Theorem xpccatid 14240

Description: The product of two categories is a category. (Contributed by Mario Carneiro, 11-Jan-2017.)

Hypotheses

Ref	Expression
xpccat.t	|- T = ( C X.c D )
xpccat.c	|- ( ph -> C e. Cat )
xpccat.d	|- ( ph -> D e. Cat )
xpccat.x	|- X = ( Base ` C )
xpccat.y	|- Y = ( Base ` D )
xpccat.i	|- I = ( Id ` C )
xpccat.j	|- J = ( Id ` D )

Assertion

Ref	Expression
xpccatid	|- ( ph -> ( T e. Cat /\ ( Id ` T ) = ( x e. X , y e. Y |-> <. ( I ` x ) , ( J ` y ) >. ) ) )

Distinct variable groups:   x, y, I   x, J, y   x, C, y   ph, x, y   x, X, y   x, D, y   x, Y, y

Allowed substitution hints:   T( x, y)

Proof of Theorem xpccatid

Dummy variables f g h s t u v are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		xpccat.t	. . . . 5 |- T = ( C X.c D )
2		xpccat.x	. . . . 5 |- X = ( Base ` C )
3		xpccat.y	. . . . 5 |- Y = ( Base ` D )
4	1, 2, 3	xpcbas 14230	. . . 4 |- ( X X. Y ) = ( Base ` T )
5	4	a1i 11	. . 3 |- ( ph -> ( X X. Y ) = ( Base ` T ) )
6		eqidd 2405	. . 3 |- ( ph -> ( Hom ` T ) = ( Hom ` T ) )
7		eqidd 2405	. . 3 |- ( ph -> (comp ` T ) = (comp ` T ) )
8		ovex 6065	. . . . 5 |- ( C X.c D ) e. _V
9	1, 8	eqeltri 2474	. . . 4 |- T e. _V
10	9	a1i 11	. . 3 |- ( ph -> T e. _V )
11		biid 228	. . 3 |- ( ( ( s e. ( X X. Y ) /\ t e. ( X X. Y ) ) /\ ( u e. ( X X. Y ) /\ v e. ( X X. Y ) ) /\ ( f e. ( s ( Hom ` T ) t ) /\ g e. ( t ( Hom ` T ) u ) /\ h e. ( u ( Hom ` T ) v ) ) ) <-> ( ( s e. ( X X. Y ) /\ t e. ( X X. Y ) ) /\ ( u e. ( X X. Y ) /\ v e. ( X X. Y ) ) /\ ( f e. ( s ( Hom ` T ) t ) /\ g e. ( t ( Hom ` T ) u ) /\ h e. ( u ( Hom ` T ) v ) ) ) )
12		eqid 2404	. . . . . 6 |- ( Hom ` C ) = ( Hom ` C )
13		xpccat.i	. . . . . 6 |- I = ( Id ` C )
14		xpccat.c	. . . . . . 7 |- ( ph -> C e. Cat )
15	14	adantr 452	. . . . . 6 |- ( ( ph /\ t e. ( X X. Y ) ) -> C e. Cat )
16		xp1st 6335	. . . . . . 7 |- ( t e. ( X X. Y ) -> ( 1st ` t ) e. X )
17	16	adantl 453	. . . . . 6 |- ( ( ph /\ t e. ( X X. Y ) ) -> ( 1st ` t ) e. X )
18	2, 12, 13, 15, 17	catidcl 13862	. . . . 5 |- ( ( ph /\ t e. ( X X. Y ) ) -> ( I ` ( 1st ` t ) ) e. ( ( 1st ` t ) ( Hom ` C ) ( 1st ` t ) ) )
19		eqid 2404	. . . . . 6 |- ( Hom ` D ) = ( Hom ` D )
20		xpccat.j	. . . . . 6 |- J = ( Id ` D )
21		xpccat.d	. . . . . . 7 |- ( ph -> D e. Cat )
22	21	adantr 452	. . . . . 6 |- ( ( ph /\ t e. ( X X. Y ) ) -> D e. Cat )
23		xp2nd 6336	. . . . . . 7 |- ( t e. ( X X. Y ) -> ( 2nd ` t ) e. Y )
24	23	adantl 453	. . . . . 6 |- ( ( ph /\ t e. ( X X. Y ) ) -> ( 2nd ` t ) e. Y )
25	3, 19, 20, 22, 24	catidcl 13862	. . . . 5 |- ( ( ph /\ t e. ( X X. Y ) ) -> ( J ` ( 2nd ` t ) ) e. ( ( 2nd ` t ) ( Hom ` D ) ( 2nd ` t ) ) )
26		opelxpi 4869	. . . . 5 |- ( ( ( I ` ( 1st ` t ) ) e. ( ( 1st ` t ) ( Hom ` C ) ( 1st ` t ) ) /\ ( J ` ( 2nd ` t ) ) e. ( ( 2nd ` t ) ( Hom ` D ) ( 2nd ` t ) ) ) -> <. ( I ` ( 1st ` t ) ) , ( J ` ( 2nd ` t ) ) >. e. ( ( ( 1st ` t ) ( Hom ` C ) ( 1st ` t ) ) X. ( ( 2nd ` t ) ( Hom ` D ) ( 2nd ` t ) ) ) )
27	18, 25, 26	syl2anc 643	. . . 4 |- ( ( ph /\ t e. ( X X. Y ) ) -> <. ( I ` ( 1st ` t ) ) , ( J ` ( 2nd ` t ) ) >. e. ( ( ( 1st ` t ) ( Hom ` C ) ( 1st ` t ) ) X. ( ( 2nd ` t ) ( Hom ` D ) ( 2nd ` t ) ) ) )
28		eqid 2404	. . . . 5 |- ( Hom ` T ) = ( Hom ` T )
29		simpr 448	. . . . 5 |- ( ( ph /\ t e. ( X X. Y ) ) -> t e. ( X X. Y ) )
30	1, 4, 12, 19, 28, 29, 29	xpchom 14232	. . . 4 |- ( ( ph /\ t e. ( X X. Y ) ) -> ( t ( Hom ` T ) t ) = ( ( ( 1st ` t ) ( Hom ` C ) ( 1st ` t ) ) X. ( ( 2nd ` t ) ( Hom ` D ) ( 2nd ` t ) ) ) )
31	27, 30	eleqtrrd 2481	. . 3 |- ( ( ph /\ t e. ( X X. Y ) ) -> <. ( I ` ( 1st ` t ) ) , ( J ` ( 2nd ` t ) ) >. e. ( t ( Hom ` T ) t ) )
32		fvex 5701	. . . . . . . 8 |- ( I ` ( 1st ` t ) ) e. _V
33		fvex 5701	. . . . . . . 8 |- ( J ` ( 2nd ` t ) ) e. _V
34	32, 33	op1st 6314	. . . . . . 7 |- ( 1st ` <. ( I ` ( 1st ` t ) ) , ( J ` ( 2nd ` t ) ) >. ) = ( I ` ( 1st ` t ) )
35	34	oveq1i 6050	. . . . . 6 |- ( ( 1st ` <. ( I ` ( 1st ` t ) ) , ( J ` ( 2nd ` t ) ) >. ) ( <. ( 1st ` s ) , ( 1st ` t ) >. (comp ` C ) ( 1st ` t ) ) ( 1st ` f ) ) = ( ( I ` ( 1st ` t ) ) ( <. ( 1st ` s ) , ( 1st ` t ) >. (comp ` C ) ( 1st ` t ) ) ( 1st ` f ) )
36	14	adantr 452	. . . . . . 7 |- ( ( ph /\ ( ( s e. ( X X. Y ) /\ t e. ( X X. Y ) ) /\ ( u e. ( X X. Y ) /\ v e. ( X X. Y ) ) /\ ( f e. ( s ( Hom ` T ) t ) /\ g e. ( t ( Hom ` T ) u ) /\ h e. ( u ( Hom ` T ) v ) ) ) ) -> C e. Cat )
37		simpr1l 1014	. . . . . . . 8 |- ( ( ph /\ ( ( s e. ( X X. Y ) /\ t e. ( X X. Y ) ) /\ ( u e. ( X X. Y ) /\ v e. ( X X. Y ) ) /\ ( f e. ( s ( Hom ` T ) t ) /\ g e. ( t ( Hom ` T ) u ) /\ h e. ( u ( Hom ` T ) v ) ) ) ) -> s e. ( X X. Y ) )
38		xp1st 6335	. . . . . . . 8 |- ( s e. ( X X. Y ) -> ( 1st ` s ) e. X )
39	37, 38	syl 16	. . . . . . 7 |- ( ( ph /\ ( ( s e. ( X X. Y ) /\ t e. ( X X. Y ) ) /\ ( u e. ( X X. Y ) /\ v e. ( X X. Y ) ) /\ ( f e. ( s ( Hom ` T ) t ) /\ g e. ( t ( Hom ` T ) u ) /\ h e. ( u ( Hom ` T ) v ) ) ) ) -> ( 1st ` s ) e. X )
40		eqid 2404	. . . . . . 7 |- (comp ` C ) = (comp ` C )
41		simpr1r 1015	. . . . . . . 8 |- ( ( ph /\ ( ( s e. ( X X. Y ) /\ t e. ( X X. Y ) ) /\ ( u e. ( X X. Y ) /\ v e. ( X X. Y ) ) /\ ( f e. ( s ( Hom ` T ) t ) /\ g e. ( t ( Hom ` T ) u ) /\ h e. ( u ( Hom ` T ) v ) ) ) ) -> t e. ( X X. Y ) )
