Theorem htpyi 24186

Description: A homotopy evaluated at its endpoints. (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))
htpyi.1	⊢ (𝜑 → 𝐻 ∈ (𝐹(𝐽 Htpy 𝐾)𝐺))

Assertion

Ref	Expression
htpyi	⊢ ((𝜑 ∧ 𝐴 ∈ 𝑋) → ((𝐴𝐻0) = (𝐹‘𝐴) ∧ (𝐴𝐻1) = (𝐺‘𝐴)))

Proof of Theorem htpyi

Dummy variable 𝑠 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		htpyi.1	. . . 4 ⊢ (𝜑 → 𝐻 ∈ (𝐹(𝐽 Htpy 𝐾)𝐺))
2		ishtpy.1	. . . . 5 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))
3		ishtpy.3	. . . . 5 ⊢ (𝜑 → 𝐹 ∈ (𝐽 Cn 𝐾))
4		ishtpy.4	. . . . 5 ⊢ (𝜑 → 𝐺 ∈ (𝐽 Cn 𝐾))
5	2, 3, 4	ishtpy 24184	. . . 4 ⊢ (𝜑 → (𝐻 ∈ (𝐹(𝐽 Htpy 𝐾)𝐺) ↔ (𝐻 ∈ ((𝐽 ×t II) Cn 𝐾) ∧ ∀𝑠 ∈ 𝑋 ((𝑠𝐻0) = (𝐹‘𝑠) ∧ (𝑠𝐻1) = (𝐺‘𝑠)))))
6	1, 5	mpbid 231	. . 3 ⊢ (𝜑 → (𝐻 ∈ ((𝐽 ×t II) Cn 𝐾) ∧ ∀𝑠 ∈ 𝑋 ((𝑠𝐻0) = (𝐹‘𝑠) ∧ (𝑠𝐻1) = (𝐺‘𝑠))))
7	6	simprd 497	. 2 ⊢ (𝜑 → ∀𝑠 ∈ 𝑋 ((𝑠𝐻0) = (𝐹‘𝑠) ∧ (𝑠𝐻1) = (𝐺‘𝑠)))
8		oveq1 7314	. . . . 5 ⊢ (𝑠 = 𝐴 → (𝑠𝐻0) = (𝐴𝐻0))
9		fveq2 6804	. . . . 5 ⊢ (𝑠 = 𝐴 → (𝐹‘𝑠) = (𝐹‘𝐴))
10	8, 9	eqeq12d 2752	. . . 4 ⊢ (𝑠 = 𝐴 → ((𝑠𝐻0) = (𝐹‘𝑠) ↔ (𝐴𝐻0) = (𝐹‘𝐴)))
11		oveq1 7314	. . . . 5 ⊢ (𝑠 = 𝐴 → (𝑠𝐻1) = (𝐴𝐻1))
12		fveq2 6804	. . . . 5 ⊢ (𝑠 = 𝐴 → (𝐺‘𝑠) = (𝐺‘𝐴))
13	11, 12	eqeq12d 2752	. . . 4 ⊢ (𝑠 = 𝐴 → ((𝑠𝐻1) = (𝐺‘𝑠) ↔ (𝐴𝐻1) = (𝐺‘𝐴)))
14	10, 13	anbi12d 632	. . 3 ⊢ (𝑠 = 𝐴 → (((𝑠𝐻0) = (𝐹‘𝑠) ∧ (𝑠𝐻1) = (𝐺‘𝑠)) ↔ ((𝐴𝐻0) = (𝐹‘𝐴) ∧ (𝐴𝐻1) = (𝐺‘𝐴))))
15	14	rspccva 3565	. 2 ⊢ ((∀𝑠 ∈ 𝑋 ((𝑠𝐻0) = (𝐹‘𝑠) ∧ (𝑠𝐻1) = (𝐺‘𝑠)) ∧ 𝐴 ∈ 𝑋) → ((𝐴𝐻0) = (𝐹‘𝐴) ∧ (𝐴𝐻1) = (𝐺‘𝐴)))
16	7, 15	sylan 581	1 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑋) → ((𝐴𝐻0) = (𝐹‘𝐴) ∧ (𝐴𝐻1) = (𝐺‘𝐴)))

Syntax hints:   → wi 4   ∧ wa 397   = wceq 1539   ∈ wcel 2104  ∀wral 3062  ‘cfv 6458  (class class class)co 7307  0cc0 10921  1c1 10922  TopOnctopon 22108   Cn ccn 22424   ×_t ctx 22760  IIcii 24087   Htpy chtpy 24179

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 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2707  ax-sep 5232  ax-nul 5239  ax-pow 5297  ax-pr 5361  ax-un 7620

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 846  df-3an 1089  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3306  df-v 3439  df-sbc 3722  df-csb 3838  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-nul 4263  df-if 4466  df-pw 4541  df-sn 4566  df-pr 4568  df-op 4572  df-uni 4845  df-iun 4933  df-br 5082  df-opab 5144  df-mpt 5165  df-id 5500  df-xp 5606  df-rel 5607  df-cnv 5608  df-co 5609  df-dm 5610  df-rn 5611  df-res 5612  df-ima 5613  df-iota 6410  df-fun 6460  df-fn 6461  df-f 6462  df-fv 6466  df-ov 7310  df-oprab 7311  df-mpo 7312  df-1st 7863  df-2nd 7864  df-map 8648  df-top 22092  df-topon 22109  df-cn 22427  df-htpy 24182

This theorem is referenced by:  htpycom  24188  htpyco1  24190  htpyco2  24191  htpycc  24192  phtpy01  24197  pcohtpylem  24231  txsconnlem  33251  cvmliftphtlem  33328