Theorem ishtpyd 24935

Description: Deduction for membership in the class of homotopies. (Contributed by Mario Carneiro, 22-Feb-2015.)

Hypotheses

Ref	Expression
ishtpy.1	⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))
ishtpy.3	⊢ (𝜑 → 𝐹 ∈ (𝐽 Cn 𝐾))
ishtpy.4	⊢ (𝜑 → 𝐺 ∈ (𝐽 Cn 𝐾))
ishtpyd.1	⊢ (𝜑 → 𝐻 ∈ ((𝐽 ×t II) Cn 𝐾))
ishtpyd.2	⊢ ((𝜑 ∧ 𝑠 ∈ 𝑋) → (𝑠𝐻0) = (𝐹‘𝑠))
ishtpyd.3	⊢ ((𝜑 ∧ 𝑠 ∈ 𝑋) → (𝑠𝐻1) = (𝐺‘𝑠))

Assertion

Ref	Expression
ishtpyd	⊢ (𝜑 → 𝐻 ∈ (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

Step	Hyp	Ref	Expression
1		ishtpyd.1	. 2 ⊢ (𝜑 → 𝐻 ∈ ((𝐽 ×t II) Cn 𝐾))
2		ishtpyd.2	. . . 4 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑋) → (𝑠𝐻0) = (𝐹‘𝑠))
3		ishtpyd.3	. . . 4 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑋) → (𝑠𝐻1) = (𝐺‘𝑠))
4	2, 3	jca 511	. . 3 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑋) → ((𝑠𝐻0) = (𝐹‘𝑠) ∧ (𝑠𝐻1) = (𝐺‘𝑠)))
5	4	ralrimiva 3129	. 2 ⊢ (𝜑 → ∀𝑠 ∈ 𝑋 ((𝑠𝐻0) = (𝐹‘𝑠) ∧ (𝑠𝐻1) = (𝐺‘𝑠)))
6		ishtpy.1	. . 3 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))
7		ishtpy.3	. . 3 ⊢ (𝜑 → 𝐹 ∈ (𝐽 Cn 𝐾))
8		ishtpy.4	. . 3 ⊢ (𝜑 → 𝐺 ∈ (𝐽 Cn 𝐾))
9	6, 7, 8	ishtpy 24932	. 2 ⊢ (𝜑 → (𝐻 ∈ (𝐹(𝐽 Htpy 𝐾)𝐺) ↔ (𝐻 ∈ ((𝐽 ×t II) Cn 𝐾) ∧ ∀𝑠 ∈ 𝑋 ((𝑠𝐻0) = (𝐹‘𝑠) ∧ (𝑠𝐻1) = (𝐺‘𝑠)))))
10	1, 5, 9	mpbir2and 714	1 ⊢ (𝜑 → 𝐻 ∈ (𝐹(𝐽 Htpy 𝐾)𝐺))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ∀wral 3052  ‘cfv 6493  (class class class)co 7361  0cc0 11031  1c1 11032  TopOnctopon 22859   Cn ccn 23173   ×_t ctx 23509  IIcii 24829   Htpy chtpy 24927

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5242  ax-nul 5252  ax-pow 5311  ax-pr 5378  ax-un 7683

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-rab 3401  df-v 3443  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4287  df-if 4481  df-pw 4557  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-iun 4949  df-br 5100  df-opab 5162  df-mpt 5181  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-fv 6501  df-ov 7364  df-oprab 7365  df-mpo 7366  df-1st 7936  df-2nd 7937  df-map 8770  df-top 22843  df-topon 22860  df-cn 23176  df-htpy 24930

This theorem is referenced by:  htpycom  24936  htpyid  24937  htpyco1  24938  htpyco2  24939  htpycc  24940  isphtpy2d  24947