Theorem hofcl 16899

Description: Closure of the Hom functor. Note that the codomain is the category SetCat ` U for any universe U which contains each Hom-set. This corresponds to the assertion that C be locally small (with respect to U). (Contributed by Mario Carneiro, 15-Jan-2017.)

Hypotheses

Ref	Expression
hofcl.m	|- M = (HomF ` C )
hofcl.o	|- O = (oppCat ` C )
hofcl.d	|- D = ( SetCat ` U )
hofcl.c	|- ( ph -> C e. Cat )
hofcl.u	|- ( ph -> U e. V )
hofcl.h	|- ( ph -> ran ( Hom f ` C ) C_ U )

Assertion

Ref	Expression
hofcl	|- ( ph -> M e. ( ( O X.c C ) Func D ) )

Proof of Theorem hofcl

Dummy variables f g x y z h are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		hofcl.m	. . . 4 |- M = (HomF ` C )
2		hofcl.c	. . . 4 |- ( ph -> C e. Cat )
3		eqid 2622	. . . 4 |- ( Base ` C ) = ( Base ` C )
4		eqid 2622	. . . 4 |- ( Hom ` C ) = ( Hom ` C )
5		eqid 2622	. . . 4 |- (comp ` C ) = (comp ` C )
6	1, 2, 3, 4, 5	hofval 16892	. . 3 |- ( ph -> M = <. ( Hom f ` C ) , ( x e. ( ( Base ` C ) X. ( Base ` C ) ) , y e. ( ( Base ` C ) X. ( Base ` C ) ) |-> ( f e. ( ( 1st ` y ) ( Hom ` C ) ( 1st ` x ) ) , g e. ( ( 2nd ` x ) ( Hom ` C ) ( 2nd ` y ) ) |-> ( h e. ( ( Hom ` C ) ` x ) |-> ( ( g ( x (comp ` C ) ( 2nd ` y ) ) h ) ( <. ( 1st ` y ) , ( 1st ` x ) >. (comp ` C ) ( 2nd ` y ) ) f ) ) ) ) >. )
7		fvex 6201	. . . . . 6 |- ( Hom f ` C ) e. _V
8		fvex 6201	. . . . . . . 8 |- ( Base ` C ) e. _V
9	8, 8	xpex 6962	. . . . . . 7 |- ( ( Base ` C ) X. ( Base ` C ) ) e. _V
10	9, 9	mpt2ex 7247	. . . . . 6 |- ( x e. ( ( Base ` C ) X. ( Base ` C ) ) , y e. ( ( Base ` C ) X. ( Base ` C ) ) |-> ( f e. ( ( 1st ` y ) ( Hom ` C ) ( 1st ` x ) ) , g e. ( ( 2nd ` x ) ( Hom ` C ) ( 2nd ` y ) ) |-> ( h e. ( ( Hom ` C ) ` x ) |-> ( ( g ( x (comp ` C ) ( 2nd ` y ) ) h ) ( <. ( 1st ` y ) , ( 1st ` x ) >. (comp ` C ) ( 2nd ` y ) ) f ) ) ) ) e. _V
11	7, 10	op2ndd 7179	. . . . 5 |- ( M = <. ( Hom f ` C ) , ( x e. ( ( Base ` C ) X. ( Base ` C ) ) , y e. ( ( Base ` C ) X. ( Base ` C ) ) |-> ( f e. ( ( 1st ` y ) ( Hom ` C ) ( 1st ` x ) ) , g e. ( ( 2nd ` x ) ( Hom ` C ) ( 2nd ` y ) ) |-> ( h e. ( ( Hom ` C ) ` x ) |-> ( ( g ( x (comp ` C ) ( 2nd ` y ) ) h ) ( <. ( 1st ` y ) , ( 1st ` x ) >. (comp ` C ) ( 2nd ` y ) ) f ) ) ) ) >. -> ( 2nd ` M ) = ( x e. ( ( Base ` C ) X. ( Base ` C ) ) , y e. ( ( Base ` C ) X. ( Base ` C ) ) |-> ( f e. ( ( 1st ` y ) ( Hom ` C ) ( 1st ` x ) ) , g e. ( ( 2nd ` x ) ( Hom ` C ) ( 2nd ` y ) ) |-> ( h e. ( ( Hom ` C ) ` x ) |-> ( ( g ( x (comp ` C ) ( 2nd ` y ) ) h ) ( <. ( 1st ` y ) , ( 1st ` x ) >. (comp ` C ) ( 2nd ` y ) ) f ) ) ) ) )
12	6, 11	syl 17	. . . 4 |- ( ph -> ( 2nd ` M ) = ( x e. ( ( Base ` C ) X. ( Base ` C ) ) , y e. ( ( Base ` C ) X. ( Base ` C ) ) |-> ( f e. ( ( 1st ` y ) ( Hom ` C ) ( 1st ` x ) ) , g e. ( ( 2nd ` x ) ( Hom ` C ) ( 2nd ` y ) ) |-> ( h e. ( ( Hom ` C ) ` x ) |-> ( ( g ( x (comp ` C ) ( 2nd ` y ) ) h ) ( <. ( 1st ` y ) , ( 1st ` x ) >. (comp ` C ) ( 2nd ` y ) ) f ) ) ) ) )
13	12	opeq2d 4409	. . 3 |- ( ph -> <. ( Hom f ` C ) , ( 2nd ` M ) >. = <. ( Hom f ` C ) , ( x e. ( ( Base ` C ) X. ( Base ` C ) ) , y e. ( ( Base ` C ) X. ( Base ` C ) ) |-> ( f e. ( ( 1st ` y ) ( Hom ` C ) ( 1st ` x ) ) , g e. ( ( 2nd ` x ) ( Hom ` C ) ( 2nd ` y ) ) |-> ( h e. ( ( Hom ` C ) ` x ) |-> ( ( g ( x (comp ` C ) ( 2nd ` y ) ) h ) ( <. ( 1st ` y ) , ( 1st ` x ) >. (comp ` C ) ( 2nd ` y ) ) f ) ) ) ) >. )
14	6, 13	eqtr4d 2659	. 2 |- ( ph -> M = <. ( Hom f ` C ) , ( 2nd ` M ) >. )
15		eqid 2622	. . . . 5 |- ( O X.c C ) = ( O X.c C )
16		hofcl.o	. . . . . 6 |- O = (oppCat ` C )
17	16, 3	oppcbas 16378	. . . . 5 |- ( Base ` C ) = ( Base ` O )
18	15, 17, 3	xpcbas 16818	. . . 4 |- ( ( Base ` C ) X. ( Base ` C ) ) = ( Base ` ( O X.c C ) )
19		eqid 2622	. . . 4 |- ( Base ` D ) = ( Base ` D )
20		eqid 2622	. . . 4 |- ( Hom ` ( O X.c C ) ) = ( Hom ` ( O X.c C ) )
21		eqid 2622	. . . 4 |- ( Hom ` D ) = ( Hom ` D )
22		eqid 2622	. . . 4 |- ( Id ` ( O X.c C ) ) = ( Id ` ( O X.c C ) )
23		eqid 2622	. . . 4 |- ( Id ` D ) = ( Id ` D )
24		eqid 2622	. . . 4 |- (comp ` ( O X.c C ) ) = (comp ` ( O X.c C ) )
25		eqid 2622	. . . 4 |- (comp ` D ) = (comp ` D )
26	16	oppccat 16382	. . . . . 6 |- ( C e. Cat -> O e. Cat )
27	2, 26	syl 17	. . . . 5 |- ( ph -> O e. Cat )
28	15, 27, 2	xpccat 16830	. . . 4 |- ( ph -> ( O X.c C ) e. Cat )