42	41, 16	syl 16	. . . . . . 7 |- ( ( ph /\ ( ( s e. ( X X. Y ) /\ t e. ( X X. Y ) ) /\ ( u e. ( X X. Y ) /\ v e. ( X X. Y ) ) /\ ( f e. ( s ( Hom ` T ) t ) /\ g e. ( t ( Hom ` T ) u ) /\ h e. ( u ( Hom ` T ) v ) ) ) ) -> ( 1st ` t ) e. X )
43		simpr31 1047	. . . . . . . . 9 |- ( ( ph /\ ( ( s e. ( X X. Y ) /\ t e. ( X X. Y ) ) /\ ( u e. ( X X. Y ) /\ v e. ( X X. Y ) ) /\ ( f e. ( s ( Hom ` T ) t ) /\ g e. ( t ( Hom ` T ) u ) /\ h e. ( u ( Hom ` T ) v ) ) ) ) -> f e. ( s ( Hom ` T ) t ) )
44	1, 4, 12, 19, 28, 37, 41	xpchom 14232	. . . . . . . . 9 |- ( ( ph /\ ( ( s e. ( X X. Y ) /\ t e. ( X X. Y ) ) /\ ( u e. ( X X. Y ) /\ v e. ( X X. Y ) ) /\ ( f e. ( s ( Hom ` T ) t ) /\ g e. ( t ( Hom ` T ) u ) /\ h e. ( u ( Hom ` T ) v ) ) ) ) -> ( s ( Hom ` T ) t ) = ( ( ( 1st ` s ) ( Hom ` C ) ( 1st ` t ) ) X. ( ( 2nd ` s ) ( Hom ` D ) ( 2nd ` t ) ) ) )
45	43, 44	eleqtrd 2480	. . . . . . . 8 |- ( ( ph /\ ( ( s e. ( X X. Y ) /\ t e. ( X X. Y ) ) /\ ( u e. ( X X. Y ) /\ v e. ( X X. Y ) ) /\ ( f e. ( s ( Hom ` T ) t ) /\ g e. ( t ( Hom ` T ) u ) /\ h e. ( u ( Hom ` T ) v ) ) ) ) -> f e. ( ( ( 1st ` s ) ( Hom ` C ) ( 1st ` t ) ) X. ( ( 2nd ` s ) ( Hom ` D ) ( 2nd ` t ) ) ) )
46		xp1st 6335	. . . . . . . 8 |- ( f e. ( ( ( 1st ` s ) ( Hom ` C ) ( 1st ` t ) ) X. ( ( 2nd ` s ) ( Hom ` D ) ( 2nd ` t ) ) ) -> ( 1st ` f ) e. ( ( 1st ` s ) ( Hom ` C ) ( 1st ` t ) ) )
47	45, 46	syl 16	. . . . . . 7 |- ( ( ph /\ ( ( s e. ( X X. Y ) /\ t e. ( X X. Y ) ) /\ ( u e. ( X X. Y ) /\ v e. ( X X. Y ) ) /\ ( f e. ( s ( Hom ` T ) t ) /\ g e. ( t ( Hom ` T ) u ) /\ h e. ( u ( Hom ` T ) v ) ) ) ) -> ( 1st ` f ) e. ( ( 1st ` s ) ( Hom ` C ) ( 1st ` t ) ) )
48	2, 12, 13, 36, 39, 40, 42, 47	catlid 13863	. . . . . 6 |- ( ( ph /\ ( ( s e. ( X X. Y ) /\ t e. ( X X. Y ) ) /\ ( u e. ( X X. Y ) /\ v e. ( X X. Y ) ) /\ ( f e. ( s ( Hom ` T ) t ) /\ g e. ( t ( Hom ` T ) u ) /\ h e. ( u ( Hom ` T ) v ) ) ) ) -> ( ( I ` ( 1st ` t ) ) ( <. ( 1st ` s ) , ( 1st ` t ) >. (comp ` C ) ( 1st ` t ) ) ( 1st ` f ) ) = ( 1st ` f ) )
49	35, 48	syl5eq 2448	. . . . 5 |- ( ( ph /\ ( ( s e. ( X X. Y ) /\ t e. ( X X. Y ) ) /\ ( u e. ( X X. Y ) /\ v e. ( X X. Y ) ) /\ ( f e. ( s ( Hom ` T ) t ) /\ g e. ( t ( Hom ` T ) u ) /\ h e. ( u ( Hom ` T ) v ) ) ) ) -> ( ( 1st ` <. ( I ` ( 1st ` t ) ) , ( J ` ( 2nd ` t ) ) >. ) ( <. ( 1st ` s ) , ( 1st ` t ) >. (comp ` C ) ( 1st ` t ) ) ( 1st ` f ) ) = ( 1st ` f ) )
50	32, 33	op2nd 6315	. . . . . . 7 |- ( 2nd ` <. ( I ` ( 1st ` t ) ) , ( J ` ( 2nd ` t ) ) >. ) = ( J ` ( 2nd ` t ) )
51	50	oveq1i 6050	. . . . . 6 |- ( ( 2nd ` <. ( I ` ( 1st ` t ) ) , ( J ` ( 2nd ` t ) ) >. ) ( <. ( 2nd ` s ) , ( 2nd ` t ) >. (comp ` D ) ( 2nd ` t ) ) ( 2nd ` f ) ) = ( ( J ` ( 2nd ` t ) ) ( <. ( 2nd ` s ) , ( 2nd ` t ) >. (comp ` D ) ( 2nd ` t ) ) ( 2nd ` f ) )
52	21	adantr 452	. . . . . . 7 |- ( ( ph /\ ( ( s e. ( X X. Y ) /\ t e. ( X X. Y ) ) /\ ( u e. ( X X. Y ) /\ v e. ( X X. Y ) ) /\ ( f e. ( s ( Hom ` T ) t ) /\ g e. ( t ( Hom ` T ) u ) /\ h e. ( u ( Hom ` T ) v ) ) ) ) -> D e. Cat )
53		xp2nd 6336	. . . . . . . 8 |- ( s e. ( X X. Y ) -> ( 2nd ` s ) e. Y )
54	37, 53	syl 16	. . . . . . 7 |- ( ( ph /\ ( ( s e. ( X X. Y ) /\ t e. ( X X. Y ) ) /\ ( u e. ( X X. Y ) /\ v e. ( X X. Y ) ) /\ ( f e. ( s ( Hom ` T ) t ) /\ g e. ( t ( Hom ` T ) u ) /\ h e. ( u ( Hom ` T ) v ) ) ) ) -> ( 2nd ` s ) e. Y )
55		eqid 2404	. . . . . . 7 |- (comp ` D ) = (comp ` D )
56	41, 23	syl 16	. . . . . . 7 |- ( ( ph /\ ( ( s e. ( X X. Y ) /\ t e. ( X X. Y ) ) /\ ( u e. ( X X. Y ) /\ v e. ( X X. Y ) ) /\ ( f e. ( s ( Hom ` T ) t ) /\ g e. ( t ( Hom ` T ) u ) /\ h e. ( u ( Hom ` T ) v ) ) ) ) -> ( 2nd ` t ) e. Y )
57		xp2nd 6336	. . . . . . . 8 |- ( f e. ( ( ( 1st ` s ) ( Hom ` C ) ( 1st ` t ) ) X. ( ( 2nd ` s ) ( Hom ` D ) ( 2nd ` t ) ) ) -> ( 2nd ` f ) e. ( ( 2nd ` s ) ( Hom ` D ) ( 2nd ` t ) ) )
58	45, 57	syl 16	. . . . . . 7 |- ( ( ph /\ ( ( s e. ( X X. Y ) /\ t e. ( X X. Y ) ) /\ ( u e. ( X X. Y ) /\ v e. ( X X. Y ) ) /\ ( f e. ( s ( Hom ` T ) t ) /\ g e. ( t ( Hom ` T ) u ) /\ h e. ( u ( Hom ` T ) v ) ) ) ) -> ( 2nd ` f ) e. ( ( 2nd ` s ) ( Hom ` D ) ( 2nd ` t ) ) )
59	3, 19, 20, 52, 54, 55, 56, 58	catlid 13863	. . . . . 6 |- ( ( ph /\ ( ( s e. ( X X. Y ) /\ t e. ( X X. Y ) ) /\ ( u e. ( X X. Y ) /\ v e. ( X X. Y ) ) /\ ( f e. ( s ( Hom ` T ) t ) /\ g e. ( t ( Hom ` T ) u ) /\ h e. ( u ( Hom ` T ) v ) ) ) ) -> ( ( J ` ( 2nd ` t ) ) ( <. ( 2nd ` s ) , ( 2nd ` t ) >. (comp ` D ) ( 2nd ` t ) ) ( 2nd ` f ) ) = ( 2nd ` f ) )
60	51, 59	syl5eq 2448	. . . . 5 |- ( ( ph /\ ( ( s e. ( X X. Y ) /\ t e. ( X X. Y ) ) /\ ( u e. ( X X. Y ) /\ v e. ( X X. Y ) ) /\ ( f e. ( s ( Hom ` T ) t ) /\ g e. ( t ( Hom ` T ) u ) /\ h e. ( u ( Hom ` T ) v ) ) ) ) -> ( ( 2nd ` <. ( I ` ( 1st ` t ) ) , ( J ` ( 2nd ` t ) ) >. ) ( <. ( 2nd ` s ) , ( 2nd ` t ) >. (comp ` D ) ( 2nd ` t ) ) ( 2nd ` f ) ) = ( 2nd ` f ) )
61	49, 60	opeq12d 3952	. . . 4 |- ( ( ph /\ ( ( s e. ( X X. Y ) /\ t e. ( X X. Y ) ) /\ ( u e. ( X X. Y ) /\ v e. ( X X. Y ) ) /\ ( f e. ( s ( Hom ` T ) t ) /\ g e. ( t ( Hom ` T ) u ) /\ h e. ( u ( Hom ` T ) v ) ) ) ) -> <. ( ( 1st ` <. ( I ` ( 1st ` t ) ) , ( J ` ( 2nd ` t ) ) >. ) ( <. ( 1st ` s ) , ( 1st ` t ) >. (comp ` C ) ( 1st ` t ) ) ( 1st ` f ) ) , ( ( 2nd ` <. ( I ` ( 1st ` t ) ) , ( J ` ( 2nd ` t ) ) >. ) ( <. ( 2nd ` s ) , ( 2nd ` t ) >. (comp ` D ) ( 2nd ` t ) ) ( 2nd ` f ) ) >. = <. ( 1st ` f ) , ( 2nd ` f ) >. )
