Theorem xpcval 16817

Description: Value of the binary product of categories. (Contributed by Mario Carneiro, 10-Jan-2017.)

Hypotheses

Ref	Expression
xpcval.t	|- T = ( C X.c D )
xpcval.x	|- X = ( Base ` C )
xpcval.y	|- Y = ( Base ` D )
xpcval.h	|- H = ( Hom ` C )
xpcval.j	|- J = ( Hom ` D )
xpcval.o1	|- .x. = (comp ` C )
xpcval.o2	|- .xb = (comp ` D )
xpcval.c	|- ( ph -> C e. V )
xpcval.d	|- ( ph -> D e. W )
xpcval.b	|- ( ph -> B = ( X X. Y ) )
xpcval.k	|- ( ph -> K = ( u e. B , v e. B |-> ( ( ( 1st ` u ) H ( 1st ` v ) ) X. ( ( 2nd ` u ) J ( 2nd ` v ) ) ) ) )
xpcval.o	|- ( ph -> O = ( x e. ( B X. B ) , y e. B |-> ( g e. ( ( 2nd ` x ) K y ) , f e. ( K ` x ) |-> <. ( ( 1st ` g ) ( <. ( 1st ` ( 1st ` x ) ) , ( 1st ` ( 2nd ` x ) ) >. .x. ( 1st ` y ) ) ( 1st ` f ) ) , ( ( 2nd ` g ) ( <. ( 2nd ` ( 1st ` x ) ) , ( 2nd ` ( 2nd ` x ) ) >. .xb ( 2nd ` y ) ) ( 2nd ` f ) ) >. ) ) )

Assertion

Ref	Expression
xpcval	|- ( ph -> T = { <. ( Base ` ndx ) , B >. , <. ( Hom ` ndx ) , K >. , <. (comp ` ndx ) , O >. } )

Distinct variable groups:   f, g, u, v, x, y, B   ph, f, g, u, v, x, y   C, f, g, u, v, x, y   D, f, g, u, v, x, y   f, K, g, x, y

Allowed substitution hints:   .xb ( x, y, v, u, f, g)   T( x, y, v, u, f, g)   .x. ( x, y, v, u, f, g)   H( x, y, v, u, f, g)   J( x, y, v, u, f, g)   K( v, u)   O( x, y, v, u, f, g)   V( x, y, v, u, f, g)   W( x, y, v, u, f, g)   X( x, y, v, u, f, g)   Y( x, y, v, u, f, g)

Proof of Theorem xpcval

Dummy variables b h r s are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		xpcval.t	. 2 |- T = ( C X.c D )
2		df-xpc 16812	. . . 4 |- X.c = ( r e. _V , s e. _V |-> [_ ( ( Base ` r ) X. ( Base ` s ) ) / b ]_ [_ ( u e. b , v e. b |-> ( ( ( 1st ` u ) ( Hom ` r ) ( 1st ` v ) ) X. ( ( 2nd ` u ) ( Hom ` s ) ( 2nd ` v ) ) ) ) / h ]_ { <. ( Base ` ndx ) , b >. , <. ( Hom ` ndx ) , h >. , <. (comp ` ndx ) , ( x e. ( b X. b ) , y e. b |-> ( g e. ( ( 2nd ` x ) h y ) , f e. ( h ` x ) |-> <. ( ( 1st ` g ) ( <. ( 1st ` ( 1st ` x ) ) , ( 1st ` ( 2nd ` x ) ) >. (comp ` r ) ( 1st ` y ) ) ( 1st ` f ) ) , ( ( 2nd ` g ) ( <. ( 2nd ` ( 1st ` x ) ) , ( 2nd ` ( 2nd ` x ) ) >. (comp ` s ) ( 2nd ` y ) ) ( 2nd ` f ) ) >. ) ) >. } )