29		hofcl.u	. . . . 5 |- ( ph -> U e. V )
30		hofcl.d	. . . . . 6 |- D = ( SetCat ` U )
31	30	setccat 16735	. . . . 5 |- ( U e. V -> D e. Cat )
32	29, 31	syl 17	. . . 4 |- ( ph -> D e. Cat )
33		eqid 2622	. . . . . . . 8 |- ( Hom f ` C ) = ( Hom f ` C )
34	33, 3	homffn 16353	. . . . . . 7 |- ( Hom f ` C ) Fn ( ( Base ` C ) X. ( Base ` C ) )
35	34	a1i 11	. . . . . 6 |- ( ph -> ( Hom f ` C ) Fn ( ( Base ` C ) X. ( Base ` C ) ) )
36		hofcl.h	. . . . . 6 |- ( ph -> ran ( Hom f ` C ) C_ U )
37		df-f 5892	. . . . . 6 |- ( ( Hom f ` C ) : ( ( Base ` C ) X. ( Base ` C ) ) --> U <-> ( ( Hom f ` C ) Fn ( ( Base ` C ) X. ( Base ` C ) ) /\ ran ( Hom f ` C ) C_ U ) )
38	35, 36, 37	sylanbrc 698	. . . . 5 |- ( ph -> ( Hom f ` C ) : ( ( Base ` C ) X. ( Base ` C ) ) --> U )
39	30, 29	setcbas 16728	. . . . . 6 |- ( ph -> U = ( Base ` D ) )
40	39	feq3d 6032	. . . . 5 |- ( ph -> ( ( Hom f ` C ) : ( ( Base ` C ) X. ( Base ` C ) ) --> U <-> ( Hom f ` C ) : ( ( Base ` C ) X. ( Base ` C ) ) --> ( Base ` D ) ) )
41	38, 40	mpbid 222	. . . 4 |- ( ph -> ( Hom f ` C ) : ( ( Base ` C ) X. ( Base ` C ) ) --> ( Base ` D ) )
42		eqid 2622	. . . . . 6 |- ( x e. ( ( Base ` C ) X. ( Base ` C ) ) , y e. ( ( Base ` C ) X. ( Base ` C ) ) |-> ( f e. ( ( 1st ` y ) ( Hom ` C ) ( 1st ` x ) ) , g e. ( ( 2nd ` x ) ( Hom ` C ) ( 2nd ` y ) ) |-> ( h e. ( ( Hom ` C ) ` x ) |-> ( ( g ( x (comp ` C ) ( 2nd ` y ) ) h ) ( <. ( 1st ` y ) , ( 1st ` x ) >. (comp ` C ) ( 2nd ` y ) ) f ) ) ) ) = ( x e. ( ( Base ` C ) X. ( Base ` C ) ) , y e. ( ( Base ` C ) X. ( Base ` C ) ) |-> ( f e. ( ( 1st ` y ) ( Hom ` C ) ( 1st ` x ) ) , g e. ( ( 2nd ` x ) ( Hom ` C ) ( 2nd ` y ) ) |-> ( h e. ( ( Hom ` C ) ` x ) |-> ( ( g ( x (comp ` C ) ( 2nd ` y ) ) h ) ( <. ( 1st ` y ) , ( 1st ` x ) >. (comp ` C ) ( 2nd ` y ) ) f ) ) ) )
43		ovex 6678	. . . . . . 7 |- ( ( 1st ` y ) ( Hom ` C ) ( 1st ` x ) ) e. _V
44		ovex 6678	. . . . . . 7 |- ( ( 2nd ` x ) ( Hom ` C ) ( 2nd ` y ) ) e. _V
45	43, 44	mpt2ex 7247	. . . . . 6 |- ( f e. ( ( 1st ` y ) ( Hom ` C ) ( 1st ` x ) ) , g e. ( ( 2nd ` x ) ( Hom ` C ) ( 2nd ` y ) ) |-> ( h e. ( ( Hom ` C ) ` x ) |-> ( ( g ( x (comp ` C ) ( 2nd ` y ) ) h ) ( <. ( 1st ` y ) , ( 1st ` x ) >. (comp ` C ) ( 2nd ` y ) ) f ) ) ) e. _V
46	42, 45	fnmpt2i 7239	. . . . 5 |- ( x e. ( ( Base ` C ) X. ( Base ` C ) ) , y e. ( ( Base ` C ) X. ( Base ` C ) ) |-> ( f e. ( ( 1st ` y ) ( Hom ` C ) ( 1st ` x ) ) , g e. ( ( 2nd ` x ) ( Hom ` C ) ( 2nd ` y ) ) |-> ( h e. ( ( Hom ` C ) ` x ) |-> ( ( g ( x (comp ` C ) ( 2nd ` y ) ) h ) ( <. ( 1st ` y ) , ( 1st ` x ) >. (comp ` C ) ( 2nd ` y ) ) f ) ) ) ) Fn ( ( ( Base ` C ) X. ( Base ` C ) ) X. ( ( Base ` C ) X. ( Base ` C ) ) )
47	12	fneq1d 5981	. . . . 5 |- ( ph -> ( ( 2nd ` M ) Fn ( ( ( Base ` C ) X. ( Base ` C ) ) X. ( ( Base ` C ) X. ( Base ` C ) ) ) <-> ( x e. ( ( Base ` C ) X. ( Base ` C ) ) , y e. ( ( Base ` C ) X. ( Base ` C ) ) |-> ( f e. ( ( 1st ` y ) ( Hom ` C ) ( 1st ` x ) ) , g e. ( ( 2nd ` x ) ( Hom ` C ) ( 2nd ` y ) ) |-> ( h e. ( ( Hom ` C ) ` x ) |-> ( ( g ( x (comp ` C ) ( 2nd ` y ) ) h ) ( <. ( 1st ` y ) , ( 1st ` x ) >. (comp ` C ) ( 2nd ` y ) ) f ) ) ) ) Fn ( ( ( Base ` C ) X. ( Base ` C ) ) X. ( ( Base ` C ) X. ( Base ` C ) ) ) ) )
48	46, 47	mpbiri 248	. . . 4 |- ( ph -> ( 2nd ` M ) Fn ( ( ( Base ` C ) X. ( Base ` C ) ) X. ( ( Base ` C ) X. ( Base ` C ) ) ) )
49	2	ad3antrrr 766	. . . . . . . . . . . 12 |- ( ( ( ( ph /\ ( x e. ( ( Base ` C ) X. ( Base ` C ) ) /\ y e. ( ( Base ` C ) X. ( Base ` C ) ) ) ) /\ ( f e. ( ( 1st ` y ) ( Hom ` C ) ( 1st ` x ) ) /\ g e. ( ( 2nd ` x ) ( Hom ` C ) ( 2nd ` y ) ) ) ) /\ h e. ( ( Hom ` C ) ` x ) ) -> C e. Cat )
50		simplrr 801	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ ( x e. ( ( Base ` C ) X. ( Base ` C ) ) /\ y e. ( ( Base ` C ) X. ( Base ` C ) ) ) ) /\ ( f e. ( ( 1st ` y ) ( Hom ` C ) ( 1st ` x ) ) /\ g e. ( ( 2nd ` x ) ( Hom ` C ) ( 2nd ` y ) ) ) ) -> y e. ( ( Base ` C ) X. ( Base ` C ) ) )
51		xp1st 7198	. . . . . . . . . . . . . 14 |- ( y e. ( ( Base ` C ) X. ( Base ` C ) ) -> ( 1st ` y ) e. ( Base ` C ) )
52	50, 51	syl 17	. . . . . . . . . . . . 13 |- ( ( ( ph /\ ( x e. ( ( Base ` C ) X. ( Base ` C ) ) /\ y e. ( ( Base ` C ) X. ( Base ` C ) ) ) ) /\ ( f e. ( ( 1st ` y ) ( Hom ` C ) ( 1st ` x ) ) /\ g e. ( ( 2nd ` x ) ( Hom ` C ) ( 2nd ` y ) ) ) ) -> ( 1st ` y ) e. ( Base ` C ) )
