Theorem phtpyi 24053

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 24050	. . . 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 231	. . 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 7263	. . . . 5 ⊢ (𝑠 = 𝐴 → (0𝐻𝑠) = (0𝐻𝐴))
8	7	eqeq1d 2740	. . . 4 ⊢ (𝑠 = 𝐴 → ((0𝐻𝑠) = (𝐹‘0) ↔ (0𝐻𝐴) = (𝐹‘0)))
9		oveq2 7263	. . . . 5 ⊢ (𝑠 = 𝐴 → (1𝐻𝑠) = (1𝐻𝐴))
10	9	eqeq1d 2740	. . . 4 ⊢ (𝑠 = 𝐴 → ((1𝐻𝑠) = (𝐹‘1) ↔ (1𝐻𝐴) = (𝐹‘1)))
11	8, 10	anbi12d 630	. . 3 ⊢ (𝑠 = 𝐴 → (((0𝐻𝑠) = (𝐹‘0) ∧ (1𝐻𝑠) = (𝐹‘1)) ↔ ((0𝐻𝐴) = (𝐹‘0) ∧ (1𝐻𝐴) = (𝐹‘1))))
12	11	rspccva 3551	. 2 ⊢ ((∀𝑠 ∈ (0[,]1)((0𝐻𝑠) = (𝐹‘0) ∧ (1𝐻𝑠) = (𝐹‘1)) ∧ 𝐴 ∈ (0[,]1)) → ((0𝐻𝐴) = (𝐹‘0) ∧ (1𝐻𝐴) = (𝐹‘1)))
13	6, 12	sylan 579	1 ⊢ ((𝜑 ∧ 𝐴 ∈ (0[,]1)) → ((0𝐻𝐴) = (𝐹‘0) ∧ (1𝐻𝐴) = (𝐹‘1)))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1539   ∈ wcel 2108  ∀wral 3063  ‘cfv 6418  (class class class)co 7255  0cc0 10802  1c1 10803  [,]cicc 13011   Cn ccn 22283  IIcii 23944   Htpy chtpy 24036  PHtpycphtpy 24037

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-reu 3070  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-ov 7258  df-oprab 7259  df-mpo 7260  df-1st 7804  df-2nd 7805  df-map 8575  df-top 21951  df-topon 21968  df-cn 22286  df-phtpy 24040

This theorem is referenced by:  phtpy01  24054  phtpycom  24057  phtpyco2  24059  phtpycc  24060  pcohtpylem  24088  txsconnlem  33102  cvmliftphtlem  33179  cvmliftpht  33180