3	2	a1i 11	. . 3 |- ( ph -> X.c = ( r e. _V , s e. _V |-> [_ ( ( Base ` r ) X. ( Base ` s ) ) / b ]_ [_ ( u e. b , v e. b |-> ( ( ( 1st ` u ) ( Hom ` r ) ( 1st ` v ) ) X. ( ( 2nd ` u ) ( Hom ` s ) ( 2nd ` v ) ) ) ) / h ]_ { <. ( Base ` ndx ) , b >. , <. ( Hom ` ndx ) , h >. , <. (comp ` ndx ) , ( x e. ( b X. b ) , y e. b |-> ( g e. ( ( 2nd ` x ) h y ) , f e. ( h ` x ) |-> <. ( ( 1st ` g ) ( <. ( 1st ` ( 1st ` x ) ) , ( 1st ` ( 2nd ` x ) ) >. (comp ` r ) ( 1st ` y ) ) ( 1st ` f ) ) , ( ( 2nd ` g ) ( <. ( 2nd ` ( 1st ` x ) ) , ( 2nd ` ( 2nd ` x ) ) >. (comp ` s ) ( 2nd ` y ) ) ( 2nd ` f ) ) >. ) ) >. } ) )
4		fvex 6201	. . . . . 6 |- ( Base ` r ) e. _V
5		fvex 6201	. . . . . 6 |- ( Base ` s ) e. _V
6	4, 5	xpex 6962	. . . . 5 |- ( ( Base ` r ) X. ( Base ` s ) ) e. _V
7	6	a1i 11	. . . 4 |- ( ( ph /\ ( r = C /\ s = D ) ) -> ( ( Base ` r ) X. ( Base ` s ) ) e. _V )
8		simprl 794	. . . . . . . 8 |- ( ( ph /\ ( r = C /\ s = D ) ) -> r = C )
9	8	fveq2d 6195	. . . . . . 7 |- ( ( ph /\ ( r = C /\ s = D ) ) -> ( Base ` r ) = ( Base ` C ) )
10		xpcval.x	. . . . . . 7 |- X = ( Base ` C )
11	9, 10	syl6eqr 2674	. . . . . 6 |- ( ( ph /\ ( r = C /\ s = D ) ) -> ( Base ` r ) = X )
12		simprr 796	. . . . . . . 8 |- ( ( ph /\ ( r = C /\ s = D ) ) -> s = D )
13	12	fveq2d 6195	. . . . . . 7 |- ( ( ph /\ ( r = C /\ s = D ) ) -> ( Base ` s ) = ( Base ` D ) )
14		xpcval.y	. . . . . . 7 |- Y = ( Base ` D )
15	13, 14	syl6eqr 2674	. . . . . 6 |- ( ( ph /\ ( r = C /\ s = D ) ) -> ( Base ` s ) = Y )
16	11, 15	xpeq12d 5140	. . . . 5 |- ( ( ph /\ ( r = C /\ s = D ) ) -> ( ( Base ` r ) X. ( Base ` s ) ) = ( X X. Y ) )
17		xpcval.b	. . . . . 6 |- ( ph -> B = ( X X. Y ) )
18	17	adantr 481	. . . . 5 |- ( ( ph /\ ( r = C /\ s = D ) ) -> B = ( X X. Y ) )
19	16, 18	eqtr4d 2659	. . . 4 |- ( ( ph /\ ( r = C /\ s = D ) ) -> ( ( Base ` r ) X. ( Base ` s ) ) = B )
20		vex 3203	. . . . . . 7 |- b e. _V
21	20, 20	mpt2ex 7247	. . . . . 6 |- ( u e. b , v e. b |-> ( ( ( 1st ` u ) ( Hom ` r ) ( 1st ` v ) ) X. ( ( 2nd ` u ) ( Hom ` s ) ( 2nd ` v ) ) ) ) e. _V
22	21	a1i 11	. . . . 5 |- ( ( ( ph /\ ( r = C /\ s = D ) ) /\ b = B ) -> ( u e. b , v e. b |-> ( ( ( 1st ` u ) ( Hom ` r ) ( 1st ` v ) ) X. ( ( 2nd ` u ) ( Hom ` s ) ( 2nd ` v ) ) ) ) e. _V )
23		simpr 477	. . . . . . 7 |- ( ( ( ph /\ ( r = C /\ s = D ) ) /\ b = B ) -> b = B )
24		simplrl 800	. . . . . . . . . . 11 |- ( ( ( ph /\ ( r = C /\ s = D ) ) /\ b = B ) -> r = C )
25	24	fveq2d 6195	. . . . . . . . . 10 |- ( ( ( ph /\ ( r = C /\ s = D ) ) /\ b = B ) -> ( Hom ` r ) = ( Hom ` C ) )