53	52	adantr 481	. . . . . . . . . . . 12 |- ( ( ( ( ph /\ ( x e. ( ( Base ` C ) X. ( Base ` C ) ) /\ y e. ( ( Base ` C ) X. ( Base ` C ) ) ) ) /\ ( f e. ( ( 1st ` y ) ( Hom ` C ) ( 1st ` x ) ) /\ g e. ( ( 2nd ` x ) ( Hom ` C ) ( 2nd ` y ) ) ) ) /\ h e. ( ( Hom ` C ) ` x ) ) -> ( 1st ` y ) e. ( Base ` C ) )
54		simplrl 800	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ ( x e. ( ( Base ` C ) X. ( Base ` C ) ) /\ y e. ( ( Base ` C ) X. ( Base ` C ) ) ) ) /\ ( f e. ( ( 1st ` y ) ( Hom ` C ) ( 1st ` x ) ) /\ g e. ( ( 2nd ` x ) ( Hom ` C ) ( 2nd ` y ) ) ) ) -> x e. ( ( Base ` C ) X. ( Base ` C ) ) )
55		xp1st 7198	. . . . . . . . . . . . . 14 |- ( x e. ( ( Base ` C ) X. ( Base ` C ) ) -> ( 1st ` x ) e. ( Base ` C ) )
56	54, 55	syl 17	. . . . . . . . . . . . 13 |- ( ( ( ph /\ ( x e. ( ( Base ` C ) X. ( Base ` C ) ) /\ y e. ( ( Base ` C ) X. ( Base ` C ) ) ) ) /\ ( f e. ( ( 1st ` y ) ( Hom ` C ) ( 1st ` x ) ) /\ g e. ( ( 2nd ` x ) ( Hom ` C ) ( 2nd ` y ) ) ) ) -> ( 1st ` x ) e. ( Base ` C ) )
57	56	adantr 481	. . . . . . . . . . . 12 |- ( ( ( ( ph /\ ( x e. ( ( Base ` C ) X. ( Base ` C ) ) /\ y e. ( ( Base ` C ) X. ( Base ` C ) ) ) ) /\ ( f e. ( ( 1st ` y ) ( Hom ` C ) ( 1st ` x ) ) /\ g e. ( ( 2nd ` x ) ( Hom ` C ) ( 2nd ` y ) ) ) ) /\ h e. ( ( Hom ` C ) ` x ) ) -> ( 1st ` x ) e. ( Base ` C ) )
58		xp2nd 7199	. . . . . . . . . . . . . 14 |- ( y e. ( ( Base ` C ) X. ( Base ` C ) ) -> ( 2nd ` y ) e. ( Base ` C ) )
59	50, 58	syl 17	. . . . . . . . . . . . 13 |- ( ( ( ph /\ ( x e. ( ( Base ` C ) X. ( Base ` C ) ) /\ y e. ( ( Base ` C ) X. ( Base ` C ) ) ) ) /\ ( f e. ( ( 1st ` y ) ( Hom ` C ) ( 1st ` x ) ) /\ g e. ( ( 2nd ` x ) ( Hom ` C ) ( 2nd ` y ) ) ) ) -> ( 2nd ` y ) e. ( Base ` C ) )
60	59	adantr 481	. . . . . . . . . . . 12 |- ( ( ( ( ph /\ ( x e. ( ( Base ` C ) X. ( Base ` C ) ) /\ y e. ( ( Base ` C ) X. ( Base ` C ) ) ) ) /\ ( f e. ( ( 1st ` y ) ( Hom ` C ) ( 1st ` x ) ) /\ g e. ( ( 2nd ` x ) ( Hom ` C ) ( 2nd ` y ) ) ) ) /\ h e. ( ( Hom ` C ) ` x ) ) -> ( 2nd ` y ) e. ( Base ` C ) )
61		simplrl 800	. . . . . . . . . . . 12 |- ( ( ( ( ph /\ ( x e. ( ( Base ` C ) X. ( Base ` C ) ) /\ y e. ( ( Base ` C ) X. ( Base ` C ) ) ) ) /\ ( f e. ( ( 1st ` y ) ( Hom ` C ) ( 1st ` x ) ) /\ g e. ( ( 2nd ` x ) ( Hom ` C ) ( 2nd ` y ) ) ) ) /\ h e. ( ( Hom ` C ) ` x ) ) -> f e. ( ( 1st ` y ) ( Hom ` C ) ( 1st ` x ) ) )
62		1st2nd2 7205	. . . . . . . . . . . . . . . . 17 |- ( x e. ( ( Base ` C ) X. ( Base ` C ) ) -> x = <. ( 1st ` x ) , ( 2nd ` x ) >. )
63	54, 62	syl 17	. . . . . . . . . . . . . . . 16 |- ( ( ( ph /\ ( x e. ( ( Base ` C ) X. ( Base ` C ) ) /\ y e. ( ( Base ` C ) X. ( Base ` C ) ) ) ) /\ ( f e. ( ( 1st ` y ) ( Hom ` C ) ( 1st ` x ) ) /\ g e. ( ( 2nd ` x ) ( Hom ` C ) ( 2nd ` y ) ) ) ) -> x = <. ( 1st ` x ) , ( 2nd ` x ) >. )
64	63	adantr 481	. . . . . . . . . . . . . . 15 |- ( ( ( ( ph /\ ( x e. ( ( Base ` C ) X. ( Base ` C ) ) /\ y e. ( ( Base ` C ) X. ( Base ` C ) ) ) ) /\ ( f e. ( ( 1st ` y ) ( Hom ` C ) ( 1st ` x ) ) /\ g e. ( ( 2nd ` x ) ( Hom ` C ) ( 2nd ` y ) ) ) ) /\ h e. ( ( Hom ` C ) ` x ) ) -> x = <. ( 1st ` x ) , ( 2nd ` x ) >. )
65	64	oveq1d 6665	. . . . . . . . . . . . . 14 |- ( ( ( ( ph /\ ( x e. ( ( Base ` C ) X. ( Base ` C ) ) /\ y e. ( ( Base ` C ) X. ( Base ` C ) ) ) ) /\ ( f e. ( ( 1st ` y ) ( Hom ` C ) ( 1st ` x ) ) /\ g e. ( ( 2nd ` x ) ( Hom ` C ) ( 2nd ` y ) ) ) ) /\ h e. ( ( Hom ` C ) ` x ) ) -> ( x (comp ` C ) ( 2nd ` y ) ) = ( <. ( 1st ` x ) , ( 2nd ` x ) >. (comp ` C ) ( 2nd ` y ) ) )
66	65	oveqd 6667	. . . . . . . . . . . . 13 |- ( ( ( ( ph /\ ( x e. ( ( Base ` C ) X. ( Base ` C ) ) /\ y e. ( ( Base ` C ) X. ( Base ` C ) ) ) ) /\ ( f e. ( ( 1st ` y ) ( Hom ` C ) ( 1st ` x ) ) /\ g e. ( ( 2nd ` x ) ( Hom ` C ) ( 2nd ` y ) ) ) ) /\ h e. ( ( Hom ` C ) ` x ) ) -> ( g ( x (comp ` C ) ( 2nd ` y ) ) h ) = ( g ( <. ( 1st ` x ) , ( 2nd ` x ) >. (comp ` C ) ( 2nd ` y ) ) h ) )
67		xp2nd 7199	. . . . . . . . . . . . . . . 16 |- ( x e. ( ( Base ` C ) X. ( Base ` C ) ) -> ( 2nd ` x ) e. ( Base ` C ) )
68	54, 67	syl 17	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ ( x e. ( ( Base ` C ) X. ( Base ` C ) ) /\ y e. ( ( Base ` C ) X. ( Base ` C ) ) ) ) /\ ( f e. ( ( 1st ` y ) ( Hom ` C ) ( 1st ` x ) ) /\ g e. ( ( 2nd ` x ) ( Hom ` C ) ( 2nd ` y ) ) ) ) -> ( 2nd ` x ) e. ( Base ` C ) )
