Theorem ishtpy 22771

Description: Membership in the class of homotopies between two continuous functions. (Contributed by Mario Carneiro, 22-Feb-2015.) (Revised by Mario Carneiro, 5-Sep-2015.)

Hypotheses

Ref	Expression
ishtpy.1	|- ( ph -> J e. (TopOn ` X ) )
ishtpy.3	|- ( ph -> F e. ( J Cn K ) )
ishtpy.4	|- ( ph -> G e. ( J Cn K ) )

Assertion

Ref	Expression
ishtpy	|- ( ph -> ( H e. ( F ( J Htpy K ) G ) <-> ( H e. ( ( J tX II ) Cn K ) /\ A. s e. X ( ( s H 0 ) = ( F ` s ) /\ ( s H 1 ) = ( G ` s ) ) ) ) )

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

Allowed substitution hint:   K( s)

Proof of Theorem ishtpy

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

Step	Hyp	Ref	Expression
1		df-htpy 22769	. . . . . 6 |- Htpy = ( j e. Top , k e. Top |-> ( f e. ( j Cn k ) , g e. ( j Cn k ) |-> { h e. ( ( j tX II ) Cn k ) | A. s e. U. j ( ( s h 0 ) = ( f ` s ) /\ ( s h 1 ) = ( g ` s ) ) } ) )
2	1	a1i 11	. . . . 5 |- ( ph -> Htpy = ( j e. Top , k e. Top |-> ( f e. ( j Cn k ) , g e. ( j Cn k ) |-> { h e. ( ( j tX II ) Cn k ) | A. s e. U. j ( ( s h 0 ) = ( f ` s ) /\ ( s h 1 ) = ( g ` s ) ) } ) ) )
3		simprl 794	. . . . . . 7 |- ( ( ph /\ ( j = J /\ k = K ) ) -> j = J )
4		simprr 796	. . . . . . 7 |- ( ( ph /\ ( j = J /\ k = K ) ) -> k = K )
5	3, 4	oveq12d 6668	. . . . . 6 |- ( ( ph /\ ( j = J /\ k = K ) ) -> ( j Cn k ) = ( J Cn K ) )
6	3	oveq1d 6665	. . . . . . . 8 |- ( ( ph /\ ( j = J /\ k = K ) ) -> ( j tX II ) = ( J tX II ) )
7	6, 4	oveq12d 6668	. . . . . . 7 |- ( ( ph /\ ( j = J /\ k = K ) ) -> ( ( j tX II ) Cn k ) = ( ( J tX II ) Cn K ) )
8	3	unieqd 4446	. . . . . . . . 9 |- ( ( ph /\ ( j = J /\ k = K ) ) -> U. j = U. J )
9		ishtpy.1	. . . . . . . . . . 11 |- ( ph -> J e. (TopOn ` X ) )
10		toponuni 20719	. . . . . . . . . . 11 |- ( J e. (TopOn ` X ) -> X = U. J )
11	9, 10	syl 17	. . . . . . . . . 10 |- ( ph -> X = U. J )
12	11	adantr 481	. . . . . . . . 9 |- ( ( ph /\ ( j = J /\ k = K ) ) -> X = U. J )
13	8, 12	eqtr4d 2659	. . . . . . . 8 |- ( ( ph /\ ( j = J /\ k = K ) ) -> U. j = X )
14	13	raleqdv 3144	. . . . . . 7 |- ( ( ph /\ ( j = J /\ k = K ) ) -> ( A. s e. U. j ( ( s h 0 ) = ( f ` s ) /\ ( s h 1 ) = ( g ` s ) ) <-> A. s e. X ( ( s h 0 ) = ( f ` s ) /\ ( s h 1 ) = ( g ` s ) ) ) )
15	7, 14	rabeqbidv 3195	. . . . . 6 |- ( ( ph /\ ( j = J /\ k = K ) ) -> { h e. ( ( j tX II ) Cn k ) | A. s e. U. j ( ( s h 0 ) = ( f ` s ) /\ ( s h 1 ) = ( g ` s ) ) } = { h e. ( ( J tX II ) Cn K ) | A. s e. X ( ( s h 0 ) = ( f ` s ) /\ ( s h 1 ) = ( g ` s ) ) } )