26		xpcval.h	. . . . . . . . . 10 |- H = ( Hom ` C )
27	25, 26	syl6eqr 2674	. . . . . . . . 9 |- ( ( ( ph /\ ( r = C /\ s = D ) ) /\ b = B ) -> ( Hom ` r ) = H )
28	27	oveqd 6667	. . . . . . . 8 |- ( ( ( ph /\ ( r = C /\ s = D ) ) /\ b = B ) -> ( ( 1st ` u ) ( Hom ` r ) ( 1st ` v ) ) = ( ( 1st ` u ) H ( 1st ` v ) ) )
29		simplrr 801	. . . . . . . . . . 11 |- ( ( ( ph /\ ( r = C /\ s = D ) ) /\ b = B ) -> s = D )
30	29	fveq2d 6195	. . . . . . . . . 10 |- ( ( ( ph /\ ( r = C /\ s = D ) ) /\ b = B ) -> ( Hom ` s ) = ( Hom ` D ) )
31		xpcval.j	. . . . . . . . . 10 |- J = ( Hom ` D )
32	30, 31	syl6eqr 2674	. . . . . . . . 9 |- ( ( ( ph /\ ( r = C /\ s = D ) ) /\ b = B ) -> ( Hom ` s ) = J )
33	32	oveqd 6667	. . . . . . . 8 |- ( ( ( ph /\ ( r = C /\ s = D ) ) /\ b = B ) -> ( ( 2nd ` u ) ( Hom ` s ) ( 2nd ` v ) ) = ( ( 2nd ` u ) J ( 2nd ` v ) ) )
34	28, 33	xpeq12d 5140	. . . . . . 7 |- ( ( ( ph /\ ( r = C /\ s = D ) ) /\ b = B ) -> ( ( ( 1st ` u ) ( Hom ` r ) ( 1st ` v ) ) X. ( ( 2nd ` u ) ( Hom ` s ) ( 2nd ` v ) ) ) = ( ( ( 1st ` u ) H ( 1st ` v ) ) X. ( ( 2nd ` u ) J ( 2nd ` v ) ) ) )
35	23, 23, 34	mpt2eq123dv 6717	. . . . . 6 |- ( ( ( ph /\ ( r = C /\ s = D ) ) /\ b = B ) -> ( u e. b , v e. b |-> ( ( ( 1st ` u ) ( Hom ` r ) ( 1st ` v ) ) X. ( ( 2nd ` u ) ( Hom ` s ) ( 2nd ` v ) ) ) ) = ( u e. B , v e. B |-> ( ( ( 1st ` u ) H ( 1st ` v ) ) X. ( ( 2nd ` u ) J ( 2nd ` v ) ) ) ) )
36		xpcval.k	. . . . . . 7 |- ( ph -> K = ( u e. B , v e. B |-> ( ( ( 1st ` u ) H ( 1st ` v ) ) X. ( ( 2nd ` u ) J ( 2nd ` v ) ) ) ) )
37	36	ad2antrr 762	. . . . . 6 |- ( ( ( ph /\ ( r = C /\ s = D ) ) /\ b = B ) -> K = ( u e. B , v e. B |-> ( ( ( 1st ` u ) H ( 1st ` v ) ) X. ( ( 2nd ` u ) J ( 2nd ` v ) ) ) ) )
38	35, 37	eqtr4d 2659	. . . . 5 |- ( ( ( ph /\ ( r = C /\ s = D ) ) /\ b = B ) -> ( u e. b , v e. b |-> ( ( ( 1st ` u ) ( Hom ` r ) ( 1st ` v ) ) X. ( ( 2nd ` u ) ( Hom ` s ) ( 2nd ` v ) ) ) ) = K )
39		simplr 792	. . . . . . 7 |- ( ( ( ( ph /\ ( r = C /\ s = D ) ) /\ b = B ) /\ h = K ) -> b = B )
40	39	opeq2d 4409	. . . . . 6 |- ( ( ( ( ph /\ ( r = C /\ s = D ) ) /\ b = B ) /\ h = K ) -> <. ( Base ` ndx ) , b >. = <. ( Base ` ndx ) , B >. )
41		simpr 477	. . . . . . 7 |- ( ( ( ( ph /\ ( r = C /\ s = D ) ) /\ b = B ) /\ h = K ) -> h = K )