69	68	adantr 481	. . . . . . . . . . . . . 14 |- ( ( ( ( ph /\ ( x e. ( ( Base ` C ) X. ( Base ` C ) ) /\ y e. ( ( Base ` C ) X. ( Base ` C ) ) ) ) /\ ( f e. ( ( 1st ` y ) ( Hom ` C ) ( 1st ` x ) ) /\ g e. ( ( 2nd ` x ) ( Hom ` C ) ( 2nd ` y ) ) ) ) /\ h e. ( ( Hom ` C ) ` x ) ) -> ( 2nd ` x ) e. ( Base ` C ) )
70	63	fveq2d 6195	. . . . . . . . . . . . . . . . 17 |- ( ( ( ph /\ ( x e. ( ( Base ` C ) X. ( Base ` C ) ) /\ y e. ( ( Base ` C ) X. ( Base ` C ) ) ) ) /\ ( f e. ( ( 1st ` y ) ( Hom ` C ) ( 1st ` x ) ) /\ g e. ( ( 2nd ` x ) ( Hom ` C ) ( 2nd ` y ) ) ) ) -> ( ( Hom ` C ) ` x ) = ( ( Hom ` C ) ` <. ( 1st ` x ) , ( 2nd ` x ) >. ) )
71		df-ov 6653	. . . . . . . . . . . . . . . . 17 |- ( ( 1st ` x ) ( Hom ` C ) ( 2nd ` x ) ) = ( ( Hom ` C ) ` <. ( 1st ` x ) , ( 2nd ` x ) >. )
72	70, 71	syl6eqr 2674	. . . . . . . . . . . . . . . 16 |- ( ( ( ph /\ ( x e. ( ( Base ` C ) X. ( Base ` C ) ) /\ y e. ( ( Base ` C ) X. ( Base ` C ) ) ) ) /\ ( f e. ( ( 1st ` y ) ( Hom ` C ) ( 1st ` x ) ) /\ g e. ( ( 2nd ` x ) ( Hom ` C ) ( 2nd ` y ) ) ) ) -> ( ( Hom ` C ) ` x ) = ( ( 1st ` x ) ( Hom ` C ) ( 2nd ` x ) ) )
73	72	eleq2d 2687	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ ( x e. ( ( Base ` C ) X. ( Base ` C ) ) /\ y e. ( ( Base ` C ) X. ( Base ` C ) ) ) ) /\ ( f e. ( ( 1st ` y ) ( Hom ` C ) ( 1st ` x ) ) /\ g e. ( ( 2nd ` x ) ( Hom ` C ) ( 2nd ` y ) ) ) ) -> ( h e. ( ( Hom ` C ) ` x ) <-> h e. ( ( 1st ` x ) ( Hom ` C ) ( 2nd ` x ) ) ) )
74	73	biimpa 501	. . . . . . . . . . . . . 14 |- ( ( ( ( ph /\ ( x e. ( ( Base ` C ) X. ( Base ` C ) ) /\ y e. ( ( Base ` C ) X. ( Base ` C ) ) ) ) /\ ( f e. ( ( 1st ` y ) ( Hom ` C ) ( 1st ` x ) ) /\ g e. ( ( 2nd ` x ) ( Hom ` C ) ( 2nd ` y ) ) ) ) /\ h e. ( ( Hom ` C ) ` x ) ) -> h e. ( ( 1st ` x ) ( Hom ` C ) ( 2nd ` x ) ) )
75		simplrr 801	. . . . . . . . . . . . . 14 |- ( ( ( ( ph /\ ( x e. ( ( Base ` C ) X. ( Base ` C ) ) /\ y e. ( ( Base ` C ) X. ( Base ` C ) ) ) ) /\ ( f e. ( ( 1st ` y ) ( Hom ` C ) ( 1st ` x ) ) /\ g e. ( ( 2nd ` x ) ( Hom ` C ) ( 2nd ` y ) ) ) ) /\ h e. ( ( Hom ` C ) ` x ) ) -> g e. ( ( 2nd ` x ) ( Hom ` C ) ( 2nd ` y ) ) )
76	3, 4, 5, 49, 57, 69, 60, 74, 75	catcocl 16346	. . . . . . . . . . . . 13 |- ( ( ( ( ph /\ ( x e. ( ( Base ` C ) X. ( Base ` C ) ) /\ y e. ( ( Base ` C ) X. ( Base ` C ) ) ) ) /\ ( f e. ( ( 1st ` y ) ( Hom ` C ) ( 1st ` x ) ) /\ g e. ( ( 2nd ` x ) ( Hom ` C ) ( 2nd ` y ) ) ) ) /\ h e. ( ( Hom ` C ) ` x ) ) -> ( g ( <. ( 1st ` x ) , ( 2nd ` x ) >. (comp ` C ) ( 2nd ` y ) ) h ) e. ( ( 1st ` x ) ( Hom ` C ) ( 2nd ` y ) ) )
77	66, 76	eqeltrd 2701	. . . . . . . . . . . 12 |- ( ( ( ( ph /\ ( x e. ( ( Base ` C ) X. ( Base ` C ) ) /\ y e. ( ( Base ` C ) X. ( Base ` C ) ) ) ) /\ ( f e. ( ( 1st ` y ) ( Hom ` C ) ( 1st ` x ) ) /\ g e. ( ( 2nd ` x ) ( Hom ` C ) ( 2nd ` y ) ) ) ) /\ h e. ( ( Hom ` C ) ` x ) ) -> ( g ( x (comp ` C ) ( 2nd ` y ) ) h ) e. ( ( 1st ` x ) ( Hom ` C ) ( 2nd ` y ) ) )
78	3, 4, 5, 49, 53, 57, 60, 61, 77	catcocl 16346	. . . . . . . . . . 11 |- ( ( ( ( ph /\ ( x e. ( ( Base ` C ) X. ( Base ` C ) ) /\ y e. ( ( Base ` C ) X. ( Base ` C ) ) ) ) /\ ( f e. ( ( 1st ` y ) ( Hom ` C ) ( 1st ` x ) ) /\ g e. ( ( 2nd ` x ) ( Hom ` C ) ( 2nd ` y ) ) ) ) /\ h e. ( ( Hom ` C ) ` x ) ) -> ( ( g ( x (comp ` C ) ( 2nd ` y ) ) h ) ( <. ( 1st ` y ) , ( 1st ` x ) >. (comp ` C ) ( 2nd ` y ) ) f ) e. ( ( 1st ` y ) ( Hom ` C ) ( 2nd ` y ) ) )
79		1st2nd2 7205	. . . . . . . . . . . . . . 15 |- ( y e. ( ( Base ` C ) X. ( Base ` C ) ) -> y = <. ( 1st ` y ) , ( 2nd ` y ) >. )
80	50, 79	syl 17	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ ( x e. ( ( Base ` C ) X. ( Base ` C ) ) /\ y e. ( ( Base ` C ) X. ( Base ` C ) ) ) ) /\ ( f e. ( ( 1st ` y ) ( Hom ` C ) ( 1st ` x ) ) /\ g e. ( ( 2nd ` x ) ( Hom ` C ) ( 2nd ` y ) ) ) ) -> y = <. ( 1st ` y ) , ( 2nd ` y ) >. )
81	80	fveq2d 6195	. . . . . . . . . . . . 13 |- ( ( ( ph /\ ( x e. ( ( Base ` C ) X. ( Base ` C ) ) /\ y e. ( ( Base ` C ) X. ( Base ` C ) ) ) ) /\ ( f e. ( ( 1st ` y ) ( Hom ` C ) ( 1st ` x ) ) /\ g e. ( ( 2nd ` x ) ( Hom ` C ) ( 2nd ` y ) ) ) ) -> ( ( Hom ` C ) ` y ) = ( ( Hom ` C ) ` <. ( 1st ` y ) , ( 2nd ` y ) >. ) )