16	5, 5, 15	mpt2eq123dv 6717	. . . . 5 |- ( ( ph /\ ( j = J /\ k = K ) ) -> ( f e. ( j Cn k ) , g e. ( j Cn k ) |-> { h e. ( ( j tX II ) Cn k ) | A. s e. U. j ( ( s h 0 ) = ( f ` s ) /\ ( s h 1 ) = ( g ` s ) ) } ) = ( f e. ( J Cn K ) , g e. ( J Cn K ) |-> { h e. ( ( J tX II ) Cn K ) | A. s e. X ( ( s h 0 ) = ( f ` s ) /\ ( s h 1 ) = ( g ` s ) ) } ) )
17		topontop 20718	. . . . . 6 |- ( J e. (TopOn ` X ) -> J e. Top )
18	9, 17	syl 17	. . . . 5 |- ( ph -> J e. Top )
19		ishtpy.3	. . . . . 6 |- ( ph -> F e. ( J Cn K ) )
20		cntop2 21045	. . . . . 6 |- ( F e. ( J Cn K ) -> K e. Top )
21	19, 20	syl 17	. . . . 5 |- ( ph -> K e. Top )
22		ssrab2 3687	. . . . . . . . . 10 |- { h e. ( ( J tX II ) Cn K ) | A. s e. X ( ( s h 0 ) = ( f ` s ) /\ ( s h 1 ) = ( g ` s ) ) } C_ ( ( J tX II ) Cn K )
23		ovex 6678	. . . . . . . . . . 11 |- ( ( J tX II ) Cn K ) e. _V
24	23	elpw2 4828	. . . . . . . . . 10 |- ( { h e. ( ( J tX II ) Cn K ) | A. s e. X ( ( s h 0 ) = ( f ` s ) /\ ( s h 1 ) = ( g ` s ) ) } e. ~P ( ( J tX II ) Cn K ) <-> { h e. ( ( J tX II ) Cn K ) | A. s e. X ( ( s h 0 ) = ( f ` s ) /\ ( s h 1 ) = ( g ` s ) ) } C_ ( ( J tX II ) Cn K ) )
25	22, 24	mpbir 221	. . . . . . . . 9 |- { h e. ( ( J tX II ) Cn K ) | A. s e. X ( ( s h 0 ) = ( f ` s ) /\ ( s h 1 ) = ( g ` s ) ) } e. ~P ( ( J tX II ) Cn K )
26	25	rgen2w 2925	. . . . . . . 8 |- A. f e. ( J Cn K ) A. g e. ( J Cn K ) { h e. ( ( J tX II ) Cn K ) | A. s e. X ( ( s h 0 ) = ( f ` s ) /\ ( s h 1 ) = ( g ` s ) ) } e. ~P ( ( J tX II ) Cn K )
27		eqid 2622	. . . . . . . . 9 |- ( f e. ( J Cn K ) , g e. ( J Cn K ) |-> { h e. ( ( J tX II ) Cn K ) | A. s e. X ( ( s h 0 ) = ( f ` s ) /\ ( s h 1 ) = ( g ` s ) ) } ) = ( f e. ( J Cn K ) , g e. ( J Cn K ) |-> { h e. ( ( J tX II ) Cn K ) | A. s e. X ( ( s h 0 ) = ( f ` s ) /\ ( s h 1 ) = ( g ` s ) ) } )
28	27	fmpt2 7237	. . . . . . . 8 |- ( A. f e. ( J Cn K ) A. g e. ( J Cn K ) { h e. ( ( J tX II ) Cn K ) | A. s e. X ( ( s h 0 ) = ( f ` s ) /\ ( s h 1 ) = ( g ` s ) ) } e. ~P ( ( J tX II ) Cn K ) <-> ( f e. ( J Cn K ) , g e. ( J Cn K ) |-> { h e. ( ( J tX II ) Cn K ) | A. s e. X ( ( s h 0 ) = ( f ` s ) /\ ( s h 1 ) = ( g ` s ) ) } ) : ( ( J Cn K ) X. ( J Cn K ) ) --> ~P ( ( J tX II ) Cn K ) )