62		eqid 2404	. . . . 5 |- (comp ` T ) = (comp ` T )
63	41, 31	syldan 457	. . . . 5 |- ( ( ph /\ ( ( s e. ( X X. Y ) /\ t e. ( X X. Y ) ) /\ ( u e. ( X X. Y ) /\ v e. ( X X. Y ) ) /\ ( f e. ( s ( Hom ` T ) t ) /\ g e. ( t ( Hom ` T ) u ) /\ h e. ( u ( Hom ` T ) v ) ) ) ) -> <. ( I ` ( 1st ` t ) ) , ( J ` ( 2nd ` t ) ) >. e. ( t ( Hom ` T ) t ) )
64	1, 4, 28, 40, 55, 62, 37, 41, 41, 43, 63	xpcco 14235	. . . 4 |- ( ( ph /\ ( ( s e. ( X X. Y ) /\ t e. ( X X. Y ) ) /\ ( u e. ( X X. Y ) /\ v e. ( X X. Y ) ) /\ ( f e. ( s ( Hom ` T ) t ) /\ g e. ( t ( Hom ` T ) u ) /\ h e. ( u ( Hom ` T ) v ) ) ) ) -> ( <. ( I ` ( 1st ` t ) ) , ( J ` ( 2nd ` t ) ) >. ( <. s , t >. (comp ` T ) t ) f ) = <. ( ( 1st ` <. ( I ` ( 1st ` t ) ) , ( J ` ( 2nd ` t ) ) >. ) ( <. ( 1st ` s ) , ( 1st ` t ) >. (comp ` C ) ( 1st ` t ) ) ( 1st ` f ) ) , ( ( 2nd ` <. ( I ` ( 1st ` t ) ) , ( J ` ( 2nd ` t ) ) >. ) ( <. ( 2nd ` s ) , ( 2nd ` t ) >. (comp ` D ) ( 2nd ` t ) ) ( 2nd ` f ) ) >. )
65		1st2nd2 6345	. . . . 5 |- ( f e. ( ( ( 1st ` s ) ( Hom ` C ) ( 1st ` t ) ) X. ( ( 2nd ` s ) ( Hom ` D ) ( 2nd ` t ) ) ) -> f = <. ( 1st ` f ) , ( 2nd ` f ) >. )
66	45, 65	syl 16	. . . 4 |- ( ( ph /\ ( ( s e. ( X X. Y ) /\ t e. ( X X. Y ) ) /\ ( u e. ( X X. Y ) /\ v e. ( X X. Y ) ) /\ ( f e. ( s ( Hom ` T ) t ) /\ g e. ( t ( Hom ` T ) u ) /\ h e. ( u ( Hom ` T ) v ) ) ) ) -> f = <. ( 1st ` f ) , ( 2nd ` f ) >. )
67	61, 64, 66	3eqtr4d 2446	. . 3 |- ( ( ph /\ ( ( s e. ( X X. Y ) /\ t e. ( X X. Y ) ) /\ ( u e. ( X X. Y ) /\ v e. ( X X. Y ) ) /\ ( f e. ( s ( Hom ` T ) t ) /\ g e. ( t ( Hom ` T ) u ) /\ h e. ( u ( Hom ` T ) v ) ) ) ) -> ( <. ( I ` ( 1st ` t ) ) , ( J ` ( 2nd ` t ) ) >. ( <. s , t >. (comp ` T ) t ) f ) = f )
68	34	oveq2i 6051	. . . . . 6 |- ( ( 1st ` g ) ( <. ( 1st ` t ) , ( 1st ` t ) >. (comp ` C ) ( 1st ` u ) ) ( 1st ` <. ( I ` ( 1st ` t ) ) , ( J ` ( 2nd ` t ) ) >. ) ) = ( ( 1st ` g ) ( <. ( 1st ` t ) , ( 1st ` t ) >. (comp ` C ) ( 1st ` u ) ) ( I ` ( 1st ` t ) ) )
69		simpr2l 1016	. . . . . . . 8 |- ( ( ph /\ ( ( s e. ( X X. Y ) /\ t e. ( X X. Y ) ) /\ ( u e. ( X X. Y ) /\ v e. ( X X. Y ) ) /\ ( f e. ( s ( Hom ` T ) t ) /\ g e. ( t ( Hom ` T ) u ) /\ h e. ( u ( Hom ` T ) v ) ) ) ) -> u e. ( X X. Y ) )
70		xp1st 6335	. . . . . . . 8 |- ( u e. ( X X. Y ) -> ( 1st ` u ) e. X )
71	69, 70	syl 16	. . . . . . 7 |- ( ( ph /\ ( ( s e. ( X X. Y ) /\ t e. ( X X. Y ) ) /\ ( u e. ( X X. Y ) /\ v e. ( X X. Y ) ) /\ ( f e. ( s ( Hom ` T ) t ) /\ g e. ( t ( Hom ` T ) u ) /\ h e. ( u ( Hom ` T ) v ) ) ) ) -> ( 1st ` u ) e. X )
72		simpr32 1048	. . . . . . . . 9 |- ( ( ph /\ ( ( s e. ( X X. Y ) /\ t e. ( X X. Y ) ) /\ ( u e. ( X X. Y ) /\ v e. ( X X. Y ) ) /\ ( f e. ( s ( Hom ` T ) t ) /\ g e. ( t ( Hom ` T ) u ) /\ h e. ( u ( Hom ` T ) v ) ) ) ) -> g e. ( t ( Hom ` T ) u ) )
73	1, 4, 12, 19, 28, 41, 69	xpchom 14232	. . . . . . . . 9 |- ( ( ph /\ ( ( s e. ( X X. Y ) /\ t e. ( X X. Y ) ) /\ ( u e. ( X X. Y ) /\ v e. ( X X. Y ) ) /\ ( f e. ( s ( Hom ` T ) t ) /\ g e. ( t ( Hom ` T ) u ) /\ h e. ( u ( Hom ` T ) v ) ) ) ) -> ( t ( Hom ` T ) u ) = ( ( ( 1st ` t ) ( Hom ` C ) ( 1st ` u ) ) X. ( ( 2nd ` t ) ( Hom ` D ) ( 2nd ` u ) ) ) )
74	72, 73	eleqtrd 2480	. . . . . . . 8 |- ( ( ph /\ ( ( s e. ( X X. Y ) /\ t e. ( X X. Y ) ) /\ ( u e. ( X X. Y ) /\ v e. ( X X. Y ) ) /\ ( f e. ( s ( Hom ` T ) t ) /\ g e. ( t ( Hom ` T ) u ) /\ h e. ( u ( Hom ` T ) v ) ) ) ) -> g e. ( ( ( 1st ` t ) ( Hom ` C ) ( 1st ` u ) ) X. ( ( 2nd ` t ) ( Hom ` D ) ( 2nd ` u ) ) ) )
75		xp1st 6335	. . . . . . . 8 |- ( g e. ( ( ( 1st ` t ) ( Hom ` C ) ( 1st ` u ) ) X. ( ( 2nd ` t ) ( Hom ` D ) ( 2nd ` u ) ) ) -> ( 1st ` g ) e. ( ( 1st ` t ) ( Hom ` C ) ( 1st ` u ) ) )
76	74, 75	syl 16	. . . . . . 7 |- ( ( ph /\ ( ( s e. ( X X. Y ) /\ t e. ( X X. Y ) ) /\ ( u e. ( X X. Y ) /\ v e. ( X X. Y ) ) /\ ( f e. ( s ( Hom ` T ) t ) /\ g e. ( t ( Hom ` T ) u ) /\ h e. ( u ( Hom ` T ) v ) ) ) ) -> ( 1st ` g ) e. ( ( 1st ` t ) ( Hom ` C ) ( 1st ` u ) ) )
77	2, 12, 13, 36, 42, 40, 71, 76	catrid 13864	. . . . . 6 |- ( ( ph /\ ( ( s e. ( X X. Y ) /\ t e. ( X X. Y ) ) /\ ( u e. ( X X. Y ) /\ v e. ( X X. Y ) ) /\ ( f e. ( s ( Hom ` T ) t ) /\ g e. ( t ( Hom ` T ) u ) /\ h e. ( u ( Hom ` T ) v ) ) ) ) -> ( ( 1st ` g ) ( <. ( 1st ` t ) , ( 1st ` t ) >. (comp ` C ) ( 1st ` u ) ) ( I ` ( 1st ` t ) ) ) = ( 1st ` g ) )
78	68, 77	syl5eq 2448	. . . . 5 |- ( ( ph /\ ( ( s e. ( X X. Y ) /\ t e. ( X X. Y ) ) /\ ( u e. ( X X. Y ) /\ v e. ( X X. Y ) ) /\ ( f e. ( s ( Hom ` T ) t ) /\ g e. ( t ( Hom ` T ) u ) /\ h e. ( u ( Hom ` T ) v ) ) ) ) -> ( ( 1st ` g ) ( <. ( 1st ` t ) , ( 1st ` t ) >. (comp ` C ) ( 1st ` u ) ) ( 1st ` <. ( I ` ( 1st ` t ) ) , ( J ` ( 2nd ` t ) ) >. ) ) = ( 1st ` g ) )
79	50	oveq2i 6051	. . . . . 6 |- ( ( 2nd ` g ) ( <. ( 2nd ` t ) , ( 2nd ` t ) >. (comp ` D ) ( 2nd ` u ) ) ( 2nd ` <. ( I ` ( 1st ` t ) ) , ( J ` ( 2nd ` t ) ) >. ) ) = ( ( 2nd ` g ) ( <. ( 2nd ` t ) , ( 2nd ` t ) >. (comp ` D ) ( 2nd ` u ) ) ( J ` ( 2nd ` t ) ) )
80		xp2nd 6336	. . . . . . . 8 |- ( u e. ( X X. Y ) -> ( 2nd ` u ) e. Y )