82		df-ov 6653	. . . . . . . . . . . . 13 |- ( ( 1st ` y ) ( Hom ` C ) ( 2nd ` y ) ) = ( ( Hom ` C ) ` <. ( 1st ` y ) , ( 2nd ` y ) >. )
83	81, 82	syl6eqr 2674	. . . . . . . . . . . 12 |- ( ( ( ph /\ ( x e. ( ( Base ` C ) X. ( Base ` C ) ) /\ y e. ( ( Base ` C ) X. ( Base ` C ) ) ) ) /\ ( f e. ( ( 1st ` y ) ( Hom ` C ) ( 1st ` x ) ) /\ g e. ( ( 2nd ` x ) ( Hom ` C ) ( 2nd ` y ) ) ) ) -> ( ( Hom ` C ) ` y ) = ( ( 1st ` y ) ( Hom ` C ) ( 2nd ` y ) ) )
84	83	adantr 481	. . . . . . . . . . 11 |- ( ( ( ( ph /\ ( x e. ( ( Base ` C ) X. ( Base ` C ) ) /\ y e. ( ( Base ` C ) X. ( Base ` C ) ) ) ) /\ ( f e. ( ( 1st ` y ) ( Hom ` C ) ( 1st ` x ) ) /\ g e. ( ( 2nd ` x ) ( Hom ` C ) ( 2nd ` y ) ) ) ) /\ h e. ( ( Hom ` C ) ` x ) ) -> ( ( Hom ` C ) ` y ) = ( ( 1st ` y ) ( Hom ` C ) ( 2nd ` y ) ) )
85	78, 84	eleqtrrd 2704	. . . . . . . . . 10 |- ( ( ( ( ph /\ ( x e. ( ( Base ` C ) X. ( Base ` C ) ) /\ y e. ( ( Base ` C ) X. ( Base ` C ) ) ) ) /\ ( f e. ( ( 1st ` y ) ( Hom ` C ) ( 1st ` x ) ) /\ g e. ( ( 2nd ` x ) ( Hom ` C ) ( 2nd ` y ) ) ) ) /\ h e. ( ( Hom ` C ) ` x ) ) -> ( ( g ( x (comp ` C ) ( 2nd ` y ) ) h ) ( <. ( 1st ` y ) , ( 1st ` x ) >. (comp ` C ) ( 2nd ` y ) ) f ) e. ( ( Hom ` C ) ` y ) )
86		eqid 2622	. . . . . . . . . 10 |- ( h e. ( ( Hom ` C ) ` x ) |-> ( ( g ( x (comp ` C ) ( 2nd ` y ) ) h ) ( <. ( 1st ` y ) , ( 1st ` x ) >. (comp ` C ) ( 2nd ` y ) ) f ) ) = ( h e. ( ( Hom ` C ) ` x ) |-> ( ( g ( x (comp ` C ) ( 2nd ` y ) ) h ) ( <. ( 1st ` y ) , ( 1st ` x ) >. (comp ` C ) ( 2nd ` y ) ) f ) )
87	85, 86	fmptd 6385	. . . . . . . . 9 |- ( ( ( ph /\ ( x e. ( ( Base ` C ) X. ( Base ` C ) ) /\ y e. ( ( Base ` C ) X. ( Base ` C ) ) ) ) /\ ( f e. ( ( 1st ` y ) ( Hom ` C ) ( 1st ` x ) ) /\ g e. ( ( 2nd ` x ) ( Hom ` C ) ( 2nd ` y ) ) ) ) -> ( h e. ( ( Hom ` C ) ` x ) |-> ( ( g ( x (comp ` C ) ( 2nd ` y ) ) h ) ( <. ( 1st ` y ) , ( 1st ` x ) >. (comp ` C ) ( 2nd ` y ) ) f ) ) : ( ( Hom ` C ) ` x ) --> ( ( Hom ` C ) ` y ) )
88	29	ad2antrr 762	. . . . . . . . . 10 |- ( ( ( ph /\ ( x e. ( ( Base ` C ) X. ( Base ` C ) ) /\ y e. ( ( Base ` C ) X. ( Base ` C ) ) ) ) /\ ( f e. ( ( 1st ` y ) ( Hom ` C ) ( 1st ` x ) ) /\ g e. ( ( 2nd ` x ) ( Hom ` C ) ( 2nd ` y ) ) ) ) -> U e. V )
89	33, 3, 4, 56, 68	homfval 16352	. . . . . . . . . . . 12 |- ( ( ( ph /\ ( x e. ( ( Base ` C ) X. ( Base ` C ) ) /\ y e. ( ( Base ` C ) X. ( Base ` C ) ) ) ) /\ ( f e. ( ( 1st ` y ) ( Hom ` C ) ( 1st ` x ) ) /\ g e. ( ( 2nd ` x ) ( Hom ` C ) ( 2nd ` y ) ) ) ) -> ( ( 1st ` x ) ( Hom f ` C ) ( 2nd ` x ) ) = ( ( 1st ` x ) ( Hom ` C ) ( 2nd ` x ) ) )
90	63	fveq2d 6195	. . . . . . . . . . . . 13 |- ( ( ( ph /\ ( x e. ( ( Base ` C ) X. ( Base ` C ) ) /\ y e. ( ( Base ` C ) X. ( Base ` C ) ) ) ) /\ ( f e. ( ( 1st ` y ) ( Hom ` C ) ( 1st ` x ) ) /\ g e. ( ( 2nd ` x ) ( Hom ` C ) ( 2nd ` y ) ) ) ) -> ( ( Hom f ` C ) ` x ) = ( ( Hom f ` C ) ` <. ( 1st ` x ) , ( 2nd ` x ) >. ) )
91		df-ov 6653	. . . . . . . . . . . . 13 |- ( ( 1st ` x ) ( Hom f ` C ) ( 2nd ` x ) ) = ( ( Hom f ` C ) ` <. ( 1st ` x ) , ( 2nd ` x ) >. )
92	90, 91	syl6eqr 2674	. . . . . . . . . . . 12 |- ( ( ( ph /\ ( x e. ( ( Base ` C ) X. ( Base ` C ) ) /\ y e. ( ( Base ` C ) X. ( Base ` C ) ) ) ) /\ ( f e. ( ( 1st ` y ) ( Hom ` C ) ( 1st ` x ) ) /\ g e. ( ( 2nd ` x ) ( Hom ` C ) ( 2nd ` y ) ) ) ) -> ( ( Hom f ` C ) ` x ) = ( ( 1st ` x ) ( Hom f ` C ) ( 2nd ` x ) ) )
93	89, 92, 72	3eqtr4d 2666	. . . . . . . . . . 11 |- ( ( ( ph /\ ( x e. ( ( Base ` C ) X. ( Base ` C ) ) /\ y e. ( ( Base ` C ) X. ( Base ` C ) ) ) ) /\ ( f e. ( ( 1st ` y ) ( Hom ` C ) ( 1st ` x ) ) /\ g e. ( ( 2nd ` x ) ( Hom ` C ) ( 2nd ` y ) ) ) ) -> ( ( Hom f ` C ) ` x ) = ( ( Hom ` C ) ` x ) )
94	38	ad2antrr 762	. . . . . . . . . . . 12 |- ( ( ( ph /\ ( x e. ( ( Base ` C ) X. ( Base ` C ) ) /\ y e. ( ( Base ` C ) X. ( Base ` C ) ) ) ) /\ ( f e. ( ( 1st ` y ) ( Hom ` C ) ( 1st ` x ) ) /\ g e. ( ( 2nd ` x ) ( Hom ` C ) ( 2nd ` y ) ) ) ) -> ( Hom f ` C ) : ( ( Base ` C ) X. ( Base ` C ) ) --> U )
95	94, 54	ffvelrnd 6360	. . . . . . . . . . 11 |- ( ( ( ph /\ ( x e. ( ( Base ` C ) X. ( Base ` C ) ) /\ y e. ( ( Base ` C ) X. ( Base ` C ) ) ) ) /\ ( f e. ( ( 1st ` y ) ( Hom ` C ) ( 1st ` x ) ) /\ g e. ( ( 2nd ` x ) ( Hom ` C ) ( 2nd ` y ) ) ) ) -> ( ( Hom f ` C ) ` x ) e. U )
