Theorem phtpyhtpy 24949

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 24948	. . 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 3928	1 ⊢ (𝜑 → (𝐹(PHtpy‘𝐽)𝐺) ⊆ (𝐹(II Htpy 𝐽)𝐺))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ∀wral 3052   ⊆ wss 3890  ‘cfv 6499  (class class class)co 7367  0cc0 11038  1c1 11039  [,]cicc 13301   Cn ccn 23189  IIcii 24842   Htpy chtpy 24934  PHtpycphtpy 24935

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5213  ax-sep 5232  ax-nul 5242  ax-pow 5308  ax-pr 5376  ax-un 7689

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6455  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  df-ov 7370  df-oprab 7371  df-mpo 7372  df-1st 7942  df-2nd 7943  df-map 8775  df-top 22859  df-topon 22876  df-cn 23192  df-phtpy 24938

This theorem is referenced by:  phtpycn  24950  phtpy01  24952  phtpycom  24955  phtpyco2  24957  phtpycc  24958  pcohtpylem  24986  txsconnlem  35422  cvmliftphtlem  35499