Theorem phtpyhtpy 22781

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

Hypotheses

Ref	Expression
isphtpy.2	|- ( ph -> F e. ( II Cn J ) )
isphtpy.3	|- ( ph -> G e. ( II Cn J ) )

Assertion

Ref	Expression
phtpyhtpy	|- ( ph -> ( F ( PHtpy ` J ) G ) C_ ( 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 |- ( ph -> F e. ( II Cn J ) )
2		isphtpy.3	. . . 4 |- ( ph -> G e. ( II Cn J ) )
3	1, 2	isphtpy 22780	. . 3 |- ( ph -> ( h e. ( F ( PHtpy ` J ) G ) <-> ( h e. ( F ( II Htpy J ) G ) /\ A. s e. ( 0 [,] 1 ) ( ( 0 h s ) = ( F ` 0 ) /\ ( 1 h s ) = ( F ` 1 ) ) ) ) )
4		simpl 473	. . 3 |- ( ( h e. ( F ( II Htpy J ) G ) /\ A. s e. ( 0 [,] 1 ) ( ( 0 h s ) = ( F ` 0 ) /\ ( 1 h s ) = ( F ` 1 ) ) ) -> h e. ( F ( II Htpy J ) G ) )
5	3, 4	syl6bi 243	. 2 |- ( ph -> ( h e. ( F ( PHtpy ` J ) G ) -> h e. ( F ( II Htpy J ) G ) ) )
6	5	ssrdv 3609	1 |- ( ph -> ( F ( PHtpy ` J ) G ) C_ ( F ( II Htpy J ) G ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   C_ wss 3574   ` cfv 5888  (class class class)co 6650   0cc0 9936   1c1 9937   [,]cicc 12178   Cn ccn 21028   IIcii 22678   Htpy chtpy 22766   PHtpycphtpy 22767

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-map 7859  df-top 20699  df-topon 20716  df-cn 21031  df-phtpy 22770

This theorem is referenced by:  phtpycn  22782  phtpy01  22784  phtpycom  22787  phtpyco2  22789  phtpycc  22790  pcohtpylem  22819  txsconnlem  31222  cvmliftphtlem  31299