96	93, 95	eqeltrrd 2702	. . . . . . . . . 10 |- ( ( ( ph /\ ( x e. ( ( Base ` C ) X. ( Base ` C ) ) /\ y e. ( ( Base ` C ) X. ( Base ` C ) ) ) ) /\ ( f e. ( ( 1st ` y ) ( Hom ` C ) ( 1st ` x ) ) /\ g e. ( ( 2nd ` x ) ( Hom ` C ) ( 2nd ` y ) ) ) ) -> ( ( Hom ` C ) ` x ) e. U )
97	33, 3, 4, 52, 59	homfval 16352	. . . . . . . . . . . 12 |- ( ( ( ph /\ ( x e. ( ( Base ` C ) X. ( Base ` C ) ) /\ y e. ( ( Base ` C ) X. ( Base ` C ) ) ) ) /\ ( f e. ( ( 1st ` y ) ( Hom ` C ) ( 1st ` x ) ) /\ g e. ( ( 2nd ` x ) ( Hom ` C ) ( 2nd ` y ) ) ) ) -> ( ( 1st ` y ) ( Hom f ` C ) ( 2nd ` y ) ) = ( ( 1st ` y ) ( Hom ` C ) ( 2nd ` y ) ) )
98	80	fveq2d 6195	. . . . . . . . . . . . 13 |- ( ( ( ph /\ ( x e. ( ( Base ` C ) X. ( Base ` C ) ) /\ y e. ( ( Base ` C ) X. ( Base ` C ) ) ) ) /\ ( f e. ( ( 1st ` y ) ( Hom ` C ) ( 1st ` x ) ) /\ g e. ( ( 2nd ` x ) ( Hom ` C ) ( 2nd ` y ) ) ) ) -> ( ( Hom f ` C ) ` y ) = ( ( Hom f ` C ) ` <. ( 1st ` y ) , ( 2nd ` y ) >. ) )
99		df-ov 6653	. . . . . . . . . . . . 13 |- ( ( 1st ` y ) ( Hom f ` C ) ( 2nd ` y ) ) = ( ( Hom f ` C ) ` <. ( 1st ` y ) , ( 2nd ` y ) >. )
100	98, 99	syl6eqr 2674	. . . . . . . . . . . 12 |- ( ( ( ph /\ ( x e. ( ( Base ` C ) X. ( Base ` C ) ) /\ y e. ( ( Base ` C ) X. ( Base ` C ) ) ) ) /\ ( f e. ( ( 1st ` y ) ( Hom ` C ) ( 1st ` x ) ) /\ g e. ( ( 2nd ` x ) ( Hom ` C ) ( 2nd ` y ) ) ) ) -> ( ( Hom f ` C ) ` y ) = ( ( 1st ` y ) ( Hom f ` C ) ( 2nd ` y ) ) )
101	97, 100, 83	3eqtr4d 2666	. . . . . . . . . . 11 |- ( ( ( ph /\ ( x e. ( ( Base ` C ) X. ( Base ` C ) ) /\ y e. ( ( Base ` C ) X. ( Base ` C ) ) ) ) /\ ( f e. ( ( 1st ` y ) ( Hom ` C ) ( 1st ` x ) ) /\ g e. ( ( 2nd ` x ) ( Hom ` C ) ( 2nd ` y ) ) ) ) -> ( ( Hom f ` C ) ` y ) = ( ( Hom ` C ) ` y ) )
102	94, 50	ffvelrnd 6360	. . . . . . . . . . 11 |- ( ( ( ph /\ ( x e. ( ( Base ` C ) X. ( Base ` C ) ) /\ y e. ( ( Base ` C ) X. ( Base ` C ) ) ) ) /\ ( f e. ( ( 1st ` y ) ( Hom ` C ) ( 1st ` x ) ) /\ g e. ( ( 2nd ` x ) ( Hom ` C ) ( 2nd ` y ) ) ) ) -> ( ( Hom f ` C ) ` y ) e. U )
103	101, 102	eqeltrrd 2702	. . . . . . . . . 10 |- ( ( ( ph /\ ( x e. ( ( Base ` C ) X. ( Base ` C ) ) /\ y e. ( ( Base ` C ) X. ( Base ` C ) ) ) ) /\ ( f e. ( ( 1st ` y ) ( Hom ` C ) ( 1st ` x ) ) /\ g e. ( ( 2nd ` x ) ( Hom ` C ) ( 2nd ` y ) ) ) ) -> ( ( Hom ` C ) ` y ) e. U )
104	30, 88, 21, 96, 103	elsetchom 16731	. . . . . . . . 9 |- ( ( ( ph /\ ( x e. ( ( Base ` C ) X. ( Base ` C ) ) /\ y e. ( ( Base ` C ) X. ( Base ` C ) ) ) ) /\ ( f e. ( ( 1st ` y ) ( Hom ` C ) ( 1st ` x ) ) /\ g e. ( ( 2nd ` x ) ( Hom ` C ) ( 2nd ` y ) ) ) ) -> ( ( h e. ( ( Hom ` C ) ` x ) |-> ( ( g ( x (comp ` C ) ( 2nd ` y ) ) h ) ( <. ( 1st ` y ) , ( 1st ` x ) >. (comp ` C ) ( 2nd ` y ) ) f ) ) e. ( ( ( Hom ` C ) ` x ) ( Hom ` D ) ( ( Hom ` C ) ` y ) ) <-> ( h e. ( ( Hom ` C ) ` x ) |-> ( ( g ( x (comp ` C ) ( 2nd ` y ) ) h ) ( <. ( 1st ` y ) , ( 1st ` x ) >. (comp ` C ) ( 2nd ` y ) ) f ) ) : ( ( Hom ` C ) ` x ) --> ( ( Hom ` C ) ` y ) ) )
105	87, 104	mpbird 247	. . . . . . . 8 |- ( ( ( ph /\ ( x e. ( ( Base ` C ) X. ( Base ` C ) ) /\ y e. ( ( Base ` C ) X. ( Base ` C ) ) ) ) /\ ( f e. ( ( 1st ` y ) ( Hom ` C ) ( 1st ` x ) ) /\ g e. ( ( 2nd ` x ) ( Hom ` C ) ( 2nd ` y ) ) ) ) -> ( h e. ( ( Hom ` C ) ` x ) |-> ( ( g ( x (comp ` C ) ( 2nd ` y ) ) h ) ( <. ( 1st ` y ) , ( 1st ` x ) >. (comp ` C ) ( 2nd ` y ) ) f ) ) e. ( ( ( Hom ` C ) ` x ) ( Hom ` D ) ( ( Hom ` C ) ` y ) ) )
106	93, 101	oveq12d 6668	. . . . . . . 8 |- ( ( ( ph /\ ( x e. ( ( Base ` C ) X. ( Base ` C ) ) /\ y e. ( ( Base ` C ) X. ( Base ` C ) ) ) ) /\ ( f e. ( ( 1st ` y ) ( Hom ` C ) ( 1st ` x ) ) /\ g e. ( ( 2nd ` x ) ( Hom ` C ) ( 2nd ` y ) ) ) ) -> ( ( ( Hom f ` C ) ` x ) ( Hom ` D ) ( ( Hom f ` C ) ` y ) ) = ( ( ( Hom ` C ) ` x ) ( Hom ` D ) ( ( Hom ` C ) ` y ) ) )
107	105, 106	eleqtrrd 2704	. . . . . . 7 |- ( ( ( ph /\ ( x e. ( ( Base ` C ) X. ( Base ` C ) ) /\ y e. ( ( Base ` C ) X. ( Base ` C ) ) ) ) /\ ( f e. ( ( 1st ` y ) ( Hom ` C ) ( 1st ` x ) ) /\ g e. ( ( 2nd ` x ) ( Hom ` C ) ( 2nd ` y ) ) ) ) -> ( h e. ( ( Hom ` C ) ` x ) |-> ( ( g ( x (comp ` C ) ( 2nd ` y ) ) h ) ( <. ( 1st ` y ) , ( 1st ` x ) >. (comp ` C ) ( 2nd ` y ) ) f ) ) e. ( ( ( Hom f ` C ) ` x ) ( Hom ` D ) ( ( Hom f ` C ) ` y ) ) )