42	41	opeq2d 4409	. . . . . 6 |- ( ( ( ( ph /\ ( r = C /\ s = D ) ) /\ b = B ) /\ h = K ) -> <. ( Hom ` ndx ) , h >. = <. ( Hom ` ndx ) , K >. )
43	39, 39	xpeq12d 5140	. . . . . . . . 9 |- ( ( ( ( ph /\ ( r = C /\ s = D ) ) /\ b = B ) /\ h = K ) -> ( b X. b ) = ( B X. B ) )
44	41	oveqd 6667	. . . . . . . . . 10 |- ( ( ( ( ph /\ ( r = C /\ s = D ) ) /\ b = B ) /\ h = K ) -> ( ( 2nd ` x ) h y ) = ( ( 2nd ` x ) K y ) )
45	41	fveq1d 6193	. . . . . . . . . 10 |- ( ( ( ( ph /\ ( r = C /\ s = D ) ) /\ b = B ) /\ h = K ) -> ( h ` x ) = ( K ` x ) )
46	24	adantr 481	. . . . . . . . . . . . . . 15 |- ( ( ( ( ph /\ ( r = C /\ s = D ) ) /\ b = B ) /\ h = K ) -> r = C )
47	46	fveq2d 6195	. . . . . . . . . . . . . 14 |- ( ( ( ( ph /\ ( r = C /\ s = D ) ) /\ b = B ) /\ h = K ) -> (comp ` r ) = (comp ` C ) )
48		xpcval.o1	. . . . . . . . . . . . . 14 |- .x. = (comp ` C )
49	47, 48	syl6eqr 2674	. . . . . . . . . . . . 13 |- ( ( ( ( ph /\ ( r = C /\ s = D ) ) /\ b = B ) /\ h = K ) -> (comp ` r ) = .x. )
50	49	oveqd 6667	. . . . . . . . . . . 12 |- ( ( ( ( ph /\ ( r = C /\ s = D ) ) /\ b = B ) /\ h = K ) -> ( <. ( 1st ` ( 1st ` x ) ) , ( 1st ` ( 2nd ` x ) ) >. (comp ` r ) ( 1st ` y ) ) = ( <. ( 1st ` ( 1st ` x ) ) , ( 1st ` ( 2nd ` x ) ) >. .x. ( 1st ` y ) ) )
51	50	oveqd 6667	. . . . . . . . . . 11 |- ( ( ( ( ph /\ ( r = C /\ s = D ) ) /\ b = B ) /\ h = K ) -> ( ( 1st ` g ) ( <. ( 1st ` ( 1st ` x ) ) , ( 1st ` ( 2nd ` x ) ) >. (comp ` r ) ( 1st ` y ) ) ( 1st ` f ) ) = ( ( 1st ` g ) ( <. ( 1st ` ( 1st ` x ) ) , ( 1st ` ( 2nd ` x ) ) >. .x. ( 1st ` y ) ) ( 1st ` f ) ) )
52	29	adantr 481	. . . . . . . . . . . . . . 15 |- ( ( ( ( ph /\ ( r = C /\ s = D ) ) /\ b = B ) /\ h = K ) -> s = D )
53	52	fveq2d 6195	. . . . . . . . . . . . . 14 |- ( ( ( ( ph /\ ( r = C /\ s = D ) ) /\ b = B ) /\ h = K ) -> (comp ` s ) = (comp ` D ) )
54		xpcval.o2	. . . . . . . . . . . . . 14 |- .xb = (comp ` D )
55	53, 54	syl6eqr 2674	. . . . . . . . . . . . 13 |- ( ( ( ( ph /\ ( r = C /\ s = D ) ) /\ b = B ) /\ h = K ) -> (comp ` s ) = .xb )
56	55	oveqd 6667	. . . . . . . . . . . 12 |- ( ( ( ( ph /\ ( r = C /\ s = D ) ) /\ b = B ) /\ h = K ) -> ( <. ( 2nd ` ( 1st ` x ) ) , ( 2nd ` ( 2nd ` x ) ) >. (comp ` s ) ( 2nd ` y ) ) = ( <. ( 2nd ` ( 1st ` x ) ) , ( 2nd ` ( 2nd ` x ) ) >. .xb ( 2nd ` y ) ) )
