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Theorem ishtpyd 24338
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 512 . . 3 ((𝜑𝑠𝑋) → ((𝑠𝐻0) = (𝐹𝑠) ∧ (𝑠𝐻1) = (𝐺𝑠)))
54ralrimiva 3143 . 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 24335 . 2 (𝜑 → (𝐻 ∈ (𝐹(𝐽 Htpy 𝐾)𝐺) ↔ (𝐻 ∈ ((𝐽 ×t II) Cn 𝐾) ∧ ∀𝑠𝑋 ((𝑠𝐻0) = (𝐹𝑠) ∧ (𝑠𝐻1) = (𝐺𝑠)))))
101, 5, 9mpbir2and 711 1 (𝜑𝐻 ∈ (𝐹(𝐽 Htpy 𝐾)𝐺))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1541  wcel 2106  wral 3064  cfv 6496  (class class class)co 7357  0cc0 11051  1c1 11052  TopOnctopon 22259   Cn ccn 22575   ×t ctx 22911  IIcii 24238   Htpy chtpy 24330
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-fv 6504  df-ov 7360  df-oprab 7361  df-mpo 7362  df-1st 7921  df-2nd 7922  df-map 8767  df-top 22243  df-topon 22260  df-cn 22578  df-htpy 24333
This theorem is referenced by:  htpycom  24339  htpyid  24340  htpyco1  24341  htpyco2  24342  htpycc  24343  isphtpy2d  24350
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