108	107	ralrimivva 2971	. . . . . 6 |- ( ( ph /\ ( x e. ( ( Base ` C ) X. ( Base ` C ) ) /\ y e. ( ( Base ` C ) X. ( Base ` C ) ) ) ) -> A. f e. ( ( 1st ` y ) ( Hom ` C ) ( 1st ` x ) ) A. g e. ( ( 2nd ` x ) ( Hom ` C ) ( 2nd ` y ) ) ( h e. ( ( Hom ` C ) ` x ) |-> ( ( g ( x (comp ` C ) ( 2nd ` y ) ) h ) ( <. ( 1st ` y ) , ( 1st ` x ) >. (comp ` C ) ( 2nd ` y ) ) f ) ) e. ( ( ( Hom f ` C ) ` x ) ( Hom ` D ) ( ( Hom f ` C ) ` y ) ) )
109		eqid 2622	. . . . . . 7 |- ( f e. ( ( 1st ` y ) ( Hom ` C ) ( 1st ` x ) ) , g e. ( ( 2nd ` x ) ( Hom ` C ) ( 2nd ` y ) ) |-> ( h e. ( ( Hom ` C ) ` x ) |-> ( ( g ( x (comp ` C ) ( 2nd ` y ) ) h ) ( <. ( 1st ` y ) , ( 1st ` x ) >. (comp ` C ) ( 2nd ` y ) ) f ) ) ) = ( f e. ( ( 1st ` y ) ( Hom ` C ) ( 1st ` x ) ) , g e. ( ( 2nd ` x ) ( Hom ` C ) ( 2nd ` y ) ) |-> ( h e. ( ( Hom ` C ) ` x ) |-> ( ( g ( x (comp ` C ) ( 2nd ` y ) ) h ) ( <. ( 1st ` y ) , ( 1st ` x ) >. (comp ` C ) ( 2nd ` y ) ) f ) ) )
110	109	fmpt2 7237	. . . . . 6 |- ( A. f e. ( ( 1st ` y ) ( Hom ` C ) ( 1st ` x ) ) A. g e. ( ( 2nd ` x ) ( Hom ` C ) ( 2nd ` y ) ) ( h e. ( ( Hom ` C ) ` x ) |-> ( ( g ( x (comp ` C ) ( 2nd ` y ) ) h ) ( <. ( 1st ` y ) , ( 1st ` x ) >. (comp ` C ) ( 2nd ` y ) ) f ) ) e. ( ( ( Hom f ` C ) ` x ) ( Hom ` D ) ( ( Hom f ` C ) ` y ) ) <-> ( f e. ( ( 1st ` y ) ( Hom ` C ) ( 1st ` x ) ) , g e. ( ( 2nd ` x ) ( Hom ` C ) ( 2nd ` y ) ) |-> ( h e. ( ( Hom ` C ) ` x ) |-> ( ( g ( x (comp ` C ) ( 2nd ` y ) ) h ) ( <. ( 1st ` y ) , ( 1st ` x ) >. (comp ` C ) ( 2nd ` y ) ) f ) ) ) : ( ( ( 1st ` y ) ( Hom ` C ) ( 1st ` x ) ) X. ( ( 2nd ` x ) ( Hom ` C ) ( 2nd ` y ) ) ) --> ( ( ( Hom f ` C ) ` x ) ( Hom ` D ) ( ( Hom f ` C ) ` y ) ) )
111	108, 110	sylib 208	. . . . 5 |- ( ( ph /\ ( x e. ( ( Base ` C ) X. ( Base ` C ) ) /\ y e. ( ( Base ` C ) X. ( Base ` C ) ) ) ) -> ( f e. ( ( 1st ` y ) ( Hom ` C ) ( 1st ` x ) ) , g e. ( ( 2nd ` x ) ( Hom ` C ) ( 2nd ` y ) ) |-> ( h e. ( ( Hom ` C ) ` x ) |-> ( ( g ( x (comp ` C ) ( 2nd ` y ) ) h ) ( <. ( 1st ` y ) , ( 1st ` x ) >. (comp ` C ) ( 2nd ` y ) ) f ) ) ) : ( ( ( 1st ` y ) ( Hom ` C ) ( 1st ` x ) ) X. ( ( 2nd ` x ) ( Hom ` C ) ( 2nd ` y ) ) ) --> ( ( ( Hom f ` C ) ` x ) ( Hom ` D ) ( ( Hom f ` C ) ` y ) ) )
112	12	oveqd 6667	. . . . . . 7 |- ( ph -> ( x ( 2nd ` M ) y ) = ( x ( x e. ( ( Base ` C ) X. ( Base ` C ) ) , y e. ( ( Base ` C ) X. ( Base ` C ) ) |-> ( f e. ( ( 1st ` y ) ( Hom ` C ) ( 1st ` x ) ) , g e. ( ( 2nd ` x ) ( Hom ` C ) ( 2nd ` y ) ) |-> ( h e. ( ( Hom ` C ) ` x ) |-> ( ( g ( x (comp ` C ) ( 2nd ` y ) ) h ) ( <. ( 1st ` y ) , ( 1st ` x ) >. (comp ` C ) ( 2nd ` y ) ) f ) ) ) ) y ) )
113	42	ovmpt4g 6783	. . . . . . . 8 |- ( ( x e. ( ( Base ` C ) X. ( Base ` C ) ) /\ y e. ( ( Base ` C ) X. ( Base ` C ) ) /\ ( f e. ( ( 1st ` y ) ( Hom ` C ) ( 1st ` x ) ) , g e. ( ( 2nd ` x ) ( Hom ` C ) ( 2nd ` y ) ) |-> ( h e. ( ( Hom ` C ) ` x ) |-> ( ( g ( x (comp ` C ) ( 2nd ` y ) ) h ) ( <. ( 1st ` y ) , ( 1st ` x ) >. (comp ` C ) ( 2nd ` y ) ) f ) ) ) e. _V ) -> ( x ( x e. ( ( Base ` C ) X. ( Base ` C ) ) , y e. ( ( Base ` C ) X. ( Base ` C ) ) |-> ( f e. ( ( 1st ` y ) ( Hom ` C ) ( 1st ` x ) ) , g e. ( ( 2nd ` x ) ( Hom ` C ) ( 2nd ` y ) ) |-> ( h e. ( ( Hom ` C ) ` x ) |-> ( ( g ( x (comp ` C ) ( 2nd ` y ) ) h ) ( <. ( 1st ` y ) , ( 1st ` x ) >. (comp ` C ) ( 2nd ` y ) ) f ) ) ) ) y ) = ( f e. ( ( 1st ` y ) ( Hom ` C ) ( 1st ` x ) ) , g e. ( ( 2nd ` x ) ( Hom ` C ) ( 2nd ` y ) ) |-> ( h e. ( ( Hom ` C ) ` x ) |-> ( ( g ( x (comp ` C ) ( 2nd ` y ) ) h ) ( <. ( 1st ` y ) , ( 1st ` x ) >. (comp ` C ) ( 2nd ` y ) ) f ) ) ) )
114	45, 113	mp3an3 1413	. . . . . . 7 |- ( ( x e. ( ( Base ` C ) X. ( Base ` C ) ) /\ y e. ( ( Base ` C ) X. ( Base ` C ) ) ) -> ( x ( x e. ( ( Base ` C ) X. ( Base ` C ) ) , y e. ( ( Base ` C ) X. ( Base ` C ) ) |-> ( f e. ( ( 1st ` y ) ( Hom ` C ) ( 1st ` x ) ) , g e. ( ( 2nd ` x ) ( Hom ` C ) ( 2nd ` y ) ) |-> ( h e. ( ( Hom ` C ) ` x ) |-> ( ( g ( x (comp ` C ) ( 2nd ` y ) ) h ) ( <. ( 1st ` y ) , ( 1st ` x ) >. (comp ` C ) ( 2nd ` y ) ) f ) ) ) ) y ) = ( f e. ( ( 1st ` y ) ( Hom ` C ) ( 1st ` x ) ) , g e. ( ( 2nd ` x ) ( Hom ` C ) ( 2nd ` y ) ) |-> ( h e. ( ( Hom ` C ) ` x ) |-> ( ( g ( x (comp ` C ) ( 2nd ` y ) ) h ) ( <. ( 1st ` y ) , ( 1st ` x ) >. (comp ` C ) ( 2nd ` y ) ) f ) ) ) )