57	56	oveqd 6667	. . . . . . . . . . 11 |- ( ( ( ( ph /\ ( r = C /\ s = D ) ) /\ b = B ) /\ h = K ) -> ( ( 2nd ` g ) ( <. ( 2nd ` ( 1st ` x ) ) , ( 2nd ` ( 2nd ` x ) ) >. (comp ` s ) ( 2nd ` y ) ) ( 2nd ` f ) ) = ( ( 2nd ` g ) ( <. ( 2nd ` ( 1st ` x ) ) , ( 2nd ` ( 2nd ` x ) ) >. .xb ( 2nd ` y ) ) ( 2nd ` f ) ) )
58	51, 57	opeq12d 4410	. . . . . . . . . 10 |- ( ( ( ( ph /\ ( r = C /\ s = D ) ) /\ b = B ) /\ h = K ) -> <. ( ( 1st ` g ) ( <. ( 1st ` ( 1st ` x ) ) , ( 1st ` ( 2nd ` x ) ) >. (comp ` r ) ( 1st ` y ) ) ( 1st ` f ) ) , ( ( 2nd ` g ) ( <. ( 2nd ` ( 1st ` x ) ) , ( 2nd ` ( 2nd ` x ) ) >. (comp ` s ) ( 2nd ` y ) ) ( 2nd ` f ) ) >. = <. ( ( 1st ` g ) ( <. ( 1st ` ( 1st ` x ) ) , ( 1st ` ( 2nd ` x ) ) >. .x. ( 1st ` y ) ) ( 1st ` f ) ) , ( ( 2nd ` g ) ( <. ( 2nd ` ( 1st ` x ) ) , ( 2nd ` ( 2nd ` x ) ) >. .xb ( 2nd ` y ) ) ( 2nd ` f ) ) >. )
59	44, 45, 58	mpt2eq123dv 6717	. . . . . . . . 9 |- ( ( ( ( ph /\ ( r = C /\ s = D ) ) /\ b = B ) /\ h = K ) -> ( g e. ( ( 2nd ` x ) h y ) , f e. ( h ` x ) |-> <. ( ( 1st ` g ) ( <. ( 1st ` ( 1st ` x ) ) , ( 1st ` ( 2nd ` x ) ) >. (comp ` r ) ( 1st ` y ) ) ( 1st ` f ) ) , ( ( 2nd ` g ) ( <. ( 2nd ` ( 1st ` x ) ) , ( 2nd ` ( 2nd ` x ) ) >. (comp ` s ) ( 2nd ` y ) ) ( 2nd ` f ) ) >. ) = ( g e. ( ( 2nd ` x ) K y ) , f e. ( K ` x ) |-> <. ( ( 1st ` g ) ( <. ( 1st ` ( 1st ` x ) ) , ( 1st ` ( 2nd ` x ) ) >. .x. ( 1st ` y ) ) ( 1st ` f ) ) , ( ( 2nd ` g ) ( <. ( 2nd ` ( 1st ` x ) ) , ( 2nd ` ( 2nd ` x ) ) >. .xb ( 2nd ` y ) ) ( 2nd ` f ) ) >. ) )
60	43, 39, 59	mpt2eq123dv 6717	. . . . . . . 8 |- ( ( ( ( ph /\ ( r = C /\ s = D ) ) /\ b = B ) /\ h = K ) -> ( x e. ( b X. b ) , y e. b |-> ( g e. ( ( 2nd ` x ) h y ) , f e. ( h ` x ) |-> <. ( ( 1st ` g ) ( <. ( 1st ` ( 1st ` x ) ) , ( 1st ` ( 2nd ` x ) ) >. (comp ` r ) ( 1st ` y ) ) ( 1st ` f ) ) , ( ( 2nd ` g ) ( <. ( 2nd ` ( 1st ` x ) ) , ( 2nd ` ( 2nd ` x ) ) >. (comp ` s ) ( 2nd ` y ) ) ( 2nd ` f ) ) >. ) ) = ( x e. ( B X. B ) , y e. B |-> ( g e. ( ( 2nd ` x ) K y ) , f e. ( K ` x ) |-> <. ( ( 1st ` g ) ( <. ( 1st ` ( 1st ` x ) ) , ( 1st ` ( 2nd ` x ) ) >. .x. ( 1st ` y ) ) ( 1st ` f ) ) , ( ( 2nd ` g ) ( <. ( 2nd ` ( 1st ` x ) ) , ( 2nd ` ( 2nd ` x ) ) >. .xb ( 2nd ` y ) ) ( 2nd ` f ) ) >. ) ) )
61		xpcval.o	. . . . . . . . 9 |- ( ph -> O = ( x e. ( B X. B ) , y e. B |-> ( g e. ( ( 2nd ` x ) K y ) , f e. ( K ` x ) |-> <. ( ( 1st ` g ) ( <. ( 1st ` ( 1st ` x ) ) , ( 1st ` ( 2nd ` x ) ) >. .x. ( 1st ` y ) ) ( 1st ` f ) ) , ( ( 2nd ` g ) ( <. ( 2nd ` ( 1st ` x ) ) , ( 2nd ` ( 2nd ` x ) ) >. .xb ( 2nd ` y ) ) ( 2nd ` f ) ) >. ) ) )
