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Theorem isphtpyd 22593
 Description: Deduction for membership in the class of path homotopies. (Contributed by Mario Carneiro, 23-Feb-2015.)
Hypotheses
Ref Expression
isphtpy.2 (𝜑𝐹 ∈ (II Cn 𝐽))
isphtpy.3 (𝜑𝐺 ∈ (II Cn 𝐽))
isphtpyd.1 (𝜑𝐻 ∈ (𝐹(II Htpy 𝐽)𝐺))
isphtpyd.2 ((𝜑𝑠 ∈ (0[,]1)) → (0𝐻𝑠) = (𝐹‘0))
isphtpyd.3 ((𝜑𝑠 ∈ (0[,]1)) → (1𝐻𝑠) = (𝐹‘1))
Assertion
Ref Expression
isphtpyd (𝜑𝐻 ∈ (𝐹(PHtpy‘𝐽)𝐺))
Distinct variable groups:   𝐹,𝑠   𝐺,𝑠   𝐻,𝑠   𝐽,𝑠   𝜑,𝑠

Proof of Theorem isphtpyd
StepHypRef Expression
1 isphtpyd.1 . 2 (𝜑𝐻 ∈ (𝐹(II Htpy 𝐽)𝐺))
2 isphtpyd.2 . . . 4 ((𝜑𝑠 ∈ (0[,]1)) → (0𝐻𝑠) = (𝐹‘0))
3 isphtpyd.3 . . . 4 ((𝜑𝑠 ∈ (0[,]1)) → (1𝐻𝑠) = (𝐹‘1))
42, 3jca 553 . . 3 ((𝜑𝑠 ∈ (0[,]1)) → ((0𝐻𝑠) = (𝐹‘0) ∧ (1𝐻𝑠) = (𝐹‘1)))
54ralrimiva 2949 . 2 (𝜑 → ∀𝑠 ∈ (0[,]1)((0𝐻𝑠) = (𝐹‘0) ∧ (1𝐻𝑠) = (𝐹‘1)))
6 isphtpy.2 . . 3 (𝜑𝐹 ∈ (II Cn 𝐽))
7 isphtpy.3 . . 3 (𝜑𝐺 ∈ (II Cn 𝐽))
86, 7isphtpy 22588 . 2 (𝜑 → (𝐻 ∈ (𝐹(PHtpy‘𝐽)𝐺) ↔ (𝐻 ∈ (𝐹(II Htpy 𝐽)𝐺) ∧ ∀𝑠 ∈ (0[,]1)((0𝐻𝑠) = (𝐹‘0) ∧ (1𝐻𝑠) = (𝐹‘1)))))
91, 5, 8mpbir2and 959 1 (𝜑𝐻 ∈ (𝐹(PHtpy‘𝐽)𝐺))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896  ‘cfv 5804  (class class class)co 6549  0cc0 9815  1c1 9816  [,]cicc 12049   Cn ccn 20838  IIcii 22486   Htpy chtpy 22574  PHtpycphtpy 22575 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-1st 7059  df-2nd 7060  df-map 7746  df-top 20521  df-topon 20523  df-cn 20841  df-phtpy 22578 This theorem is referenced by:  isphtpy2d  22594  phtpycom  22595  phtpyid  22596  phtpyco2  22597  phtpycc  22598
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