81	69, 80	syl 16	. . . . . . 7 |- ( ( ph /\ ( ( s e. ( X X. Y ) /\ t e. ( X X. Y ) ) /\ ( u e. ( X X. Y ) /\ v e. ( X X. Y ) ) /\ ( f e. ( s ( Hom ` T ) t ) /\ g e. ( t ( Hom ` T ) u ) /\ h e. ( u ( Hom ` T ) v ) ) ) ) -> ( 2nd ` u ) e. Y )
82		xp2nd 6336	. . . . . . . 8 |- ( g e. ( ( ( 1st ` t ) ( Hom ` C ) ( 1st ` u ) ) X. ( ( 2nd ` t ) ( Hom ` D ) ( 2nd ` u ) ) ) -> ( 2nd ` g ) e. ( ( 2nd ` t ) ( Hom ` D ) ( 2nd ` u ) ) )
83	74, 82	syl 16	. . . . . . 7 |- ( ( ph /\ ( ( s e. ( X X. Y ) /\ t e. ( X X. Y ) ) /\ ( u e. ( X X. Y ) /\ v e. ( X X. Y ) ) /\ ( f e. ( s ( Hom ` T ) t ) /\ g e. ( t ( Hom ` T ) u ) /\ h e. ( u ( Hom ` T ) v ) ) ) ) -> ( 2nd ` g ) e. ( ( 2nd ` t ) ( Hom ` D ) ( 2nd ` u ) ) )
84	3, 19, 20, 52, 56, 55, 81, 83	catrid 13864	. . . . . 6 |- ( ( ph /\ ( ( s e. ( X X. Y ) /\ t e. ( X X. Y ) ) /\ ( u e. ( X X. Y ) /\ v e. ( X X. Y ) ) /\ ( f e. ( s ( Hom ` T ) t ) /\ g e. ( t ( Hom ` T ) u ) /\ h e. ( u ( Hom ` T ) v ) ) ) ) -> ( ( 2nd ` g ) ( <. ( 2nd ` t ) , ( 2nd ` t ) >. (comp ` D ) ( 2nd ` u ) ) ( J ` ( 2nd ` t ) ) ) = ( 2nd ` g ) )
85	79, 84	syl5eq 2448	. . . . 5 |- ( ( ph /\ ( ( s e. ( X X. Y ) /\ t e. ( X X. Y ) ) /\ ( u e. ( X X. Y ) /\ v e. ( X X. Y ) ) /\ ( f e. ( s ( Hom ` T ) t ) /\ g e. ( t ( Hom ` T ) u ) /\ h e. ( u ( Hom ` T ) v ) ) ) ) -> ( ( 2nd ` g ) ( <. ( 2nd ` t ) , ( 2nd ` t ) >. (comp ` D ) ( 2nd ` u ) ) ( 2nd ` <. ( I ` ( 1st ` t ) ) , ( J ` ( 2nd ` t ) ) >. ) ) = ( 2nd ` g ) )
86	78, 85	opeq12d 3952	. . . 4 |- ( ( ph /\ ( ( s e. ( X X. Y ) /\ t e. ( X X. Y ) ) /\ ( u e. ( X X. Y ) /\ v e. ( X X. Y ) ) /\ ( f e. ( s ( Hom ` T ) t ) /\ g e. ( t ( Hom ` T ) u ) /\ h e. ( u ( Hom ` T ) v ) ) ) ) -> <. ( ( 1st ` g ) ( <. ( 1st ` t ) , ( 1st ` t ) >. (comp ` C ) ( 1st ` u ) ) ( 1st ` <. ( I ` ( 1st ` t ) ) , ( J ` ( 2nd ` t ) ) >. ) ) , ( ( 2nd ` g ) ( <. ( 2nd ` t ) , ( 2nd ` t ) >. (comp ` D ) ( 2nd ` u ) ) ( 2nd ` <. ( I ` ( 1st ` t ) ) , ( J ` ( 2nd ` t ) ) >. ) ) >. = <. ( 1st ` g ) , ( 2nd ` g ) >. )
87	1, 4, 28, 40, 55, 62, 41, 41, 69, 63, 72	xpcco 14235	. . . 4 |- ( ( ph /\ ( ( s e. ( X X. Y ) /\ t e. ( X X. Y ) ) /\ ( u e. ( X X. Y ) /\ v e. ( X X. Y ) ) /\ ( f e. ( s ( Hom ` T ) t ) /\ g e. ( t ( Hom ` T ) u ) /\ h e. ( u ( Hom ` T ) v ) ) ) ) -> ( g ( <. t , t >. (comp ` T ) u ) <. ( I ` ( 1st ` t ) ) , ( J ` ( 2nd ` t ) ) >. ) = <. ( ( 1st ` g ) ( <. ( 1st ` t ) , ( 1st ` t ) >. (comp ` C ) ( 1st ` u ) ) ( 1st ` <. ( I ` ( 1st ` t ) ) , ( J ` ( 2nd ` t ) ) >. ) ) , ( ( 2nd ` g ) ( <. ( 2nd ` t ) , ( 2nd ` t ) >. (comp ` D ) ( 2nd ` u ) ) ( 2nd ` <. ( I ` ( 1st ` t ) ) , ( J ` ( 2nd ` t ) ) >. ) ) >. )
88		1st2nd2 6345	. . . . 5 |- ( g e. ( ( ( 1st ` t ) ( Hom ` C ) ( 1st ` u ) ) X. ( ( 2nd ` t ) ( Hom ` D ) ( 2nd ` u ) ) ) -> g = <. ( 1st ` g ) , ( 2nd ` g ) >. )
89	74, 88	syl 16	. . . 4 |- ( ( ph /\ ( ( s e. ( X X. Y ) /\ t e. ( X X. Y ) ) /\ ( u e. ( X X. Y ) /\ v e. ( X X. Y ) ) /\ ( f e. ( s ( Hom ` T ) t ) /\ g e. ( t ( Hom ` T ) u ) /\ h e. ( u ( Hom ` T ) v ) ) ) ) -> g = <. ( 1st ` g ) , ( 2nd ` g ) >. )
90	86, 87, 89	3eqtr4d 2446	. . 3 |- ( ( ph /\ ( ( s e. ( X X. Y ) /\ t e. ( X X. Y ) ) /\ ( u e. ( X X. Y ) /\ v e. ( X X. Y ) ) /\ ( f e. ( s ( Hom ` T ) t ) /\ g e. ( t ( Hom ` T ) u ) /\ h e. ( u ( Hom ` T ) v ) ) ) ) -> ( g ( <. t , t >. (comp ` T ) u ) <. ( I ` ( 1st ` t ) ) , ( J ` ( 2nd ` t ) ) >. ) = g )
91	2, 12, 40, 36, 39, 42, 71, 47, 76	catcocl 13865	. . . . 5 |- ( ( ph /\ ( ( s e. ( X X. Y ) /\ t e. ( X X. Y ) ) /\ ( u e. ( X X. Y ) /\ v e. ( X X. Y ) ) /\ ( f e. ( s ( Hom ` T ) t ) /\ g e. ( t ( Hom ` T ) u ) /\ h e. ( u ( Hom ` T ) v ) ) ) ) -> ( ( 1st ` g ) ( <. ( 1st ` s ) , ( 1st ` t ) >. (comp ` C ) ( 1st ` u ) ) ( 1st ` f ) ) e. ( ( 1st ` s ) ( Hom ` C ) ( 1st ` u ) ) )
92	3, 19, 55, 52, 54, 56, 81, 58, 83	catcocl 13865	. . . . 5 |- ( ( ph /\ ( ( s e. ( X X. Y ) /\ t e. ( X X. Y ) ) /\ ( u e. ( X X. Y ) /\ v e. ( X X. Y ) ) /\ ( f e. ( s ( Hom ` T ) t ) /\ g e. ( t ( Hom ` T ) u ) /\ h e. ( u ( Hom ` T ) v ) ) ) ) -> ( ( 2nd ` g ) ( <. ( 2nd ` s ) , ( 2nd ` t ) >. (comp ` D ) ( 2nd ` u ) ) ( 2nd ` f ) ) e. ( ( 2nd ` s ) ( Hom ` D ) ( 2nd ` u ) ) )
93		opelxpi 4869	. . . . 5 |- ( ( ( ( 1st ` g ) ( <. ( 1st ` s ) , ( 1st ` t ) >. (comp ` C ) ( 1st ` u ) ) ( 1st ` f ) ) e. ( ( 1st ` s ) ( Hom ` C ) ( 1st ` u ) ) /\ ( ( 2nd ` g ) ( <. ( 2nd ` s ) , ( 2nd ` t ) >. (comp ` D ) ( 2nd ` u ) ) ( 2nd ` f ) ) e. ( ( 2nd ` s ) ( Hom ` D ) ( 2nd ` u ) ) ) -> <. ( ( 1st ` g ) ( <. ( 1st ` s ) , ( 1st ` t ) >. (comp ` C ) ( 1st ` u ) ) ( 1st ` f ) ) , ( ( 2nd ` g ) ( <. ( 2nd ` s ) , ( 2nd ` t ) >. (comp ` D ) ( 2nd ` u ) ) ( 2nd ` f ) ) >. e. ( ( ( 1st ` s ) ( Hom ` C ) ( 1st ` u ) ) X. ( ( 2nd ` s ) ( Hom ` D ) ( 2nd ` u ) ) ) )
94	91, 92, 93	syl2anc 643	. . . 4 |- ( ( ph /\ ( ( s e. ( X X. Y ) /\ t e. ( X X. Y ) ) /\ ( u e. ( X X. Y ) /\ v e. ( X X. Y ) ) /\ ( f e. ( s ( Hom ` T ) t ) /\ g e. ( t ( Hom ` T ) u ) /\ h e. ( u ( Hom ` T ) v ) ) ) ) -> <. ( ( 1st ` g ) ( <. ( 1st ` s ) , ( 1st ` t ) >. (comp ` C ) ( 1st ` u ) ) ( 1st ` f ) ) , ( ( 2nd ` g ) ( <. ( 2nd ` s ) , ( 2nd ` t ) >. (comp ` D ) ( 2nd ` u ) ) ( 2nd ` f ) ) >. e. ( ( ( 1st ` s ) ( Hom ` C ) ( 1st ` u ) ) X. ( ( 2nd ` s ) ( Hom ` D ) ( 2nd ` u ) ) ) )