62	61	ad3antrrr 766	. . . . . . . 8 |- ( ( ( ( ph /\ ( r = C /\ s = D ) ) /\ b = B ) /\ h = K ) -> O = ( x e. ( B X. B ) , y e. B |-> ( g e. ( ( 2nd ` x ) K y ) , f e. ( K ` x ) |-> <. ( ( 1st ` g ) ( <. ( 1st ` ( 1st ` x ) ) , ( 1st ` ( 2nd ` x ) ) >. .x. ( 1st ` y ) ) ( 1st ` f ) ) , ( ( 2nd ` g ) ( <. ( 2nd ` ( 1st ` x ) ) , ( 2nd ` ( 2nd ` x ) ) >. .xb ( 2nd ` y ) ) ( 2nd ` f ) ) >. ) ) )
63	60, 62	eqtr4d 2659	. . . . . . 7 |- ( ( ( ( ph /\ ( r = C /\ s = D ) ) /\ b = B ) /\ h = K ) -> ( x e. ( b X. b ) , y e. b |-> ( g e. ( ( 2nd ` x ) h y ) , f e. ( h ` x ) |-> <. ( ( 1st ` g ) ( <. ( 1st ` ( 1st ` x ) ) , ( 1st ` ( 2nd ` x ) ) >. (comp ` r ) ( 1st ` y ) ) ( 1st ` f ) ) , ( ( 2nd ` g ) ( <. ( 2nd ` ( 1st ` x ) ) , ( 2nd ` ( 2nd ` x ) ) >. (comp ` s ) ( 2nd ` y ) ) ( 2nd ` f ) ) >. ) ) = O )
64	63	opeq2d 4409	. . . . . 6 |- ( ( ( ( ph /\ ( r = C /\ s = D ) ) /\ b = B ) /\ h = K ) -> <. (comp ` ndx ) , ( x e. ( b X. b ) , y e. b |-> ( g e. ( ( 2nd ` x ) h y ) , f e. ( h ` x ) |-> <. ( ( 1st ` g ) ( <. ( 1st ` ( 1st ` x ) ) , ( 1st ` ( 2nd ` x ) ) >. (comp ` r ) ( 1st ` y ) ) ( 1st ` f ) ) , ( ( 2nd ` g ) ( <. ( 2nd ` ( 1st ` x ) ) , ( 2nd ` ( 2nd ` x ) ) >. (comp ` s ) ( 2nd ` y ) ) ( 2nd ` f ) ) >. ) ) >. = <. (comp ` ndx ) , O >. )
65	40, 42, 64	tpeq123d 4283	. . . . 5 |- ( ( ( ( ph /\ ( r = C /\ s = D ) ) /\ b = B ) /\ h = K ) -> { <. ( Base ` ndx ) , b >. , <. ( Hom ` ndx ) , h >. , <. (comp ` ndx ) , ( x e. ( b X. b ) , y e. b |-> ( g e. ( ( 2nd ` x ) h y ) , f e. ( h ` x ) |-> <. ( ( 1st ` g ) ( <. ( 1st ` ( 1st ` x ) ) , ( 1st ` ( 2nd ` x ) ) >. (comp ` r ) ( 1st ` y ) ) ( 1st ` f ) ) , ( ( 2nd ` g ) ( <. ( 2nd ` ( 1st ` x ) ) , ( 2nd ` ( 2nd ` x ) ) >. (comp ` s ) ( 2nd ` y ) ) ( 2nd ` f ) ) >. ) ) >. } = { <. ( Base ` ndx ) , B >. , <. ( Hom ` ndx ) , K >. , <. (comp ` ndx ) , O >. } )
