Theorem htpyi 22773

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 22771	. . . 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 222	. . 3 ⊢ (𝜑 → (𝐻 ∈ ((𝐽 ×t II) Cn 𝐾) ∧ ∀𝑠 ∈ 𝑋 ((𝑠𝐻0) = (𝐹‘𝑠) ∧ (𝑠𝐻1) = (𝐺‘𝑠))))
7	6	simprd 479	. 2 ⊢ (𝜑 → ∀𝑠 ∈ 𝑋 ((𝑠𝐻0) = (𝐹‘𝑠) ∧ (𝑠𝐻1) = (𝐺‘𝑠)))
8		oveq1 6657	. . . . 5 ⊢ (𝑠 = 𝐴 → (𝑠𝐻0) = (𝐴𝐻0))
9		fveq2 6191	. . . . 5 ⊢ (𝑠 = 𝐴 → (𝐹‘𝑠) = (𝐹‘𝐴))
10	8, 9	eqeq12d 2637	. . . 4 ⊢ (𝑠 = 𝐴 → ((𝑠𝐻0) = (𝐹‘𝑠) ↔ (𝐴𝐻0) = (𝐹‘𝐴)))
11		oveq1 6657	. . . . 5 ⊢ (𝑠 = 𝐴 → (𝑠𝐻1) = (𝐴𝐻1))
12		fveq2 6191	. . . . 5 ⊢ (𝑠 = 𝐴 → (𝐺‘𝑠) = (𝐺‘𝐴))
13	11, 12	eqeq12d 2637	. . . 4 ⊢ (𝑠 = 𝐴 → ((𝑠𝐻1) = (𝐺‘𝑠) ↔ (𝐴𝐻1) = (𝐺‘𝐴)))
14	10, 13	anbi12d 747	. . 3 ⊢ (𝑠 = 𝐴 → (((𝑠𝐻0) = (𝐹‘𝑠) ∧ (𝑠𝐻1) = (𝐺‘𝑠)) ↔ ((𝐴𝐻0) = (𝐹‘𝐴) ∧ (𝐴𝐻1) = (𝐺‘𝐴))))
15	14	rspccva 3308	. 2 ⊢ ((∀𝑠 ∈ 𝑋 ((𝑠𝐻0) = (𝐹‘𝑠) ∧ (𝑠𝐻1) = (𝐺‘𝑠)) ∧ 𝐴 ∈ 𝑋) → ((𝐴𝐻0) = (𝐹‘𝐴) ∧ (𝐴𝐻1) = (𝐺‘𝐴)))
16	7, 15	sylan 488	1 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑋) → ((𝐴𝐻0) = (𝐹‘𝐴) ∧ (𝐴𝐻1) = (𝐺‘𝐴)))

Syntax hints:   → wi 4   ∧ wa 384   = wceq 1483   ∈ wcel 1990  ∀wral 2912  ‘cfv 5888  (class class class)co 6650  0cc0 9936  1c1 9937  TopOnctopon 20715   Cn ccn 21028   ×_t ctx 21363  IIcii 22678   Htpy chtpy 22766

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-map 7859  df-top 20699  df-topon 20716  df-cn 21031  df-htpy 22769

This theorem is referenced by:  htpycom  22775  htpyco1  22777  htpyco2  22778  htpycc  22779  phtpy01  22784  pcohtpylem  22819  txsconnlem  31222  cvmliftphtlem  31299