Theorem isphtpy 22780

Description: Membership in the class of path homotopies between two continuous functions. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised 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
isphtpy	|- ( 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 ) ) ) ) )

Distinct variable groups:   F, s   G, s   H, s   J, s   ph, s

Proof of Theorem isphtpy

Dummy variables f g h j are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		isphtpy.2	. . . . 5 |- ( ph -> F e. ( II Cn J ) )
2		cntop2 21045	. . . . 5 |- ( F e. ( II Cn J ) -> J e. Top )
3		oveq2 6658	. . . . . . 7 |- ( j = J -> ( II Cn j ) = ( II Cn J ) )
4		oveq2 6658	. . . . . . . . 9 |- ( j = J -> ( II Htpy j ) = ( II Htpy J ) )
5	4	oveqd 6667	. . . . . . . 8 |- ( j = J -> ( f ( II Htpy j ) g ) = ( f ( II Htpy J ) g ) )
6		rabeq 3192	. . . . . . . 8 |- ( ( f ( II Htpy j ) g ) = ( f ( II Htpy J ) g ) -> { 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 ) | A. s e. ( 0 [,] 1 ) ( ( 0 h s ) = ( f ` 0 ) /\ ( 1 h s ) = ( f ` 1 ) ) } )
7	5, 6	syl 17	. . . . . . 7 |- ( j = J -> { 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 ) | A. s e. ( 0 [,] 1 ) ( ( 0 h s ) = ( f ` 0 ) /\ ( 1 h s ) = ( f ` 1 ) ) } )
8	3, 3, 7	mpt2eq123dv 6717	. . . . . 6 |- ( j = J -> ( f e. ( II Cn j ) , g e. ( II Cn j ) |-> { h e. ( f ( II Htpy j ) g ) | A. s e. ( 0 [,] 1 ) ( ( 0 h s ) = ( f ` 0 ) /\ ( 1 h s ) = ( f ` 1 ) ) } ) = ( f e. ( II Cn J ) , g e. ( II Cn J ) |-> { h e. ( f ( II Htpy J ) g ) | A. s e. ( 0 [,] 1 ) ( ( 0 h s ) = ( f ` 0 ) /\ ( 1 h s ) = ( f ` 1 ) ) } ) )
9		df-phtpy 22770	. . . . . 6 |- PHtpy = ( j e. Top |-> ( f e. ( II Cn j ) , g e. ( II Cn j ) |-> { h e. ( f ( II Htpy j ) g ) | A. s e. ( 0 [,] 1 ) ( ( 0 h s ) = ( f ` 0 ) /\ ( 1 h s ) = ( f ` 1 ) ) } ) )
10		ovex 6678	. . . . . . 7 |- ( II Cn J ) e. _V
11	10, 10	mpt2ex 7247	. . . . . 6 |- ( f e. ( II Cn J ) , g e. ( II Cn J ) |-> { h e. ( f ( II Htpy J ) g ) | A. s e. ( 0 [,] 1 ) ( ( 0 h s ) = ( f ` 0 ) /\ ( 1 h s ) = ( f ` 1 ) ) } ) e. _V
12	8, 9, 11	fvmpt 6282	. . . . 5 |- ( J e. Top -> ( PHtpy ` J ) = ( f e. ( II Cn J ) , g e. ( II Cn J ) |-> { h e. ( f ( II Htpy J ) g ) | A. s e. ( 0 [,] 1 ) ( ( 0 h s ) = ( f ` 0 ) /\ ( 1 h s ) = ( f ` 1 ) ) } ) )
13	1, 2, 12	3syl 18	. . . 4 |- ( ph -> ( PHtpy ` J ) = ( f e. ( II Cn J ) , g e. ( II Cn J ) |-> { h e. ( f ( II Htpy J ) g ) | A. s e. ( 0 [,] 1 ) ( ( 0 h s ) = ( f ` 0 ) /\ ( 1 h s ) = ( f ` 1 ) ) } ) )
14		oveq12 6659	. . . . . 6 |- ( ( f = F /\ g = G ) -> ( f ( II Htpy J ) g ) = ( F ( II Htpy J ) G ) )
15		simpl 473	. . . . . . . . . 10 |- ( ( f = F /\ g = G ) -> f = F )
16	15	fveq1d 6193	. . . . . . . . 9 |- ( ( f = F /\ g = G ) -> ( f ` 0 ) = ( F ` 0 ) )
17	16	eqeq2d 2632	. . . . . . . 8 |- ( ( f = F /\ g = G ) -> ( ( 0 h s ) = ( f ` 0 ) <-> ( 0 h s ) = ( F ` 0 ) ) )
