(𝑓:{𝐴}⟶𝐷𝜑) → ∃𝑥𝐷 𝜓))
109imp 445 . 2 ((𝐴𝑉 ∧ ∃𝑓(𝑓:{𝐴}⟶𝐷𝜑)) → ∃𝑥𝐷 𝜓)
11 nfv 1840 . . . 4 𝑥 𝐴𝑉
12 nfre1 3001 . . . 4 𝑥𝑥𝐷 𝜓
1311, 12nfan 1825 . . 3 𝑥(𝐴𝑉 ∧ ∃𝑥𝐷 𝜓)
14 vex 3193 . . . . . . . . 9 𝑥 ∈ V
15 f1osng 6144 . . . . . . . . 9 ((𝐴𝑉𝑥 ∈ V) → {⟨𝐴, 𝑥⟩}:{𝐴}–1-1-onto→{𝑥})
1614, 15mpan2 706 . . . . . . . 8 (𝐴𝑉 → {⟨𝐴, 𝑥⟩}:{𝐴}–1-1-onto→{𝑥})
1716ad3antrrr 765 . . . . . . 7 ((((𝐴𝑉 ∧ ∃𝑥𝐷 𝜓) ∧ 𝑥𝐷) ∧ 𝜓) → {⟨𝐴, 𝑥⟩}:{𝐴}–1-1-onto→{𝑥})
18 f1of 6104 . . . . . . 7 ({⟨𝐴, 𝑥⟩}:{𝐴}–1-1-onto→{𝑥} → {⟨𝐴, 𝑥⟩}:{𝐴}⟶{𝑥})
1917, 18syl 17 . . . . . 6 ((((𝐴𝑉 ∧ ∃𝑥𝐷 𝜓) ∧ 𝑥𝐷) ∧ 𝜓) → {⟨𝐴, 𝑥⟩}:{𝐴}⟶{𝑥})
20 simplr 791 . . . . . . 7 ((((𝐴𝑉 ∧ ∃𝑥𝐷 𝜓) ∧ 𝑥𝐷) ∧ 𝜓) → 𝑥𝐷)
2120snssd 4316 . . . . . 6 ((((𝐴𝑉 ∧ ∃𝑥𝐷 𝜓) ∧ 𝑥𝐷) ∧ 𝜓) → {𝑥} ⊆ 𝐷)
22 fss 6023 . . . . . 6 (({⟨𝐴, 𝑥⟩}:{𝐴}⟶{𝑥} ∧ {𝑥} ⊆ 𝐷) → {⟨𝐴, 𝑥⟩}:{𝐴}⟶𝐷)
2319, 21, 22syl2anc 692 . . . . 5 ((((𝐴𝑉 ∧ ∃𝑥𝐷 𝜓) ∧ 𝑥𝐷) ∧ 𝜓) → {⟨𝐴, 𝑥⟩}:{𝐴}⟶𝐷)
24 fvsng 6412 . . . . . . . 8 ((𝐴𝑉𝑥 ∈ V) → ({⟨𝐴, 𝑥⟩}‘𝐴) = 𝑥)
2514, 24mpan2 706 . . . . . . 7 (𝐴𝑉 → ({⟨𝐴, 𝑥⟩}‘𝐴) = 𝑥)
2625eqcomd 2627 . . . . . 6 (𝐴𝑉𝑥 = ({⟨𝐴, 𝑥⟩}‘𝐴))
2726ad3antrrr 765 . . . . 5 ((((𝐴𝑉 ∧ ∃𝑥𝐷 𝜓) ∧ 𝑥𝐷) ∧ 𝜓) → 𝑥 = ({⟨𝐴, 𝑥⟩}‘𝐴))
28 snex 4879 . . . . . 6 {⟨𝐴, 𝑥⟩} ∈ V
29 feq1 5993 . . . . . . 7 (𝑓 = {⟨𝐴, 𝑥⟩} → (𝑓:{𝐴}⟶𝐷 ↔ {⟨𝐴, 𝑥⟩}:{𝐴}⟶𝐷))
30 fveq1 6157 . . . . . . . 8 (𝑓 = {⟨𝐴, 𝑥⟩} → (𝑓𝐴) = ({⟨𝐴, 𝑥⟩}‘𝐴))
3130eqeq2d 2631 . . . . . . 7 (𝑓 = {⟨𝐴, 𝑥⟩} → (𝑥 = (𝑓𝐴) ↔ 𝑥 = ({⟨𝐴, 𝑥⟩}‘𝐴)))
3229, 31anbi12d 746 . . . . . 6 (𝑓 = {⟨𝐴, 𝑥⟩} → ((𝑓:{𝐴}⟶𝐷𝑥 = (𝑓𝐴)) ↔ ({⟨𝐴, 𝑥⟩}:{𝐴}⟶𝐷𝑥 = ({⟨𝐴, 𝑥⟩}‘𝐴))))
3328, 32spcev 3290 . . . . 5 (({⟨𝐴, 𝑥⟩}:{𝐴}⟶𝐷𝑥 = ({⟨𝐴, 𝑥⟩}‘𝐴)) → ∃𝑓(𝑓:{𝐴}⟶𝐷𝑥 = (𝑓𝐴)))
3423, 27, 33syl2anc 692 . . . 4 ((((𝐴𝑉 ∧ ∃𝑥𝐷 𝜓) ∧ 𝑥𝐷) ∧ 𝜓) → ∃𝑓(𝑓:{𝐴}⟶𝐷𝑥 = (𝑓𝐴)))
35 simprl 793 . . . . . . 7 (((((𝐴𝑉 ∧ ∃𝑥𝐷 𝜓) ∧ 𝑥𝐷) ∧ 𝜓) ∧ (𝑓:{𝐴}⟶𝐷𝑥 = (𝑓𝐴))) → 𝑓:{𝐴}⟶𝐷)