95	1, 4, 28, 40, 55, 62, 37, 41, 69, 43, 72	xpcco 14235	. . . 4 |- ( ( ph /\ ( ( s e. ( X X. Y ) /\ t e. ( X X. Y ) ) /\ ( u e. ( X X. Y ) /\ v e. ( X X. Y ) ) /\ ( f e. ( s ( Hom ` T ) t ) /\ g e. ( t ( Hom ` T ) u ) /\ h e. ( u ( Hom ` T ) v ) ) ) ) -> ( g ( <. s , t >. (comp ` T ) u ) f ) = <. ( ( 1st ` g ) ( <. ( 1st ` s ) , ( 1st ` t ) >. (comp ` C ) ( 1st ` u ) ) ( 1st ` f ) ) , ( ( 2nd ` g ) ( <. ( 2nd ` s ) , ( 2nd ` t ) >. (comp ` D ) ( 2nd ` u ) ) ( 2nd ` f ) ) >. )
96	1, 4, 12, 19, 28, 37, 69	xpchom 14232	. . . 4 |- ( ( ph /\ ( ( s e. ( X X. Y ) /\ t e. ( X X. Y ) ) /\ ( u e. ( X X. Y ) /\ v e. ( X X. Y ) ) /\ ( f e. ( s ( Hom ` T ) t ) /\ g e. ( t ( Hom ` T ) u ) /\ h e. ( u ( Hom ` T ) v ) ) ) ) -> ( s ( Hom ` T ) u ) = ( ( ( 1st ` s ) ( Hom ` C ) ( 1st ` u ) ) X. ( ( 2nd ` s ) ( Hom ` D ) ( 2nd ` u ) ) ) )
97	94, 95, 96	3eltr4d 2485	. . 3 |- ( ( ph /\ ( ( s e. ( X X. Y ) /\ t e. ( X X. Y ) ) /\ ( u e. ( X X. Y ) /\ v e. ( X X. Y ) ) /\ ( f e. ( s ( Hom ` T ) t ) /\ g e. ( t ( Hom ` T ) u ) /\ h e. ( u ( Hom ` T ) v ) ) ) ) -> ( g ( <. s , t >. (comp ` T ) u ) f ) e. ( s ( Hom ` T ) u ) )
98		simpr2r 1017	. . . . . . . 8 |- ( ( ph /\ ( ( s e. ( X X. Y ) /\ t e. ( X X. Y ) ) /\ ( u e. ( X X. Y ) /\ v e. ( X X. Y ) ) /\ ( f e. ( s ( Hom ` T ) t ) /\ g e. ( t ( Hom ` T ) u ) /\ h e. ( u ( Hom ` T ) v ) ) ) ) -> v e. ( X X. Y ) )
99		xp1st 6335	. . . . . . . 8 |- ( v e. ( X X. Y ) -> ( 1st ` v ) e. X )
100	98, 99	syl 16	. . . . . . 7 |- ( ( ph /\ ( ( s e. ( X X. Y ) /\ t e. ( X X. Y ) ) /\ ( u e. ( X X. Y ) /\ v e. ( X X. Y ) ) /\ ( f e. ( s ( Hom ` T ) t ) /\ g e. ( t ( Hom ` T ) u ) /\ h e. ( u ( Hom ` T ) v ) ) ) ) -> ( 1st ` v ) e. X )
101		simpr33 1049	. . . . . . . . 9 |- ( ( ph /\ ( ( s e. ( X X. Y ) /\ t e. ( X X. Y ) ) /\ ( u e. ( X X. Y ) /\ v e. ( X X. Y ) ) /\ ( f e. ( s ( Hom ` T ) t ) /\ g e. ( t ( Hom ` T ) u ) /\ h e. ( u ( Hom ` T ) v ) ) ) ) -> h e. ( u ( Hom ` T ) v ) )
102	1, 4, 12, 19, 28, 69, 98	xpchom 14232	. . . . . . . . 9 |- ( ( ph /\ ( ( s e. ( X X. Y ) /\ t e. ( X X. Y ) ) /\ ( u e. ( X X. Y ) /\ v e. ( X X. Y ) ) /\ ( f e. ( s ( Hom ` T ) t ) /\ g e. ( t ( Hom ` T ) u ) /\ h e. ( u ( Hom ` T ) v ) ) ) ) -> ( u ( Hom ` T ) v ) = ( ( ( 1st ` u ) ( Hom ` C ) ( 1st ` v ) ) X. ( ( 2nd ` u ) ( Hom ` D ) ( 2nd ` v ) ) ) )
103	101, 102	eleqtrd 2480	. . . . . . . 8 |- ( ( ph /\ ( ( s e. ( X X. Y ) /\ t e. ( X X. Y ) ) /\ ( u e. ( X X. Y ) /\ v e. ( X X. Y ) ) /\ ( f e. ( s ( Hom ` T ) t ) /\ g e. ( t ( Hom ` T ) u ) /\ h e. ( u ( Hom ` T ) v ) ) ) ) -> h e. ( ( ( 1st ` u ) ( Hom ` C ) ( 1st ` v ) ) X. ( ( 2nd ` u ) ( Hom ` D ) ( 2nd ` v ) ) ) )
104		xp1st 6335	. . . . . . . 8 |- ( h e. ( ( ( 1st ` u ) ( Hom ` C ) ( 1st ` v ) ) X. ( ( 2nd ` u ) ( Hom ` D ) ( 2nd ` v ) ) ) -> ( 1st ` h ) e. ( ( 1st ` u ) ( Hom ` C ) ( 1st ` v ) ) )
105	103, 104	syl 16	. . . . . . 7 |- ( ( ph /\ ( ( s e. ( X X. Y ) /\ t e. ( X X. Y ) ) /\ ( u e. ( X X. Y ) /\ v e. ( X X. Y ) ) /\ ( f e. ( s ( Hom ` T ) t ) /\ g e. ( t ( Hom ` T ) u ) /\ h e. ( u ( Hom ` T ) v ) ) ) ) -> ( 1st ` h ) e. ( ( 1st ` u ) ( Hom ` C ) ( 1st ` v ) ) )
106	2, 12, 40, 36, 39, 42, 71, 47, 76, 100, 105	catass 13866	. . . . . 6 |- ( ( ph /\ ( ( s e. ( X X. Y ) /\ t e. ( X X. Y ) ) /\ ( u e. ( X X. Y ) /\ v e. ( X X. Y ) ) /\ ( f e. ( s ( Hom ` T ) t ) /\ g e. ( t ( Hom ` T ) u ) /\ h e. ( u ( Hom ` T ) v ) ) ) ) -> ( ( ( 1st ` h ) ( <. ( 1st ` t ) , ( 1st ` u ) >. (comp ` C ) ( 1st ` v ) ) ( 1st ` g ) ) ( <. ( 1st ` s ) , ( 1st ` t ) >. (comp ` C ) ( 1st ` v ) ) ( 1st ` f ) ) = ( ( 1st ` h ) ( <. ( 1st ` s ) , ( 1st ` u ) >. (comp ` C ) ( 1st ` v ) ) ( ( 1st ` g ) ( <. ( 1st ` s ) , ( 1st ` t ) >. (comp ` C ) ( 1st ` u ) ) ( 1st ` f ) ) ) )
107	1, 4, 28, 40, 55, 62, 41, 69, 98, 72, 101	xpcco 14235	. . . . . . . . 9 |- ( ( ph /\ ( ( s e. ( X X. Y ) /\ t e. ( X X. Y ) ) /\ ( u e. ( X X. Y ) /\ v e. ( X X. Y ) ) /\ ( f e. ( s ( Hom ` T ) t ) /\ g e. ( t ( Hom ` T ) u ) /\ h e. ( u ( Hom ` T ) v ) ) ) ) -> ( h ( <. t , u >. (comp ` T ) v ) g ) = <. ( ( 1st ` h ) ( <. ( 1st ` t ) , ( 1st ` u ) >. (comp ` C ) ( 1st ` v ) ) ( 1st ` g ) ) , ( ( 2nd ` h ) ( <. ( 2nd ` t ) , ( 2nd ` u ) >. (comp ` D ) ( 2nd ` v ) ) ( 2nd ` g ) ) >. )
108	107	fveq2d 5691	. . . . . . . 8 |- ( ( ph /\ ( ( s e. ( X X. Y ) /\ t e. ( X X. Y ) ) /\ ( u e. ( X X. Y ) /\ v e. ( X X. Y ) ) /\ ( f e. ( s ( Hom ` T ) t ) /\ g e. ( t ( Hom ` T ) u ) /\ h e. ( u ( Hom ` T ) v ) ) ) ) -> ( 1st ` ( h ( <. t , u >. (comp ` T ) v ) g ) ) = ( 1st ` <. ( ( 1st ` h ) ( <. ( 1st ` t ) , ( 1st ` u ) >. (comp ` C ) ( 1st ` v ) ) ( 1st ` g ) ) , ( ( 2nd ` h ) ( <. ( 2nd ` t ) , ( 2nd ` u ) >. (comp ` D ) ( 2nd ` v ) ) ( 2nd ` g ) ) >. ) )
109		ovex 6065	. . . . . . . . 9 |- ( ( 1st ` h ) ( <. ( 1st ` t ) , ( 1st ` u ) >. (comp ` C ) ( 1st ` v ) ) ( 1st ` g ) ) e. _V
110		ovex 6065	. . . . . . . . 9 |- ( ( 2nd ` h ) ( <. ( 2nd ` t ) , ( 2nd ` u ) >. (comp ` D ) ( 2nd ` v ) ) ( 2nd ` g ) ) e. _V
111	109, 110	op1st 6314	. . . . . . . 8 |- ( 1st ` <. ( ( 1st ` h ) ( <. ( 1st ` t ) , ( 1st ` u ) >. (comp ` C ) ( 1st ` v ) ) ( 1st ` g ) ) , ( ( 2nd ` h ) ( <. ( 2nd ` t ) , ( 2nd ` u ) >. (comp ` D ) ( 2nd ` v ) ) ( 2nd ` g ) ) >. ) = ( ( 1st ` h ) ( <. ( 1st ` t ) , ( 1st ` u ) >. (comp ` C ) ( 1st ` v ) ) ( 1st ` g ) )