18	15	fveq1d 6193	. . . . . . . . 9 |- ( ( f = F /\ g = G ) -> ( f ` 1 ) = ( F ` 1 ) )
19	18	eqeq2d 2632	. . . . . . . 8 |- ( ( f = F /\ g = G ) -> ( ( 1 h s ) = ( f ` 1 ) <-> ( 1 h s ) = ( F ` 1 ) ) )
20	17, 19	anbi12d 747	. . . . . . 7 |- ( ( f = F /\ g = G ) -> ( ( ( 0 h s ) = ( f ` 0 ) /\ ( 1 h s ) = ( f ` 1 ) ) <-> ( ( 0 h s ) = ( F ` 0 ) /\ ( 1 h s ) = ( F ` 1 ) ) ) )
21	20	ralbidv 2986	. . . . . 6 |- ( ( f = F /\ g = G ) -> ( A. s e. ( 0 [,] 1 ) ( ( 0 h s ) = ( f ` 0 ) /\ ( 1 h s ) = ( f ` 1 ) ) <-> A. s e. ( 0 [,] 1 ) ( ( 0 h s ) = ( F ` 0 ) /\ ( 1 h s ) = ( F ` 1 ) ) ) )
22	14, 21	rabeqbidv 3195	. . . . 5 |- ( ( f = F /\ g = G ) -> { 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 ) | A. s e. ( 0 [,] 1 ) ( ( 0 h s ) = ( F ` 0 ) /\ ( 1 h s ) = ( F ` 1 ) ) } )
23	22	adantl 482	. . . 4 |- ( ( ph /\ ( f = F /\ g = G ) ) -> { 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 ) | A. s e. ( 0 [,] 1 ) ( ( 0 h s ) = ( F ` 0 ) /\ ( 1 h s ) = ( F ` 1 ) ) } )
24		isphtpy.3	. . . 4 |- ( ph -> G e. ( II Cn J ) )
25		ovex 6678	. . . . . 6 |- ( F ( II Htpy J ) G ) e. _V
26	25	rabex 4813	. . . . 5 |- { h e. ( F ( II Htpy J ) G ) | A. s e. ( 0 [,] 1 ) ( ( 0 h s ) = ( F ` 0 ) /\ ( 1 h s ) = ( F ` 1 ) ) } e. _V
27	26	a1i 11	. . . 4 |- ( ph -> { h e. ( F ( II Htpy J ) G ) | A. s e. ( 0 [,] 1 ) ( ( 0 h s ) = ( F ` 0 ) /\ ( 1 h s ) = ( F ` 1 ) ) } e. _V )
28	13, 23, 1, 24, 27	ovmpt2d 6788	. . 3 |- ( ph -> ( 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 ) ) } )
29	28	eleq2d 2687	. 2 |- ( ph -> ( H e. ( F ( PHtpy ` J ) G ) <-> H e. { h e. ( F ( II Htpy J ) G ) | A. s e. ( 0 [,] 1 ) ( ( 0 h s ) = ( F ` 0 ) /\ ( 1 h s ) = ( F ` 1 ) ) } ) )
30		oveq 6656	. . . . . 6 |- ( h = H -> ( 0 h s ) = ( 0 H s ) )
31	30	eqeq1d 2624	. . . . 5 |- ( h = H -> ( ( 0 h s ) = ( F ` 0 ) <-> ( 0 H s ) = ( F ` 0 ) ) )
32		oveq 6656	. . . . . 6 |- ( h = H -> ( 1 h s ) = ( 1 H s ) )
33	32	eqeq1d 2624	. . . . 5 |- ( h = H -> ( ( 1 h s ) = ( F ` 1 ) <-> ( 1 H s ) = ( F ` 1 ) ) )
34	31, 33	anbi12d 747	. . . 4 |- ( h = H -> ( ( ( 0 h s ) = ( F ` 0 ) /\ ( 1 h s ) = ( F ` 1 ) ) <-> ( ( 0 H s ) = ( F ` 0 ) /\ ( 1 H s ) = ( F ` 1 ) ) ) )
35	34	ralbidv 2986	. . 3 |- ( h = H -> ( A. s e. ( 0 [,] 1 ) ( ( 0 h s ) = ( F ` 0 ) /\ ( 1 h s ) = ( F ` 1 ) ) <-> A. s e. ( 0 [,] 1 ) ( ( 0 H s ) = ( F ` 0 ) /\ ( 1 H s ) = ( F ` 1 ) ) ) )
36	35	elrab 3363	. 2 |- ( H e. { 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 ) /\ A. s e. ( 0 [,] 1 ) ( ( 0 H s ) = ( F ` 0 ) /\ ( 1 H s ) = ( F ` 1 ) ) ) )
37	29, 36	syl6bb 276	1 |- ( 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 ) ) ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   {crab 2916   _Vcvv 3200   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   0cc0 9936   1c1 9937   [,]cicc 12178   Topctop 20698   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:  phtpyhtpy  22781  phtpyi  22783  isphtpyd  22785