Theorem phtpyhtpy 25014

Description: A path homotopy is a homotopy. (Contributed by Mario Carneiro, 23-Feb-2015.)

Hypotheses

Ref	Expression
isphtpy.2	⊢ (𝜑 → 𝐹 ∈ (II Cn 𝐽))
isphtpy.3	⊢ (𝜑 → 𝐺 ∈ (II Cn 𝐽))

Assertion

Ref	Expression
phtpyhtpy	⊢ (𝜑 → (𝐹(PHtpy‘𝐽)𝐺) ⊆ (𝐹(II Htpy 𝐽)𝐺))

Proof of Theorem phtpyhtpy

Dummy variables 𝑠 ℎ are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		isphtpy.2	. . . 4 ⊢ (𝜑 → 𝐹 ∈ (II Cn 𝐽))
2		isphtpy.3	. . . 4 ⊢ (𝜑 → 𝐺 ∈ (II Cn 𝐽))
3	1, 2	isphtpy 25013	. . 3 ⊢ (𝜑 → (ℎ ∈ (𝐹(PHtpy‘𝐽)𝐺) ↔ (ℎ ∈ (𝐹(II Htpy 𝐽)𝐺) ∧ ∀𝑠 ∈ (0[,]1)((0ℎ𝑠) = (𝐹‘0) ∧ (1ℎ𝑠) = (𝐹‘1)))))
4		simpl 482	. . 3 ⊢ ((ℎ ∈ (𝐹(II Htpy 𝐽)𝐺) ∧ ∀𝑠 ∈ (0[,]1)((0ℎ𝑠) = (𝐹‘0) ∧ (1ℎ𝑠) = (𝐹‘1))) → ℎ ∈ (𝐹(II Htpy 𝐽)𝐺))
5	3, 4	biimtrdi 253	. 2 ⊢ (𝜑 → (ℎ ∈ (𝐹(PHtpy‘𝐽)𝐺) → ℎ ∈ (𝐹(II Htpy 𝐽)𝐺)))
6	5	ssrdv 3989	1 ⊢ (𝜑 → (𝐹(PHtpy‘𝐽)𝐺) ⊆ (𝐹(II Htpy 𝐽)𝐺))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2108  ∀wral 3061   ⊆ wss 3951  ‘cfv 6561  (class class class)co 7431  0cc0 11155  1c1 11156  [,]cicc 13390   Cn ccn 23232  IIcii 24901   Htpy chtpy 24999  PHtpycphtpy 25000

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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8014  df-2nd 8015  df-map 8868  df-top 22900  df-topon 22917  df-cn 23235  df-phtpy 25003

This theorem is referenced by:  phtpycn  25015  phtpy01  25017  phtpycom  25020  phtpyco2  25022  phtpycc  25023  pcohtpylem  25052  txsconnlem  35245  cvmliftphtlem  35322