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Theorem ishtpyd 23182
 Description: Deduction for membership in the class of homotopies. (Contributed by Mario Carneiro, 22-Feb-2015.)
Hypotheses
Ref Expression
ishtpy.1 (𝜑𝐽 ∈ (TopOn‘𝑋))
ishtpy.3 (𝜑𝐹 ∈ (𝐽 Cn 𝐾))
ishtpy.4 (𝜑𝐺 ∈ (𝐽 Cn 𝐾))
ishtpyd.1 (𝜑𝐻 ∈ ((𝐽 ×t II) Cn 𝐾))
ishtpyd.2 ((𝜑𝑠𝑋) → (𝑠𝐻0) = (𝐹𝑠))
ishtpyd.3 ((𝜑𝑠𝑋) → (𝑠𝐻1) = (𝐺𝑠))
Assertion
Ref Expression
ishtpyd (𝜑𝐻 ∈ (𝐹(𝐽 Htpy 𝐾)𝐺))
Distinct variable groups:   𝐹,𝑠   𝐺,𝑠   𝐻,𝑠   𝐽,𝑠   𝜑,𝑠   𝑋,𝑠
Allowed substitution hint:   𝐾(𝑠)

Proof of Theorem ishtpyd
StepHypRef Expression
1 ishtpyd.1 . 2 (𝜑𝐻 ∈ ((𝐽 ×t II) Cn 𝐾))
2 ishtpyd.2 . . . 4 ((𝜑𝑠𝑋) → (𝑠𝐻0) = (𝐹𝑠))
3 ishtpyd.3 . . . 4 ((𝜑𝑠𝑋) → (𝑠𝐻1) = (𝐺𝑠))
42, 3jca 507 . . 3 ((𝜑𝑠𝑋) → ((𝑠𝐻0) = (𝐹𝑠) ∧ (𝑠𝐻1) = (𝐺𝑠)))
54ralrimiva 3148 . 2 (𝜑 → ∀𝑠𝑋 ((𝑠𝐻0) = (𝐹𝑠) ∧ (𝑠𝐻1) = (𝐺𝑠)))
6 ishtpy.1 . . 3 (𝜑𝐽 ∈ (TopOn‘𝑋))
7 ishtpy.3 . . 3 (𝜑𝐹 ∈ (𝐽 Cn 𝐾))
8 ishtpy.4 . . 3 (𝜑𝐺 ∈ (𝐽 Cn 𝐾))
96, 7, 8ishtpy 23179 . 2 (𝜑 → (𝐻 ∈ (𝐹(𝐽 Htpy 𝐾)𝐺) ↔ (𝐻 ∈ ((𝐽 ×t II) Cn 𝐾) ∧ ∀𝑠𝑋 ((𝑠𝐻0) = (𝐹𝑠) ∧ (𝑠𝐻1) = (𝐺𝑠)))))
101, 5, 9mpbir2and 703 1 (𝜑𝐻 ∈ (𝐹(𝐽 Htpy 𝐾)𝐺))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 386   = wceq 1601   ∈ wcel 2107  ∀wral 3090  ‘cfv 6135  (class class class)co 6922  0cc0 10272  1c1 10273  TopOnctopon 21122   Cn ccn 21436   ×t ctx 21772  IIcii 23086   Htpy chtpy 23174 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138  ax-un 7226 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-ral 3095  df-rex 3096  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4672  df-iun 4755  df-br 4887  df-opab 4949  df-mpt 4966  df-id 5261  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-res 5367  df-ima 5368  df-iota 6099  df-fun 6137  df-fn 6138  df-f 6139  df-fv 6143  df-ov 6925  df-oprab 6926  df-mpt2 6927  df-1st 7445  df-2nd 7446  df-map 8142  df-top 21106  df-topon 21123  df-cn 21439  df-htpy 23177 This theorem is referenced by:  htpycom  23183  htpyid  23184  htpyco1  23185  htpyco2  23186  htpycc  23187  isphtpy2d  23194
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