66	22, 38, 65	csbied2 3561	. . . 4 |- ( ( ( ph /\ ( r = C /\ s = D ) ) /\ b = B ) -> [_ ( u e. b , v e. b |-> ( ( ( 1st ` u ) ( Hom ` r ) ( 1st ` v ) ) X. ( ( 2nd ` u ) ( Hom ` s ) ( 2nd ` v ) ) ) ) / h ]_ { <. ( Base ` ndx ) , b >. , <. ( Hom ` ndx ) , h >. , <. (comp ` ndx ) , ( x e. ( b X. b ) , y e. b |-> ( g e. ( ( 2nd ` x ) h y ) , f e. ( h ` x ) |-> <. ( ( 1st ` g ) ( <. ( 1st ` ( 1st ` x ) ) , ( 1st ` ( 2nd ` x ) ) >. (comp ` r ) ( 1st ` y ) ) ( 1st ` f ) ) , ( ( 2nd ` g ) ( <. ( 2nd ` ( 1st ` x ) ) , ( 2nd ` ( 2nd ` x ) ) >. (comp ` s ) ( 2nd ` y ) ) ( 2nd ` f ) ) >. ) ) >. } = { <. ( Base ` ndx ) , B >. , <. ( Hom ` ndx ) , K >. , <. (comp ` ndx ) , O >. } )
67	7, 19, 66	csbied2 3561	. . 3 |- ( ( ph /\ ( r = C /\ s = D ) ) -> [_ ( ( Base ` r ) X. ( Base ` s ) ) / b ]_ [_ ( u e. b , v e. b |-> ( ( ( 1st ` u ) ( Hom ` r ) ( 1st ` v ) ) X. ( ( 2nd ` u ) ( Hom ` s ) ( 2nd ` v ) ) ) ) / h ]_ { <. ( Base ` ndx ) , b >. , <. ( Hom ` ndx ) , h >. , <. (comp ` ndx ) , ( x e. ( b X. b ) , y e. b |-> ( g e. ( ( 2nd ` x ) h y ) , f e. ( h ` x ) |-> <. ( ( 1st ` g ) ( <. ( 1st ` ( 1st ` x ) ) , ( 1st ` ( 2nd ` x ) ) >. (comp ` r ) ( 1st ` y ) ) ( 1st ` f ) ) , ( ( 2nd ` g ) ( <. ( 2nd ` ( 1st ` x ) ) , ( 2nd ` ( 2nd ` x ) ) >. (comp ` s ) ( 2nd ` y ) ) ( 2nd ` f ) ) >. ) ) >. } = { <. ( Base ` ndx ) , B >. , <. ( Hom ` ndx ) , K >. , <. (comp ` ndx ) , O >. } )
68		xpcval.c	. . . 4 |- ( ph -> C e. V )
69		elex 3212	. . . 4 |- ( C e. V -> C e. _V )
70	68, 69	syl 17	. . 3 |- ( ph -> C e. _V )
71		xpcval.d	. . . 4 |- ( ph -> D e. W )
72		elex 3212	. . . 4 |- ( D e. W -> D e. _V )
73	71, 72	syl 17	. . 3 |- ( ph -> D e. _V )
74		tpex 6957	. . . 4 |- { <. ( Base ` ndx ) , B >. , <. ( Hom ` ndx ) , K >. , <. (comp ` ndx ) , O >. } e. _V
75	74	a1i 11	. . 3 |- ( ph -> { <. ( Base ` ndx ) , B >. , <. ( Hom ` ndx ) , K >. , <. (comp ` ndx ) , O >. } e. _V )
76	3, 67, 70, 73, 75	ovmpt2d 6788	. 2 |- ( ph -> ( C X.c D ) = { <. ( Base ` ndx ) , B >. , <. ( Hom ` ndx ) , K >. , <. (comp ` ndx ) , O >. } )
77	1, 76	syl5eq 2668	1 |- ( ph -> T = { <. ( Base ` ndx ) , B >. , <. ( Hom ` ndx ) , K >. , <. (comp ` ndx ) , O >. } )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   [_csb 3533   {ctp 4181   <.cop 4183   X. cxp 5112   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   1stc1st 7166   2ndc2nd 7167   ndxcnx 15854   Basecbs 15857   Hom chom 15952  compcco 15953   X._c cxpc 16808

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-tp 4182  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-xpc 16812

This theorem is referenced by:  xpcbas  16818  xpchomfval  16819  xpccofval  16822  catcxpccl  16847  xpcpropd  16848