112	108, 111	syl6eq 2452	. . . . . . 7 |- ( ( ph /\ ( ( s e. ( X X. Y ) /\ t e. ( X X. Y ) ) /\ ( u e. ( X X. Y ) /\ v e. ( X X. Y ) ) /\ ( f e. ( s ( Hom ` T ) t ) /\ g e. ( t ( Hom ` T ) u ) /\ h e. ( u ( Hom ` T ) v ) ) ) ) -> ( 1st ` ( h ( <. t , u >. (comp ` T ) v ) g ) ) = ( ( 1st ` h ) ( <. ( 1st ` t ) , ( 1st ` u ) >. (comp ` C ) ( 1st ` v ) ) ( 1st ` g ) ) )
113	112	oveq1d 6055	. . . . . 6 |- ( ( ph /\ ( ( s e. ( X X. Y ) /\ t e. ( X X. Y ) ) /\ ( u e. ( X X. Y ) /\ v e. ( X X. Y ) ) /\ ( f e. ( s ( Hom ` T ) t ) /\ g e. ( t ( Hom ` T ) u ) /\ h e. ( u ( Hom ` T ) v ) ) ) ) -> ( ( 1st ` ( h ( <. t , u >. (comp ` T ) v ) g ) ) ( <. ( 1st ` s ) , ( 1st ` t ) >. (comp ` C ) ( 1st ` v ) ) ( 1st ` f ) ) = ( ( ( 1st ` h ) ( <. ( 1st ` t ) , ( 1st ` u ) >. (comp ` C ) ( 1st ` v ) ) ( 1st ` g ) ) ( <. ( 1st ` s ) , ( 1st ` t ) >. (comp ` C ) ( 1st ` v ) ) ( 1st ` f ) ) )
114	95	fveq2d 5691	. . . . . . . 8 |- ( ( ph /\ ( ( s e. ( X X. Y ) /\ t e. ( X X. Y ) ) /\ ( u e. ( X X. Y ) /\ v e. ( X X. Y ) ) /\ ( f e. ( s ( Hom ` T ) t ) /\ g e. ( t ( Hom ` T ) u ) /\ h e. ( u ( Hom ` T ) v ) ) ) ) -> ( 1st ` ( g ( <. s , t >. (comp ` T ) u ) f ) ) = ( 1st ` <. ( ( 1st ` g ) ( <. ( 1st ` s ) , ( 1st ` t ) >. (comp ` C ) ( 1st ` u ) ) ( 1st ` f ) ) , ( ( 2nd ` g ) ( <. ( 2nd ` s ) , ( 2nd ` t ) >. (comp ` D ) ( 2nd ` u ) ) ( 2nd ` f ) ) >. ) )
115		ovex 6065	. . . . . . . . 9 |- ( ( 1st ` g ) ( <. ( 1st ` s ) , ( 1st ` t ) >. (comp ` C ) ( 1st ` u ) ) ( 1st ` f ) ) e. _V
116		ovex 6065	. . . . . . . . 9 |- ( ( 2nd ` g ) ( <. ( 2nd ` s ) , ( 2nd ` t ) >. (comp ` D ) ( 2nd ` u ) ) ( 2nd ` f ) ) e. _V
117	115, 116	op1st 6314	. . . . . . . 8 |- ( 1st ` <. ( ( 1st ` g ) ( <. ( 1st ` s ) , ( 1st ` t ) >. (comp ` C ) ( 1st ` u ) ) ( 1st ` f ) ) , ( ( 2nd ` g ) ( <. ( 2nd ` s ) , ( 2nd ` t ) >. (comp ` D ) ( 2nd ` u ) ) ( 2nd ` f ) ) >. ) = ( ( 1st ` g ) ( <. ( 1st ` s ) , ( 1st ` t ) >. (comp ` C ) ( 1st ` u ) ) ( 1st ` f ) )
118	114, 117	syl6eq 2452	. . . . . . 7 |- ( ( ph /\ ( ( s e. ( X X. Y ) /\ t e. ( X X. Y ) ) /\ ( u e. ( X X. Y ) /\ v e. ( X X. Y ) ) /\ ( f e. ( s ( Hom ` T ) t ) /\ g e. ( t ( Hom ` T ) u ) /\ h e. ( u ( Hom ` T ) v ) ) ) ) -> ( 1st ` ( g ( <. s , t >. (comp ` T ) u ) f ) ) = ( ( 1st ` g ) ( <. ( 1st ` s ) , ( 1st ` t ) >. (comp ` C ) ( 1st ` u ) ) ( 1st ` f ) ) )
119	118	oveq2d 6056	. . . . . 6 |- ( ( ph /\ ( ( s e. ( X X. Y ) /\ t e. ( X X. Y ) ) /\ ( u e. ( X X. Y ) /\ v e. ( X X. Y ) ) /\ ( f e. ( s ( Hom ` T ) t ) /\ g e. ( t ( Hom ` T ) u ) /\ h e. ( u ( Hom ` T ) v ) ) ) ) -> ( ( 1st ` h ) ( <. ( 1st ` s ) , ( 1st ` u ) >. (comp ` C ) ( 1st ` v ) ) ( 1st ` ( g ( <. s , t >. (comp ` T ) u ) f ) ) ) = ( ( 1st ` h ) ( <. ( 1st ` s ) , ( 1st ` u ) >. (comp ` C ) ( 1st ` v ) ) ( ( 1st ` g ) ( <. ( 1st ` s ) , ( 1st ` t ) >. (comp ` C ) ( 1st ` u ) ) ( 1st ` f ) ) ) )
120	106, 113, 119	3eqtr4d 2446	. . . . 5 |- ( ( ph /\ ( ( s e. ( X X. Y ) /\ t e. ( X X. Y ) ) /\ ( u e. ( X X. Y ) /\ v e. ( X X. Y ) ) /\ ( f e. ( s ( Hom ` T ) t ) /\ g e. ( t ( Hom ` T ) u ) /\ h e. ( u ( Hom ` T ) v ) ) ) ) -> ( ( 1st ` ( h ( <. t , u >. (comp ` T ) v ) g ) ) ( <. ( 1st ` s ) , ( 1st ` t ) >. (comp ` C ) ( 1st ` v ) ) ( 1st ` f ) ) = ( ( 1st ` h ) ( <. ( 1st ` s ) , ( 1st ` u ) >. (comp ` C ) ( 1st ` v ) ) ( 1st ` ( g ( <. s , t >. (comp ` T ) u ) f ) ) ) )
121		xp2nd 6336	. . . . . . . 8 |- ( v e. ( X X. Y ) -> ( 2nd ` v ) e. Y )
122	98, 121	syl 16	. . . . . . 7 |- ( ( ph /\ ( ( s e. ( X X. Y ) /\ t e. ( X X. Y ) ) /\ ( u e. ( X X. Y ) /\ v e. ( X X. Y ) ) /\ ( f e. ( s ( Hom ` T ) t ) /\ g e. ( t ( Hom ` T ) u ) /\ h e. ( u ( Hom ` T ) v ) ) ) ) -> ( 2nd ` v ) e. Y )
123		xp2nd 6336	. . . . . . . 8 |- ( h e. ( ( ( 1st ` u ) ( Hom ` C ) ( 1st ` v ) ) X. ( ( 2nd ` u ) ( Hom ` D ) ( 2nd ` v ) ) ) -> ( 2nd ` h ) e. ( ( 2nd ` u ) ( Hom ` D ) ( 2nd ` v ) ) )
124	103, 123	syl 16	. . . . . . 7 |- ( ( ph /\ ( ( s e. ( X X. Y ) /\ t e. ( X X. Y ) ) /\ ( u e. ( X X. Y ) /\ v e. ( X X. Y ) ) /\ ( f e. ( s ( Hom ` T ) t ) /\ g e. ( t ( Hom ` T ) u ) /\ h e. ( u ( Hom ` T ) v ) ) ) ) -> ( 2nd ` h ) e. ( ( 2nd ` u ) ( Hom ` D ) ( 2nd ` v ) ) )
125	3, 19, 55, 52, 54, 56, 81, 58, 83, 122, 124	catass 13866	. . . . . 6 |- ( ( ph /\ ( ( s e. ( X X. Y ) /\ t e. ( X X. Y ) ) /\ ( u e. ( X X. Y ) /\ v e. ( X X. Y ) ) /\ ( f e. ( s ( Hom ` T ) t ) /\ g e. ( t ( Hom ` T ) u ) /\ h e. ( u ( Hom ` T ) v ) ) ) ) -> ( ( ( 2nd ` h ) ( <. ( 2nd ` t ) , ( 2nd ` u ) >. (comp ` D ) ( 2nd ` v ) ) ( 2nd ` g ) ) ( <. ( 2nd ` s ) , ( 2nd ` t ) >. (comp ` D ) ( 2nd ` v ) ) ( 2nd ` f ) ) = ( ( 2nd ` h ) ( <. ( 2nd ` s ) , ( 2nd ` u ) >. (comp ` D ) ( 2nd ` v ) ) ( ( 2nd ` g ) ( <. ( 2nd ` s ) , ( 2nd ` t ) >. (comp ` D ) ( 2nd ` u ) ) ( 2nd ` f ) ) ) )
126	107	fveq2d 5691	. . . . . . . 8 |- ( ( ph /\ ( ( s e. ( X X. Y ) /\ t e. ( X X. Y ) ) /\ ( u e. ( X X. Y ) /\ v e. ( X X. Y ) ) /\ ( f e. ( s ( Hom ` T ) t ) /\ g e. ( t ( Hom ` T ) u ) /\ h e. ( u ( Hom ` T ) v ) ) ) ) -> ( 2nd ` ( h ( <. t , u >. (comp ` T ) v ) g ) ) = ( 2nd ` <. ( ( 1st ` h ) ( <. ( 1st ` t ) , ( 1st ` u ) >. (comp ` C ) ( 1st ` v ) ) ( 1st ` g ) ) , ( ( 2nd ` h ) ( <. ( 2nd ` t ) , ( 2nd ` u ) >. (comp ` D ) ( 2nd ` v ) ) ( 2nd ` g ) ) >. ) )
127	109, 110	op2nd 6315	. . . . . . . 8 |- ( 2nd ` <. ( ( 1st ` h ) ( <. ( 1st ` t ) , ( 1st ` u ) >. (comp ` C ) ( 1st ` v ) ) ( 1st ` g ) ) , ( ( 2nd ` h ) ( <. ( 2nd ` t ) , ( 2nd ` u ) >. (comp ` D ) ( 2nd ` v ) ) ( 2nd ` g ) ) >. ) = ( ( 2nd ` h ) ( <. ( 2nd ` t ) , ( 2nd ` u ) >. (comp ` D ) ( 2nd ` v ) ) ( 2nd ` g ) )
