Theorem ishtpyd 24855

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 3121	. 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 24852	. 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 713	1 ⊢ (𝜑 → 𝐻 ∈ (𝐹(𝐽 Htpy 𝐾)𝐺))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∀wral 3044  ‘cfv 6476  (class class class)co 7340  0cc0 10997  1c1 10998  TopOnctopon 22779   Cn ccn 23093   ×_t ctx 23429  IIcii 24749   Htpy chtpy 24847

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5231  ax-nul 5241  ax-pow 5300  ax-pr 5367  ax-un 7662

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3393  df-v 3435  df-sbc 3739  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4281  df-if 4473  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-iun 4940  df-br 5089  df-opab 5151  df-mpt 5170  df-id 5508  df-xp 5619  df-rel 5620  df-cnv 5621  df-co 5622  df-dm 5623  df-rn 5624  df-res 5625  df-ima 5626  df-iota 6432  df-fun 6478  df-fn 6479  df-f 6480  df-fv 6484  df-ov 7343  df-oprab 7344  df-mpo 7345  df-1st 7915  df-2nd 7916  df-map 8746  df-top 22763  df-topon 22780  df-cn 23096  df-htpy 24850

This theorem is referenced by:  htpycom  24856  htpyid  24857  htpyco1  24858  htpyco2  24859  htpycc  24860  isphtpy2d  24867