29	26, 28	mpbi 220	. . . . . . 7 |- ( f e. ( J Cn K ) , g e. ( J Cn K ) |-> { h e. ( ( J tX II ) Cn K ) | A. s e. X ( ( s h 0 ) = ( f ` s ) /\ ( s h 1 ) = ( g ` s ) ) } ) : ( ( J Cn K ) X. ( J Cn K ) ) --> ~P ( ( J tX II ) Cn K )
30		ovex 6678	. . . . . . . 8 |- ( J Cn K ) e. _V
31	30, 30	xpex 6962	. . . . . . 7 |- ( ( J Cn K ) X. ( J Cn K ) ) e. _V
32	23	pwex 4848	. . . . . . 7 |- ~P ( ( J tX II ) Cn K ) e. _V
33		fex2 7121	. . . . . . 7 |- ( ( ( f e. ( J Cn K ) , g e. ( J Cn K ) |-> { h e. ( ( J tX II ) Cn K ) | A. s e. X ( ( s h 0 ) = ( f ` s ) /\ ( s h 1 ) = ( g ` s ) ) } ) : ( ( J Cn K ) X. ( J Cn K ) ) --> ~P ( ( J tX II ) Cn K ) /\ ( ( J Cn K ) X. ( J Cn K ) ) e. _V /\ ~P ( ( J tX II ) Cn K ) e. _V ) -> ( f e. ( J Cn K ) , g e. ( J Cn K ) |-> { h e. ( ( J tX II ) Cn K ) | A. s e. X ( ( s h 0 ) = ( f ` s ) /\ ( s h 1 ) = ( g ` s ) ) } ) e. _V )
34	29, 31, 32, 33	mp3an 1424	. . . . . 6 |- ( f e. ( J Cn K ) , g e. ( J Cn K ) |-> { h e. ( ( J tX II ) Cn K ) | A. s e. X ( ( s h 0 ) = ( f ` s ) /\ ( s h 1 ) = ( g ` s ) ) } ) e. _V
35	34	a1i 11	. . . . 5 |- ( ph -> ( f e. ( J Cn K ) , g e. ( J Cn K ) |-> { h e. ( ( J tX II ) Cn K ) | A. s e. X ( ( s h 0 ) = ( f ` s ) /\ ( s h 1 ) = ( g ` s ) ) } ) e. _V )
36	2, 16, 18, 21, 35	ovmpt2d 6788	. . . 4 |- ( ph -> ( J Htpy K ) = ( f e. ( J Cn K ) , g e. ( J Cn K ) |-> { h e. ( ( J tX II ) Cn K ) | A. s e. X ( ( s h 0 ) = ( f ` s ) /\ ( s h 1 ) = ( g ` s ) ) } ) )
37		fveq1 6190	. . . . . . . . 9 |- ( f = F -> ( f ` s ) = ( F ` s ) )
38	37	eqeq2d 2632	. . . . . . . 8 |- ( f = F -> ( ( s h 0 ) = ( f ` s ) <-> ( s h 0 ) = ( F ` s ) ) )
39		fveq1 6190	. . . . . . . . 9 |- ( g = G -> ( g ` s ) = ( G ` s ) )
40	39	eqeq2d 2632	. . . . . . . 8 |- ( g = G -> ( ( s h 1 ) = ( g ` s ) <-> ( s h 1 ) = ( G ` s ) ) )
41	38, 40	bi2anan9 917	. . . . . . 7 |- ( ( f = F /\ g = G ) -> ( ( ( s h 0 ) = ( f ` s ) /\ ( s h 1 ) = ( g ` s ) ) <-> ( ( s h 0 ) = ( F ` s ) /\ ( s h 1 ) = ( G ` s ) ) ) )
42	41	adantl 482	. . . . . 6 |- ( ( ph /\ ( f = F /\ g = G ) ) -> ( ( ( s h 0 ) = ( f ` s ) /\ ( s h 1 ) = ( g ` s ) ) <-> ( ( s h 0 ) = ( F ` s ) /\ ( s h 1 ) = ( G ` s ) ) ) )
43	42	ralbidv 2986	. . . . 5 |- ( ( ph /\ ( f = F /\ g = G ) ) -> ( A. s e. X ( ( s h 0 ) = ( f ` s ) /\ ( s h 1 ) = ( g ` s ) ) <-> A. s e. X ( ( s h 0 ) = ( F ` s ) /\ ( s h 1 ) = ( G ` s ) ) ) )
44	43	rabbidv 3189	. . . 4 |- ( ( ph /\ ( f = F /\ g = G ) ) -> { h e. ( ( J tX II ) Cn K ) | A. s e. X ( ( s h 0 ) = ( f ` s ) /\ ( s h 1 ) = ( g ` s ) ) } = { h e. ( ( J tX II ) Cn K ) | A. s e. X ( ( s h 0 ) = ( F ` s ) /\ ( s h 1 ) = ( G ` s ) ) } )
