Theorem ishtpy 22081

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 22079	. . . . . 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 772	. . . . . . 7 |- ( ( ph /\ ( j = J /\ k = K ) ) -> j = J )
4		simprr 774	. . . . . . 7 |- ( ( ph /\ ( j = J /\ k = K ) ) -> k = K )
5	3, 4	oveq12d 6326	. . . . . 6 |- ( ( ph /\ ( j = J /\ k = K ) ) -> ( j Cn k ) = ( J Cn K ) )
6	3	oveq1d 6323	. . . . . . . 8 |- ( ( ph /\ ( j = J /\ k = K ) ) -> ( j tX II ) = ( J tX II ) )
7	6, 4	oveq12d 6326	. . . . . . 7 |- ( ( ph /\ ( j = J /\ k = K ) ) -> ( ( j tX II ) Cn k ) = ( ( J tX II ) Cn K ) )
8	3	unieqd 4200	. . . . . . . . 9 |- ( ( ph /\ ( j = J /\ k = K ) ) -> U. j = U. J )
9		ishtpy.1	. . . . . . . . . . 11 |- ( ph -> J e. (TopOn ` X ) )
10		toponuni 20019	. . . . . . . . . . 11 |- ( J e. (TopOn ` X ) -> X = U. J )
11	9, 10	syl 17	. . . . . . . . . 10 |- ( ph -> X = U. J )
12	11	adantr 472	. . . . . . . . 9 |- ( ( ph /\ ( j = J /\ k = K ) ) -> X = U. J )
13	8, 12	eqtr4d 2508	. . . . . . . 8 |- ( ( ph /\ ( j = J /\ k = K ) ) -> U. j = X )
14	13	raleqdv 2979	. . . . . . 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 3026	. . . . . 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 6372	. . . . 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 20018	. . . . . 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 20334	. . . . . 6 |- ( F e. ( J Cn K ) -> K e. Top )
21	19, 20	syl 17	. . . . 5 |- ( ph -> K e. Top )
22		ssrab2 3500	. . . . . . . . . 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 6336	. . . . . . . . . . 11 |- ( ( J tX II ) Cn K ) e. _V
24	23	elpw2 4565	. . . . . . . . . 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 214	. . . . . . . . 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 2769	. . . . . . . 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 2471	. . . . . . . . 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 6879	. . . . . . . 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 213	. . . . . . 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 6336	. . . . . . . 8 |- ( J Cn K ) e. _V
31	30, 30	xpex 6614	. . . . . . 7 |- ( ( J Cn K ) X. ( J Cn K ) ) e. _V
32	23	pwex 4584	. . . . . . 7 |- ~P ( ( J tX II ) Cn K ) e. _V
33		fex2 6767	. . . . . . 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 1390	. . . . . 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 6443	. . . 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 5878	. . . . . . . . 9 |- ( f = F -> ( f ` s ) = ( F ` s ) )
38	37	eqeq2d 2481	. . . . . . . 8 |- ( f = F -> ( ( s h 0 ) = ( f ` s ) <-> ( s h 0 ) = ( F ` s ) ) )
39		fveq1 5878	. . . . . . . . 9 |- ( g = G -> ( g ` s ) = ( G ` s ) )
40	39	eqeq2d 2481	. . . . . . . 8 |- ( g = G -> ( ( s h 1 ) = ( g ` s ) <-> ( s h 1 ) = ( G ` s ) ) )
41	38, 40	bi2anan9 890	. . . . . . 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 473	. . . . . 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 3022	. . . 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 4550	. . . . 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 6443	. . 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 2534	. 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 6314	. . . . . 6 |- ( h = H -> ( s h 0 ) = ( s H 0 ) )
51	50	eqeq1d 2473	. . . . 5 |- ( h = H -> ( ( s h 0 ) = ( F ` s ) <-> ( s H 0 ) = ( F ` s ) ) )
52		oveq 6314	. . . . . 6 |- ( h = H -> ( s h 1 ) = ( s H 1 ) )
53	52	eqeq1d 2473	. . . . 5 |- ( h = H -> ( ( s h 1 ) = ( G ` s ) <-> ( s H 1 ) = ( G ` s ) ) )
54	51, 53	anbi12d 725	. . . 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 3184	. 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 269	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 189   /\ wa 376   = wceq 1452   e. wcel 1904   A.wral 2756   {crab 2760   _Vcvv 3031   C_ wss 3390   ~Pcpw 3942   U.cuni 4190   X. cxp 4837   -->wf 5585   ` cfv 5589  (class class class)co 6308   |-> cmpt2 6310   0cc0 9557   1c1 9558   Topctop 19994  TopOnctopon 19995   Cn ccn 20317   tX ctx 20652   IIcii 21985   Htpy chtpy 22076

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602

This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-id 4754  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-fv 5597  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-1st 6812  df-2nd 6813  df-map 7492  df-top 19998  df-topon 20000  df-cn 20320  df-htpy 22079

This theorem is referenced by:  htpycn  22082  htpyi  22083  ishtpyd  22084