36 simpllr 798 . . . . . . . . 9 ((((((𝐴𝑉 ∧ ∃𝑥𝐷 𝜓) ∧ 𝑥𝐷) ∧ 𝜓) ∧ (𝑓:{𝐴}⟶𝐷𝑥 = (𝑓𝐴))) ∧ 𝑓:{𝐴}⟶𝐷) → 𝜓)
37 simplrr 800 . . . . . . . . . 10 ((((((𝐴𝑉 ∧ ∃𝑥𝐷 𝜓) ∧ 𝑥𝐷) ∧ 𝜓) ∧ (𝑓:{𝐴}⟶𝐷𝑥 = (𝑓𝐴))) ∧ 𝑓:{𝐴}⟶𝐷) → 𝑥 = (𝑓𝐴))
3837, 4syl 17 . . . . . . . . 9 ((((((𝐴𝑉 ∧ ∃𝑥𝐷 𝜓) ∧ 𝑥𝐷) ∧ 𝜓) ∧ (𝑓:{𝐴}⟶𝐷𝑥 = (𝑓𝐴))) ∧ 𝑓:{𝐴}⟶𝐷) → (𝜓𝜑))
3936, 38mpbid 222 . . . . . . . 8 ((((((𝐴𝑉 ∧ ∃𝑥𝐷 𝜓) ∧ 𝑥𝐷) ∧ 𝜓) ∧ (𝑓:{𝐴}⟶𝐷𝑥 = (𝑓𝐴))) ∧ 𝑓:{𝐴}⟶𝐷) → 𝜑)
4035, 39mpdan 701 . . . . . . 7 (((((𝐴𝑉 ∧ ∃𝑥𝐷 𝜓) ∧ 𝑥𝐷) ∧ 𝜓) ∧ (𝑓:{𝐴}⟶𝐷𝑥 = (𝑓𝐴))) → 𝜑)
4135, 40jca 554 . . . . . 6 (((((𝐴𝑉 ∧ ∃𝑥𝐷 𝜓) ∧ 𝑥𝐷) ∧ 𝜓) ∧ (𝑓:{𝐴}⟶𝐷𝑥 = (𝑓𝐴))) → (𝑓:{𝐴}⟶𝐷𝜑))
4241ex 450 . . . . 5 ((((𝐴𝑉 ∧ ∃𝑥𝐷 𝜓) ∧ 𝑥𝐷) ∧ 𝜓) → ((𝑓:{𝐴}⟶𝐷𝑥 = (𝑓𝐴)) → (𝑓:{𝐴}⟶𝐷𝜑)))
4342eximdv 1843 . . . 4 ((((𝐴𝑉 ∧ ∃𝑥𝐷 𝜓) ∧ 𝑥𝐷) ∧ 𝜓) → (∃𝑓(𝑓:{𝐴}⟶𝐷𝑥 = (𝑓𝐴)) → ∃𝑓(𝑓:{𝐴}⟶𝐷𝜑)))
4434, 43mpd 15 . . 3 ((((𝐴𝑉 ∧ ∃𝑥𝐷 𝜓) ∧ 𝑥𝐷) ∧ 𝜓) → ∃𝑓(𝑓:{𝐴}⟶𝐷𝜑))
45 simpr 477 . . 3 ((𝐴𝑉 ∧ ∃𝑥𝐷 𝜓) → ∃𝑥𝐷 𝜓)
4613, 44, 45r19.29af 3071 . 2 ((𝐴𝑉 ∧ ∃𝑥𝐷 𝜓) → ∃𝑓(𝑓:{𝐴}⟶𝐷𝜑))
4710, 46impbida 876 1 (𝐴𝑉 → (∃𝑓(𝑓:{𝐴}⟶𝐷𝜑) ↔ ∃𝑥𝐷 𝜓))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1480  ∃wex 1701   ∈ wcel 1987  ∃wrex 2909  Vcvv 3190   ⊆ wss 3560  {csn 4155  ⟨cop 4161  ⟶wf 5853  –1-1-onto→wf1o 5856  ‘cfv 5857 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751  ax-nul 4759  ax-pr 4877 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2913  df-rex 2914  df-reu 2915  df-rab 2917  df-v 3192  df-sbc 3423  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-br 4624  df-opab 4684  df-id 4999  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-fo 5863  df-f1o 5864  df-fv 5865 This theorem is referenced by: (None)
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