45		ishtpy.4	. . . 4 |- ( ph -> G e. ( J Cn K ) )
46	23	rabex 4813	. . . . 5 |- { h e. ( ( J tX II ) Cn K ) | A. s e. X ( ( s h 0 ) = ( F ` s ) /\ ( s h 1 ) = ( G ` s ) ) } e. _V
47	46	a1i 11	. . . 4 |- ( ph -> { h e. ( ( J tX II ) Cn K ) | A. s e. X ( ( s h 0 ) = ( F ` s ) /\ ( s h 1 ) = ( G ` s ) ) } e. _V )
48	36, 44, 19, 45, 47	ovmpt2d 6788	. . 3 |- ( ph -> ( F ( J Htpy K ) G ) = { h e. ( ( J tX II ) Cn K ) | A. s e. X ( ( s h 0 ) = ( F ` s ) /\ ( s h 1 ) = ( G ` s ) ) } )
49	48	eleq2d 2687	. 2 |- ( ph -> ( H e. ( F ( J Htpy K ) G ) <-> H e. { h e. ( ( J tX II ) Cn K ) | A. s e. X ( ( s h 0 ) = ( F ` s ) /\ ( s h 1 ) = ( G ` s ) ) } ) )
50		oveq 6656	. . . . . 6 |- ( h = H -> ( s h 0 ) = ( s H 0 ) )
51	50	eqeq1d 2624	. . . . 5 |- ( h = H -> ( ( s h 0 ) = ( F ` s ) <-> ( s H 0 ) = ( F ` s ) ) )
52		oveq 6656	. . . . . 6 |- ( h = H -> ( s h 1 ) = ( s H 1 ) )
53	52	eqeq1d 2624	. . . . 5 |- ( h = H -> ( ( s h 1 ) = ( G ` s ) <-> ( s H 1 ) = ( G ` s ) ) )
54	51, 53	anbi12d 747	. . . 4 |- ( h = H -> ( ( ( s h 0 ) = ( F ` s ) /\ ( s h 1 ) = ( G ` s ) ) <-> ( ( s H 0 ) = ( F ` s ) /\ ( s H 1 ) = ( G ` s ) ) ) )
55	54	ralbidv 2986	. . 3 |- ( h = H -> ( A. s e. X ( ( s h 0 ) = ( F ` s ) /\ ( s h 1 ) = ( G ` s ) ) <-> A. s e. X ( ( s H 0 ) = ( F ` s ) /\ ( s H 1 ) = ( G ` s ) ) ) )
56	55	elrab 3363	. 2 |- ( H e. { h e. ( ( J tX II ) Cn K ) | A. s e. X ( ( s h 0 ) = ( F ` s ) /\ ( s h 1 ) = ( G ` s ) ) } <-> ( H e. ( ( J tX II ) Cn K ) /\ A. s e. X ( ( s H 0 ) = ( F ` s ) /\ ( s H 1 ) = ( G ` s ) ) ) )
57	49, 56	syl6bb 276	1 |- ( ph -> ( H e. ( F ( J Htpy K ) G ) <-> ( H e. ( ( J tX II ) Cn K ) /\ A. s e. X ( ( s H 0 ) = ( F ` s ) /\ ( s H 1 ) = ( G ` s ) ) ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   {crab 2916   _Vcvv 3200   C_ wss 3574   ~Pcpw 4158   U.cuni 4436   X. cxp 5112   -->wf 5884   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   0cc0 9936   1c1 9937   Topctop 20698  TopOnctopon 20715   Cn ccn 21028   tX ctx 21363   IIcii 22678   Htpy chtpy 22766

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-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-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-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-htpy 22769

This theorem is referenced by:  htpycn  22772  htpyi  22773  ishtpyd  22774