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Theorem fvmpt2d 5257
 Description: Deduction version of fvmpt2 5254. (Contributed by Thierry Arnoux, 8-Dec-2016.)
Hypotheses
Ref Expression
fvmpt2d.1 (𝜑𝐹 = (𝑥𝐴𝐵))
fvmpt2d.4 ((𝜑𝑥𝐴) → 𝐵𝑉)
Assertion
Ref Expression
fvmpt2d ((𝜑𝑥𝐴) → (𝐹𝑥) = 𝐵)
Distinct variable group:   𝑥,𝐴
Allowed substitution hints:   𝜑(𝑥)   𝐵(𝑥)   𝐹(𝑥)   𝑉(𝑥)

Proof of Theorem fvmpt2d
StepHypRef Expression
1 fvmpt2d.1 . . . 4 (𝜑𝐹 = (𝑥𝐴𝐵))
21fveq1d 5180 . . 3 (𝜑 → (𝐹𝑥) = ((𝑥𝐴𝐵)‘𝑥))
32adantr 261 . 2 ((𝜑𝑥𝐴) → (𝐹𝑥) = ((𝑥𝐴𝐵)‘𝑥))
4 simpr 103 . . 3 ((𝜑𝑥𝐴) → 𝑥𝐴)
5 fvmpt2d.4 . . 3 ((𝜑𝑥𝐴) → 𝐵𝑉)
6 eqid 2040 . . . 4 (𝑥𝐴𝐵) = (𝑥𝐴𝐵)
76fvmpt2 5254 . . 3 ((𝑥𝐴𝐵𝑉) → ((𝑥𝐴𝐵)‘𝑥) = 𝐵)
84, 5, 7syl2anc 391 . 2 ((𝜑𝑥𝐴) → ((𝑥𝐴𝐵)‘𝑥) = 𝐵)
93, 8eqtrd 2072 1 ((𝜑𝑥𝐴) → (𝐹𝑥) = 𝐵)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   = wceq 1243   ∈ wcel 1393   ↦ cmpt 3818  ‘cfv 4902 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944 This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-sbc 2765  df-csb 2853  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-br 3765  df-opab 3819  df-mpt 3820  df-id 4030  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-iota 4867  df-fun 4904  df-fv 4910 This theorem is referenced by: (None)
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