Theorem ishtpy 22015

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 22013	. . . . . 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 765	. . . . . . 7 |- ( ( ph /\ ( j = J /\ k = K ) ) -> j = J )
4		simprr 767	. . . . . . 7 |- ( ( ph /\ ( j = J /\ k = K ) ) -> k = K )
5	3, 4	oveq12d 6313	. . . . . 6 |- ( ( ph /\ ( j = J /\ k = K ) ) -> ( j Cn k ) = ( J Cn K ) )
6	3	oveq1d 6310	. . . . . . . 8 |- ( ( ph /\ ( j = J /\ k = K ) ) -> ( j tX II ) = ( J tX II ) )
7	6, 4	oveq12d 6313	. . . . . . 7 |- ( ( ph /\ ( j = J /\ k = K ) ) -> ( ( j tX II ) Cn k ) = ( ( J tX II ) Cn K ) )
8	3	unieqd 4211	. . . . . . . . 9 |- ( ( ph /\ ( j = J /\ k = K ) ) -> U. j = U. J )
9		ishtpy.1	. . . . . . . . . . 11 |- ( ph -> J e. (TopOn ` X ) )
10		toponuni 19954	. . . . . . . . . . 11 |- ( J e. (TopOn ` X ) -> X = U. J )
11	9, 10	syl 17	. . . . . . . . . 10 |- ( ph -> X = U. J )
12	11	adantr 467	. . . . . . . . 9 |- ( ( ph /\ ( j = J /\ k = K ) ) -> X = U. J )
13	8, 12	eqtr4d 2490	. . . . . . . 8 |- ( ( ph /\ ( j = J /\ k = K ) ) -> U. j = X )
14	13	raleqdv 2995	. . . . . . 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 3042	. . . . . 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 6358	. . . . 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 19953	. . . . . 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 20269	. . . . . 6 |- ( F e. ( J Cn K ) -> K e. Top )
21	19, 20	syl 17	. . . . 5 |- ( ph -> K e. Top )
22		ssrab2 3516	. . . . . . . . . 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 6323	. . . . . . . . . . 11 |- ( ( J tX II ) Cn K ) e. _V
24	23	elpw2 4570	. . . . . . . . . 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 213	. . . . . . . . 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 2752	. . . . . . . 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 2453	. . . . . . . . 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 6865	. . . . . . . 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 212	. . . . . . 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 6323	. . . . . . . 8 |- ( J Cn K ) e. _V
31	30, 30	xpex 6600	. . . . . . 7 |- ( ( J Cn K ) X. ( J Cn K ) ) e. _V
32	23	pwex 4589	. . . . . . 7 |- ~P ( ( J tX II ) Cn K ) e. _V
33		fex2 6753	. . . . . . 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 1366	. . . . . 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 6429	. . . 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 5869	. . . . . . . . 9 |- ( f = F -> ( f ` s ) = ( F ` s ) )
38	37	eqeq2d 2463	. . . . . . . 8 |- ( f = F -> ( ( s h 0 ) = ( f ` s ) <-> ( s h 0 ) = ( F ` s ) ) )
39		fveq1 5869	. . . . . . . . 9 |- ( g = G -> ( g ` s ) = ( G ` s ) )
40	39	eqeq2d 2463	. . . . . . . 8 |- ( g = G -> ( ( s h 1 ) = ( g ` s ) <-> ( s h 1 ) = ( G ` s ) ) )
41	38, 40	bi2anan9 885	. . . . . . 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 468	. . . . . 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 2829	. . . . 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 3038	. . . 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 4557	. . . . 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 6429	. . 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 2516	. 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 6301	. . . . . 6 |- ( h = H -> ( s h 0 ) = ( s H 0 ) )
51	50	eqeq1d 2455	. . . . 5 |- ( h = H -> ( ( s h 0 ) = ( F ` s ) <-> ( s H 0 ) = ( F ` s ) ) )
52		oveq 6301	. . . . . 6 |- ( h = H -> ( s h 1 ) = ( s H 1 ) )
53	52	eqeq1d 2455	. . . . 5 |- ( h = H -> ( ( s h 1 ) = ( G ` s ) <-> ( s H 1 ) = ( G ` s ) ) )
54	51, 53	anbi12d 718	. . . 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 2829	. . 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 3198	. 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 265	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 188   /\ wa 371   = wceq 1446   e. wcel 1889   A.wral 2739   {crab 2743   _Vcvv 3047   C_ wss 3406   ~Pcpw 3953   U.cuni 4201   X. cxp 4835   -->wf 5581   ` cfv 5585  (class class class)co 6295   |-> cmpt2 6297   0cc0 9544   1c1 9545   Topctop 19929  TopOnctopon 19930   Cn ccn 20252   tX ctx 20587   IIcii 21919   Htpy chtpy 22010

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1671  ax-4 1684  ax-5 1760  ax-6 1807  ax-7 1853  ax-8 1891  ax-9 1898  ax-10 1917  ax-11 1922  ax-12 1935  ax-13 2093  ax-ext 2433  ax-sep 4528  ax-nul 4537  ax-pow 4584  ax-pr 4642  ax-un 6588

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 988  df-tru 1449  df-ex 1666  df-nf 1670  df-sb 1800  df-eu 2305  df-mo 2306  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2583  df-ne 2626  df-ral 2744  df-rex 2745  df-rab 2748  df-v 3049  df-sbc 3270  df-csb 3366  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-nul 3734  df-if 3884  df-pw 3955  df-sn 3971  df-pr 3973  df-op 3977  df-uni 4202  df-iun 4283  df-br 4406  df-opab 4465  df-mpt 4466  df-id 4752  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5549  df-fun 5587  df-fn 5588  df-f 5589  df-fv 5593  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-1st 6798  df-2nd 6799  df-map 7479  df-top 19933  df-topon 19935  df-cn 20255  df-htpy 22013

This theorem is referenced by:  htpycn  22016  htpyi  22017  ishtpyd  22018