Theorem phtpyi 24939

Description: Membership in the class of path homotopies between two continuous functions. (Contributed by Mario Carneiro, 23-Feb-2015.)

Hypotheses

Ref	Expression
isphtpy.2	⊢ (𝜑 → 𝐹 ∈ (II Cn 𝐽))
isphtpy.3	⊢ (𝜑 → 𝐺 ∈ (II Cn 𝐽))
phtpyi.1	⊢ (𝜑 → 𝐻 ∈ (𝐹(PHtpy‘𝐽)𝐺))

Assertion

Ref	Expression
phtpyi	⊢ ((𝜑 ∧ 𝐴 ∈ (0[,]1)) → ((0𝐻𝐴) = (𝐹‘0) ∧ (1𝐻𝐴) = (𝐹‘1)))

Proof of Theorem phtpyi

Dummy variable 𝑠 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		phtpyi.1	. . . 4 ⊢ (𝜑 → 𝐻 ∈ (𝐹(PHtpy‘𝐽)𝐺))
2		isphtpy.2	. . . . 5 ⊢ (𝜑 → 𝐹 ∈ (II Cn 𝐽))
3		isphtpy.3	. . . . 5 ⊢ (𝜑 → 𝐺 ∈ (II Cn 𝐽))
4	2, 3	isphtpy 24936	. . . 4 ⊢ (𝜑 → (𝐻 ∈ (𝐹(PHtpy‘𝐽)𝐺) ↔ (𝐻 ∈ (𝐹(II Htpy 𝐽)𝐺) ∧ ∀𝑠 ∈ (0[,]1)((0𝐻𝑠) = (𝐹‘0) ∧ (1𝐻𝑠) = (𝐹‘1)))))
5	1, 4	mpbid 232	. . 3 ⊢ (𝜑 → (𝐻 ∈ (𝐹(II Htpy 𝐽)𝐺) ∧ ∀𝑠 ∈ (0[,]1)((0𝐻𝑠) = (𝐹‘0) ∧ (1𝐻𝑠) = (𝐹‘1))))
6	5	simprd 495	. 2 ⊢ (𝜑 → ∀𝑠 ∈ (0[,]1)((0𝐻𝑠) = (𝐹‘0) ∧ (1𝐻𝑠) = (𝐹‘1)))
7		oveq2 7366	. . . . 5 ⊢ (𝑠 = 𝐴 → (0𝐻𝑠) = (0𝐻𝐴))
8	7	eqeq1d 2738	. . . 4 ⊢ (𝑠 = 𝐴 → ((0𝐻𝑠) = (𝐹‘0) ↔ (0𝐻𝐴) = (𝐹‘0)))
9		oveq2 7366	. . . . 5 ⊢ (𝑠 = 𝐴 → (1𝐻𝑠) = (1𝐻𝐴))
10	9	eqeq1d 2738	. . . 4 ⊢ (𝑠 = 𝐴 → ((1𝐻𝑠) = (𝐹‘1) ↔ (1𝐻𝐴) = (𝐹‘1)))
11	8, 10	anbi12d 632	. . 3 ⊢ (𝑠 = 𝐴 → (((0𝐻𝑠) = (𝐹‘0) ∧ (1𝐻𝑠) = (𝐹‘1)) ↔ ((0𝐻𝐴) = (𝐹‘0) ∧ (1𝐻𝐴) = (𝐹‘1))))
12	11	rspccva 3575	. 2 ⊢ ((∀𝑠 ∈ (0[,]1)((0𝐻𝑠) = (𝐹‘0) ∧ (1𝐻𝑠) = (𝐹‘1)) ∧ 𝐴 ∈ (0[,]1)) → ((0𝐻𝐴) = (𝐹‘0) ∧ (1𝐻𝐴) = (𝐹‘1)))
13	6, 12	sylan 580	1 ⊢ ((𝜑 ∧ 𝐴 ∈ (0[,]1)) → ((0𝐻𝐴) = (𝐹‘0) ∧ (1𝐻𝐴) = (𝐹‘1)))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2113  ∀wral 3051  ‘cfv 6492  (class class class)co 7358  0cc0 11026  1c1 11027  [,]cicc 13264   Cn ccn 23168  IIcii 24824   Htpy chtpy 24922  PHtpycphtpy 24923

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-rep 5224  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-ov 7361  df-oprab 7362  df-mpo 7363  df-1st 7933  df-2nd 7934  df-map 8765  df-top 22838  df-topon 22855  df-cn 23171  df-phtpy 24926

This theorem is referenced by:  phtpy01  24940  phtpycom  24943  phtpyco2  24945  phtpycc  24946  pcohtpylem  24975  txsconnlem  35434  cvmliftphtlem  35511  cvmliftpht  35512