Theorem phtpyi 24883

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 24880	. . . 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 7395	. . . . 5 ⊢ (𝑠 = 𝐴 → (0𝐻𝑠) = (0𝐻𝐴))
8	7	eqeq1d 2731	. . . 4 ⊢ (𝑠 = 𝐴 → ((0𝐻𝑠) = (𝐹‘0) ↔ (0𝐻𝐴) = (𝐹‘0)))
9		oveq2 7395	. . . . 5 ⊢ (𝑠 = 𝐴 → (1𝐻𝑠) = (1𝐻𝐴))
10	9	eqeq1d 2731	. . . 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 3587	. 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 1540   ∈ wcel 2109  ∀wral 3044  ‘cfv 6511  (class class class)co 7387  0cc0 11068  1c1 11069  [,]cicc 13309   Cn ccn 23111  IIcii 24768   Htpy chtpy 24866  PHtpycphtpy 24867

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-rep 5234  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711

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-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-iun 4957  df-br 5108  df-opab 5170  df-mpt 5189  df-id 5533  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-ov 7390  df-oprab 7391  df-mpo 7392  df-1st 7968  df-2nd 7969  df-map 8801  df-top 22781  df-topon 22798  df-cn 23114  df-phtpy 24870

This theorem is referenced by:  phtpy01  24884  phtpycom  24887  phtpyco2  24889  phtpycc  24890  pcohtpylem  24919  txsconnlem  35227  cvmliftphtlem  35304  cvmliftpht  35305