128	126, 127	syl6eq 2452	. . . . . . 7 |- ( ( ph /\ ( ( s e. ( X X. Y ) /\ t e. ( X X. Y ) ) /\ ( u e. ( X X. Y ) /\ v e. ( X X. Y ) ) /\ ( f e. ( s ( Hom ` T ) t ) /\ g e. ( t ( Hom ` T ) u ) /\ h e. ( u ( Hom ` T ) v ) ) ) ) -> ( 2nd ` ( h ( <. t , u >. (comp ` T ) v ) g ) ) = ( ( 2nd ` h ) ( <. ( 2nd ` t ) , ( 2nd ` u ) >. (comp ` D ) ( 2nd ` v ) ) ( 2nd ` g ) ) )
129	128	oveq1d 6055	. . . . . 6 |- ( ( ph /\ ( ( s e. ( X X. Y ) /\ t e. ( X X. Y ) ) /\ ( u e. ( X X. Y ) /\ v e. ( X X. Y ) ) /\ ( f e. ( s ( Hom ` T ) t ) /\ g e. ( t ( Hom ` T ) u ) /\ h e. ( u ( Hom ` T ) v ) ) ) ) -> ( ( 2nd ` ( h ( <. t , u >. (comp ` T ) v ) g ) ) ( <. ( 2nd ` s ) , ( 2nd ` t ) >. (comp ` D ) ( 2nd ` v ) ) ( 2nd ` f ) ) = ( ( ( 2nd ` h ) ( <. ( 2nd ` t ) , ( 2nd ` u ) >. (comp ` D ) ( 2nd ` v ) ) ( 2nd ` g ) ) ( <. ( 2nd ` s ) , ( 2nd ` t ) >. (comp ` D ) ( 2nd ` v ) ) ( 2nd ` f ) ) )
130	95	fveq2d 5691	. . . . . . . 8 |- ( ( ph /\ ( ( s e. ( X X. Y ) /\ t e. ( X X. Y ) ) /\ ( u e. ( X X. Y ) /\ v e. ( X X. Y ) ) /\ ( f e. ( s ( Hom ` T ) t ) /\ g e. ( t ( Hom ` T ) u ) /\ h e. ( u ( Hom ` T ) v ) ) ) ) -> ( 2nd ` ( g ( <. s , t >. (comp ` T ) u ) f ) ) = ( 2nd ` <. ( ( 1st ` g ) ( <. ( 1st ` s ) , ( 1st ` t ) >. (comp ` C ) ( 1st ` u ) ) ( 1st ` f ) ) , ( ( 2nd ` g ) ( <. ( 2nd ` s ) , ( 2nd ` t ) >. (comp ` D ) ( 2nd ` u ) ) ( 2nd ` f ) ) >. ) )
131	115, 116	op2nd 6315	. . . . . . . 8 |- ( 2nd ` <. ( ( 1st ` g ) ( <. ( 1st ` s ) , ( 1st ` t ) >. (comp ` C ) ( 1st ` u ) ) ( 1st ` f ) ) , ( ( 2nd ` g ) ( <. ( 2nd ` s ) , ( 2nd ` t ) >. (comp ` D ) ( 2nd ` u ) ) ( 2nd ` f ) ) >. ) = ( ( 2nd ` g ) ( <. ( 2nd ` s ) , ( 2nd ` t ) >. (comp ` D ) ( 2nd ` u ) ) ( 2nd ` f ) )
132	130, 131	syl6eq 2452	. . . . . . 7 |- ( ( ph /\ ( ( s e. ( X X. Y ) /\ t e. ( X X. Y ) ) /\ ( u e. ( X X. Y ) /\ v e. ( X X. Y ) ) /\ ( f e. ( s ( Hom ` T ) t ) /\ g e. ( t ( Hom ` T ) u ) /\ h e. ( u ( Hom ` T ) v ) ) ) ) -> ( 2nd ` ( g ( <. s , t >. (comp ` T ) u ) f ) ) = ( ( 2nd ` g ) ( <. ( 2nd ` s ) , ( 2nd ` t ) >. (comp ` D ) ( 2nd ` u ) ) ( 2nd ` f ) ) )
133	132	oveq2d 6056	. . . . . 6 |- ( ( ph /\ ( ( s e. ( X X. Y ) /\ t e. ( X X. Y ) ) /\ ( u e. ( X X. Y ) /\ v e. ( X X. Y ) ) /\ ( f e. ( s ( Hom ` T ) t ) /\ g e. ( t ( Hom ` T ) u ) /\ h e. ( u ( Hom ` T ) v ) ) ) ) -> ( ( 2nd ` h ) ( <. ( 2nd ` s ) , ( 2nd ` u ) >. (comp ` D ) ( 2nd ` v ) ) ( 2nd ` ( g ( <. s , t >. (comp ` T ) u ) f ) ) ) = ( ( 2nd ` h ) ( <. ( 2nd ` s ) , ( 2nd ` u ) >. (comp ` D ) ( 2nd ` v ) ) ( ( 2nd ` g ) ( <. ( 2nd ` s ) , ( 2nd ` t ) >. (comp ` D ) ( 2nd ` u ) ) ( 2nd ` f ) ) ) )
134	125, 129, 133	3eqtr4d 2446	. . . . 5 |- ( ( ph /\ ( ( s e. ( X X. Y ) /\ t e. ( X X. Y ) ) /\ ( u e. ( X X. Y ) /\ v e. ( X X. Y ) ) /\ ( f e. ( s ( Hom ` T ) t ) /\ g e. ( t ( Hom ` T ) u ) /\ h e. ( u ( Hom ` T ) v ) ) ) ) -> ( ( 2nd ` ( h ( <. t , u >. (comp ` T ) v ) g ) ) ( <. ( 2nd ` s ) , ( 2nd ` t ) >. (comp ` D ) ( 2nd ` v ) ) ( 2nd ` f ) ) = ( ( 2nd ` h ) ( <. ( 2nd ` s ) , ( 2nd ` u ) >. (comp ` D ) ( 2nd ` v ) ) ( 2nd ` ( g ( <. s , t >. (comp ` T ) u ) f ) ) ) )
135	120, 134	opeq12d 3952	. . . 4 |- ( ( ph /\ ( ( s e. ( X X. Y ) /\ t e. ( X X. Y ) ) /\ ( u e. ( X X. Y ) /\ v e. ( X X. Y ) ) /\ ( f e. ( s ( Hom ` T ) t ) /\ g e. ( t ( Hom ` T ) u ) /\ h e. ( u ( Hom ` T ) v ) ) ) ) -> <. ( ( 1st ` ( h ( <. t , u >. (comp ` T ) v ) g ) ) ( <. ( 1st ` s ) , ( 1st ` t ) >. (comp ` C ) ( 1st ` v ) ) ( 1st ` f ) ) , ( ( 2nd ` ( h ( <. t , u >. (comp ` T ) v ) g ) ) ( <. ( 2nd ` s ) , ( 2nd ` t ) >. (comp ` D ) ( 2nd ` v ) ) ( 2nd ` f ) ) >. = <. ( ( 1st ` h ) ( <. ( 1st ` s ) , ( 1st ` u ) >. (comp ` C ) ( 1st ` v ) ) ( 1st ` ( g ( <. s , t >. (comp ` T ) u ) f ) ) ) , ( ( 2nd ` h ) ( <. ( 2nd ` s ) , ( 2nd ` u ) >. (comp ` D ) ( 2nd ` v ) ) ( 2nd ` ( g ( <. s , t >. (comp ` T ) u ) f ) ) ) >. )
136	2, 12, 40, 36, 42, 71, 100, 76, 105	catcocl 13865	. . . . . . 7 |- ( ( ph /\ ( ( s e. ( X X. Y ) /\ t e. ( X X. Y ) ) /\ ( u e. ( X X. Y ) /\ v e. ( X X. Y ) ) /\ ( f e. ( s ( Hom ` T ) t ) /\ g e. ( t ( Hom ` T ) u ) /\ h e. ( u ( Hom ` T ) v ) ) ) ) -> ( ( 1st ` h ) ( <. ( 1st ` t ) , ( 1st ` u ) >. (comp ` C ) ( 1st ` v ) ) ( 1st ` g ) ) e. ( ( 1st ` t ) ( Hom ` C ) ( 1st ` v ) ) )
137	3, 19, 55, 52, 56, 81, 122, 83, 124	catcocl 13865	. . . . . . 7 |- ( ( ph /\ ( ( s e. ( X X. Y ) /\ t e. ( X X. Y ) ) /\ ( u e. ( X X. Y ) /\ v e. ( X X. Y ) ) /\ ( f e. ( s ( Hom ` T ) t ) /\ g e. ( t ( Hom ` T ) u ) /\ h e. ( u ( Hom ` T ) v ) ) ) ) -> ( ( 2nd ` h ) ( <. ( 2nd ` t ) , ( 2nd ` u ) >. (comp ` D ) ( 2nd ` v ) ) ( 2nd ` g ) ) e. ( ( 2nd ` t ) ( Hom ` D ) ( 2nd ` v ) ) )
138		opelxpi 4869	. . . . . . 7 |- ( ( ( ( 1st ` h ) ( <. ( 1st ` t ) , ( 1st ` u ) >. (comp ` C ) ( 1st ` v ) ) ( 1st ` g ) ) e. ( ( 1st ` t ) ( Hom ` C ) ( 1st ` v ) ) /\ ( ( 2nd ` h ) ( <. ( 2nd ` t ) , ( 2nd ` u ) >. (comp ` D ) ( 2nd ` v ) ) ( 2nd ` g ) ) e. ( ( 2nd ` t ) ( Hom ` D ) ( 2nd ` v ) ) ) -> <. ( ( 1st ` h ) ( <. ( 1st ` t ) , ( 1st ` u ) >. (comp ` C ) ( 1st ` v ) ) ( 1st ` g ) ) , ( ( 2nd ` h ) ( <. ( 2nd ` t ) , ( 2nd ` u ) >. (comp ` D ) ( 2nd ` v ) ) ( 2nd ` g ) ) >. e. ( ( ( 1st ` t ) ( Hom ` C ) ( 1st ` v ) ) X. ( ( 2nd ` t ) ( Hom ` D ) ( 2nd ` v ) ) ) )
