Theorem phtpyhtpy 23580

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 23579	. . 3 ⊢ (𝜑 → (ℎ ∈ (𝐹(PHtpy‘𝐽)𝐺) ↔ (ℎ ∈ (𝐹(II Htpy 𝐽)𝐺) ∧ ∀𝑠 ∈ (0[,]1)((0ℎ𝑠) = (𝐹‘0) ∧ (1ℎ𝑠) = (𝐹‘1)))))
4		simpl 485	. . 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	syl6bi 255	. 2 ⊢ (𝜑 → (ℎ ∈ (𝐹(PHtpy‘𝐽)𝐺) → ℎ ∈ (𝐹(II Htpy 𝐽)𝐺)))
6	5	ssrdv 3973	1 ⊢ (𝜑 → (𝐹(PHtpy‘𝐽)𝐺) ⊆ (𝐹(II Htpy 𝐽)𝐺))

Syntax hints:   → wi 4   ∧ wa 398   = wceq 1533   ∈ wcel 2110  ∀wral 3138   ⊆ wss 3936  ‘cfv 6350  (class class class)co 7150  0cc0 10531  1c1 10532  [,]cicc 12735   Cn ccn 21826  IIcii 23477   Htpy chtpy 23565  PHtpycphtpy 23566

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2156  ax-12 2172  ax-ext 2793  ax-rep 5183  ax-sep 5196  ax-nul 5203  ax-pow 5259  ax-pr 5322  ax-un 7455

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3497  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4833  df-iun 4914  df-br 5060  df-opab 5122  df-mpt 5140  df-id 5455  df-xp 5556  df-rel 5557  df-cnv 5558  df-co 5559  df-dm 5560  df-rn 5561  df-res 5562  df-ima 5563  df-iota 6309  df-fun 6352  df-fn 6353  df-f 6354  df-f1 6355  df-fo 6356  df-f1o 6357  df-fv 6358  df-ov 7153  df-oprab 7154  df-mpo 7155  df-1st 7683  df-2nd 7684  df-map 8402  df-top 21496  df-topon 21513  df-cn 21829  df-phtpy 23569

This theorem is referenced by:  phtpycn  23581  phtpy01  23583  phtpycom  23586  phtpyco2  23588  phtpycc  23589  pcohtpylem  23617  txsconnlem  32482  cvmliftphtlem  32559