139	136, 137, 138	syl2anc 643	. . . . . 6 |- ( ( ph /\ ( ( s e. ( X X. Y ) /\ t e. ( X X. Y ) ) /\ ( u e. ( X X. Y ) /\ v e. ( X X. Y ) ) /\ ( f e. ( s ( Hom ` T ) t ) /\ g e. ( t ( Hom ` T ) u ) /\ h e. ( u ( Hom ` T ) v ) ) ) ) -> <. ( ( 1st ` h ) ( <. ( 1st ` t ) , ( 1st ` u ) >. (comp ` C ) ( 1st ` v ) ) ( 1st ` g ) ) , ( ( 2nd ` h ) ( <. ( 2nd ` t ) , ( 2nd ` u ) >. (comp ` D ) ( 2nd ` v ) ) ( 2nd ` g ) ) >. e. ( ( ( 1st ` t ) ( Hom ` C ) ( 1st ` v ) ) X. ( ( 2nd ` t ) ( Hom ` D ) ( 2nd ` v ) ) ) )
140	1, 4, 12, 19, 28, 41, 98	xpchom 14232	. . . . . 6 |- ( ( ph /\ ( ( s e. ( X X. Y ) /\ t e. ( X X. Y ) ) /\ ( u e. ( X X. Y ) /\ v e. ( X X. Y ) ) /\ ( f e. ( s ( Hom ` T ) t ) /\ g e. ( t ( Hom ` T ) u ) /\ h e. ( u ( Hom ` T ) v ) ) ) ) -> ( t ( Hom ` T ) v ) = ( ( ( 1st ` t ) ( Hom ` C ) ( 1st ` v ) ) X. ( ( 2nd ` t ) ( Hom ` D ) ( 2nd ` v ) ) ) )
141	139, 107, 140	3eltr4d 2485	. . . . 5 |- ( ( ph /\ ( ( s e. ( X X. Y ) /\ t e. ( X X. Y ) ) /\ ( u e. ( X X. Y ) /\ v e. ( X X. Y ) ) /\ ( f e. ( s ( Hom ` T ) t ) /\ g e. ( t ( Hom ` T ) u ) /\ h e. ( u ( Hom ` T ) v ) ) ) ) -> ( h ( <. t , u >. (comp ` T ) v ) g ) e. ( t ( Hom ` T ) v ) )
142	1, 4, 28, 40, 55, 62, 37, 41, 98, 43, 141	xpcco 14235	. . . 4 |- ( ( ph /\ ( ( s e. ( X X. Y ) /\ t e. ( X X. Y ) ) /\ ( u e. ( X X. Y ) /\ v e. ( X X. Y ) ) /\ ( f e. ( s ( Hom ` T ) t ) /\ g e. ( t ( Hom ` T ) u ) /\ h e. ( u ( Hom ` T ) v ) ) ) ) -> ( ( h ( <. t , u >. (comp ` T ) v ) g ) ( <. s , t >. (comp ` T ) v ) f ) = <. ( ( 1st ` ( h ( <. t , u >. (comp ` T ) v ) g ) ) ( <. ( 1st ` s ) , ( 1st ` t ) >. (comp ` C ) ( 1st ` v ) ) ( 1st ` f ) ) , ( ( 2nd ` ( h ( <. t , u >. (comp ` T ) v ) g ) ) ( <. ( 2nd ` s ) , ( 2nd ` t ) >. (comp ` D ) ( 2nd ` v ) ) ( 2nd ` f ) ) >. )
143	1, 4, 28, 40, 55, 62, 37, 69, 98, 97, 101	xpcco 14235	. . . 4 |- ( ( ph /\ ( ( s e. ( X X. Y ) /\ t e. ( X X. Y ) ) /\ ( u e. ( X X. Y ) /\ v e. ( X X. Y ) ) /\ ( f e. ( s ( Hom ` T ) t ) /\ g e. ( t ( Hom ` T ) u ) /\ h e. ( u ( Hom ` T ) v ) ) ) ) -> ( h ( <. s , u >. (comp ` T ) v ) ( g ( <. s , t >. (comp ` T ) u ) f ) ) = <. ( ( 1st ` h ) ( <. ( 1st ` s ) , ( 1st ` u ) >. (comp ` C ) ( 1st ` v ) ) ( 1st ` ( g ( <. s , t >. (comp ` T ) u ) f ) ) ) , ( ( 2nd ` h ) ( <. ( 2nd ` s ) , ( 2nd ` u ) >. (comp ` D ) ( 2nd ` v ) ) ( 2nd ` ( g ( <. s , t >. (comp ` T ) u ) f ) ) ) >. )
144	135, 142, 143	3eqtr4d 2446	. . 3 |- ( ( ph /\ ( ( s e. ( X X. Y ) /\ t e. ( X X. Y ) ) /\ ( u e. ( X X. Y ) /\ v e. ( X X. Y ) ) /\ ( f e. ( s ( Hom ` T ) t ) /\ g e. ( t ( Hom ` T ) u ) /\ h e. ( u ( Hom ` T ) v ) ) ) ) -> ( ( h ( <. t , u >. (comp ` T ) v ) g ) ( <. s , t >. (comp ` T ) v ) f ) = ( h ( <. s , u >. (comp ` T ) v ) ( g ( <. s , t >. (comp ` T ) u ) f ) ) )
145	5, 6, 7, 10, 11, 31, 67, 90, 97, 144	iscatd2 13861	. 2 |- ( ph -> ( T e. Cat /\ ( Id ` T ) = ( t e. ( X X. Y ) |-> <. ( I ` ( 1st ` t ) ) , ( J ` ( 2nd ` t ) ) >. ) ) )
146		vex 2919	. . . . . . . 8 |- x e. _V
147		vex 2919	. . . . . . . 8 |- y e. _V
148	146, 147	op1std 6316	. . . . . . 7 |- ( t = <. x , y >. -> ( 1st ` t ) = x )
149	148	fveq2d 5691	. . . . . 6 |- ( t = <. x , y >. -> ( I ` ( 1st ` t ) ) = ( I ` x ) )
150	146, 147	op2ndd 6317	. . . . . . 7 |- ( t = <. x , y >. -> ( 2nd ` t ) = y )
151	150	fveq2d 5691	. . . . . 6 |- ( t = <. x , y >. -> ( J ` ( 2nd ` t ) ) = ( J ` y ) )
152	149, 151	opeq12d 3952	. . . . 5 |- ( t = <. x , y >. -> <. ( I ` ( 1st ` t ) ) , ( J ` ( 2nd ` t ) ) >. = <. ( I ` x ) , ( J ` y ) >. )
153	152	mpt2mpt 6124	. . . 4 |- ( t e. ( X X. Y ) |-> <. ( I ` ( 1st ` t ) ) , ( J ` ( 2nd ` t ) ) >. ) = ( x e. X , y e. Y |-> <. ( I ` x ) , ( J ` y ) >. )
154	153	eqeq2i 2414	. . 3 |- ( ( Id ` T ) = ( t e. ( X X. Y ) |-> <. ( I ` ( 1st ` t ) ) , ( J ` ( 2nd ` t ) ) >. ) <-> ( Id ` T ) = ( x e. X , y e. Y |-> <. ( I ` x ) , ( J ` y ) >. ) )
155	154	anbi2i 676	. 2 |- ( ( T e. Cat /\ ( Id ` T ) = ( t e. ( X X. Y ) |-> <. ( I ` ( 1st ` t ) ) , ( J ` ( 2nd ` t ) ) >. ) ) <-> ( T e. Cat /\ ( Id ` T ) = ( x e. X , y e. Y |-> <. ( I ` x ) , ( J ` y ) >. ) ) )
156	145, 155	sylib 189	1 |- ( ph -> ( T e. Cat /\ ( Id ` T ) = ( x e. X , y e. Y |-> <. ( I ` x ) , ( J ` y ) >. ) ) )

Syntax hints:   -> wi 4   /\ wa 359   /\ w3a 936   = wceq 1649   e. wcel 1721   _Vcvv 2916   <.cop 3777   e. cmpt 4226   X. cxp 4835   ` cfv 5413  (class class class)co 6040   e. cmpt2 6042   1stc1st 6306   2ndc2nd 6307   Basecbs 13424   Hom chom 13495  compcco 13496   Catccat 13844   Idccid 13845   X._c cxpc 14220

This theorem is referenced by:  xpcid  14241  xpccat  14242

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023

This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  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 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-oadd 6687  df-er 6864  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-nn 9957  df-2 10014  df-3 10015  df-4 10016  df-5 10017  df-6 10018  df-7 10019  df-8 10020  df-9 10021  df-10 10022  df-n0 10178  df-z 10239  df-dec 10339  df-uz 10445  df-fz 11000  df-struct 13426  df-ndx 13427  df-slot 13428  df-base 13429  df-hom 13508  df-cco 13509  df-cat 13848  df-cid 13849  df-xpc 14224