115	112, 114	sylan9eq 2676	. . . . . 6 |- ( ( ph /\ ( x e. ( ( Base ` C ) X. ( Base ` C ) ) /\ y e. ( ( Base ` C ) X. ( Base ` C ) ) ) ) -> ( x ( 2nd ` M ) y ) = ( f e. ( ( 1st ` y ) ( Hom ` C ) ( 1st ` x ) ) , g e. ( ( 2nd ` x ) ( Hom ` C ) ( 2nd ` y ) ) |-> ( h e. ( ( Hom ` C ) ` x ) |-> ( ( g ( x (comp ` C ) ( 2nd ` y ) ) h ) ( <. ( 1st ` y ) , ( 1st ` x ) >. (comp ` C ) ( 2nd ` y ) ) f ) ) ) )
116		eqid 2622	. . . . . . . 8 |- ( Hom ` O ) = ( Hom ` O )
117		simprl 794	. . . . . . . 8 |- ( ( ph /\ ( x e. ( ( Base ` C ) X. ( Base ` C ) ) /\ y e. ( ( Base ` C ) X. ( Base ` C ) ) ) ) -> x e. ( ( Base ` C ) X. ( Base ` C ) ) )
118		simprr 796	. . . . . . . 8 |- ( ( ph /\ ( x e. ( ( Base ` C ) X. ( Base ` C ) ) /\ y e. ( ( Base ` C ) X. ( Base ` C ) ) ) ) -> y e. ( ( Base ` C ) X. ( Base ` C ) ) )
119	15, 18, 116, 4, 20, 117, 118	xpchom 16820	. . . . . . 7 |- ( ( ph /\ ( x e. ( ( Base ` C ) X. ( Base ` C ) ) /\ y e. ( ( Base ` C ) X. ( Base ` C ) ) ) ) -> ( x ( Hom ` ( O X.c C ) ) y ) = ( ( ( 1st ` x ) ( Hom ` O ) ( 1st ` y ) ) X. ( ( 2nd ` x ) ( Hom ` C ) ( 2nd ` y ) ) ) )
120	4, 16	oppchom 16375	. . . . . . . 8 |- ( ( 1st ` x ) ( Hom ` O ) ( 1st ` y ) ) = ( ( 1st ` y ) ( Hom ` C ) ( 1st ` x ) )
121	120	xpeq1i 5135	. . . . . . 7 |- ( ( ( 1st ` x ) ( Hom ` O ) ( 1st ` y ) ) X. ( ( 2nd ` x ) ( Hom ` C ) ( 2nd ` y ) ) ) = ( ( ( 1st ` y ) ( Hom ` C ) ( 1st ` x ) ) X. ( ( 2nd ` x ) ( Hom ` C ) ( 2nd ` y ) ) )
122	119, 121	syl6eq 2672	. . . . . 6 |- ( ( ph /\ ( x e. ( ( Base ` C ) X. ( Base ` C ) ) /\ y e. ( ( Base ` C ) X. ( Base ` C ) ) ) ) -> ( x ( Hom ` ( O X.c C ) ) y ) = ( ( ( 1st ` y ) ( Hom ` C ) ( 1st ` x ) ) X. ( ( 2nd ` x ) ( Hom ` C ) ( 2nd ` y ) ) ) )
123	115, 122	feq12d 6033	. . . . 5 |- ( ( ph /\ ( x e. ( ( Base ` C ) X. ( Base ` C ) ) /\ y e. ( ( Base ` C ) X. ( Base ` C ) ) ) ) -> ( ( x ( 2nd ` M ) y ) : ( x ( Hom ` ( O X.c C ) ) y ) --> ( ( ( Hom f ` C ) ` x ) ( Hom ` D ) ( ( Hom f ` C ) ` y ) ) <-> ( f e. ( ( 1st ` y ) ( Hom ` C ) ( 1st ` x ) ) , g e. ( ( 2nd ` x ) ( Hom ` C ) ( 2nd ` y ) ) |-> ( h e. ( ( Hom ` C ) ` x ) |-> ( ( g ( x (comp ` C ) ( 2nd ` y ) ) h ) ( <. ( 1st ` y ) , ( 1st ` x ) >. (comp ` C ) ( 2nd ` y ) ) f ) ) ) : ( ( ( 1st ` y ) ( Hom ` C ) ( 1st ` x ) ) X. ( ( 2nd ` x ) ( Hom ` C ) ( 2nd ` y ) ) ) --> ( ( ( Hom f ` C ) ` x ) ( Hom ` D ) ( ( Hom f ` C ) ` y ) ) ) )
124	111, 123	mpbird 247	. . . 4 |- ( ( ph /\ ( x e. ( ( Base ` C ) X. ( Base ` C ) ) /\ y e. ( ( Base ` C ) X. ( Base ` C ) ) ) ) -> ( x ( 2nd ` M ) y ) : ( x ( Hom ` ( O X.c C ) ) y ) --> ( ( ( Hom f ` C ) ` x ) ( Hom ` D ) ( ( Hom f ` C ) ` y ) ) )
125		eqid 2622	. . . . . . . . . 10 |- ( Id ` C ) = ( Id ` C )
126	2	ad2antrr 762	. . . . . . . . . 10 |- ( ( ( ph /\ x e. ( ( Base ` C ) X. ( Base ` C ) ) ) /\ f e. ( ( 1st ` x ) ( Hom ` C ) ( 2nd ` x ) ) ) -> C e. Cat )
127	55	adantl 482	. . . . . . . . . . 11 |- ( ( ph /\ x e. ( ( Base ` C ) X. ( Base ` C ) ) ) -> ( 1st ` x ) e. ( Base ` C ) )
128	127	adantr 481	. . . . . . . . . 10 |- ( ( ( ph /\ x e. ( ( Base ` C ) X. ( Base ` C ) ) ) /\ f e. ( ( 1st ` x ) ( Hom ` C ) ( 2nd ` x ) ) ) -> ( 1st ` x ) e. ( Base ` C ) )
129	67	adantl 482	. . . . . . . . . . 11 |- ( ( ph /\ x e. ( ( Base ` C ) X. ( Base ` C ) ) ) -> ( 2nd ` x ) e. ( Base ` C ) )
130	129	adantr 481	. . . . . . . . . 10 |- ( ( ( ph /\ x e. ( ( Base ` C ) X. ( Base ` C ) ) ) /\ f e. ( ( 1st ` x ) ( Hom ` C ) ( 2nd ` x ) ) ) -> ( 2nd ` x ) e. ( Base ` C ) )
131		simpr 477	. . . . . . . . . 10 |- ( ( ( ph /\ x e. ( ( Base ` C ) X. ( Base ` C ) ) ) /\ f e. ( ( 1st ` x ) ( Hom ` C ) ( 2nd ` x ) ) ) -> f e. ( ( 1st ` x ) ( Hom ` C ) ( 2nd ` x ) ) )
132	3, 4, 125, 126, 128, 5, 130, 131	catlid 16344	. . . . . . . . 9 |- ( ( ( ph /\ x e. ( ( Base ` C ) X. ( Base ` C ) ) ) /\ f e. ( ( 1st ` x ) ( Hom ` C ) ( 2nd ` x ) ) ) -> ( ( ( Id ` C ) ` ( 2nd ` x ) ) ( <. ( 1st ` x ) , ( 2nd ` x ) >. (comp ` C ) ( 2nd ` x ) ) f ) = f )
133	132	oveq1d 6665	. . . . . . . 8 |- ( ( ( ph /\ x e. ( ( Base ` C ) X. ( Base ` C ) ) ) /\ f e. ( ( 1st ` x ) ( Hom ` C ) ( 2nd ` x ) ) ) -> ( ( ( ( Id ` C ) ` ( 2nd ` x ) ) ( <. ( 1st ` x ) , ( 2nd ` x ) >. (comp ` C ) ( 2nd ` x ) ) f ) ( <. ( 1st ` x ) , ( 1st ` x ) >. (comp ` C ) ( 2nd ` x ) ) ( ( Id ` C ) ` ( 1st ` x ) ) ) = ( f ( <. ( 1st ` x ) , ( 1st ` x ) >. (comp ` C ) ( 2nd ` x ) ) ( ( Id ` C ) ` ( 1st ` x ) ) ) )
134	3, 4, 125, 126, 128, 5, 130, 131	catrid 16345	. . . . . . . 8 |- ( ( ( ph /\ x e. ( ( Base ` C ) X. ( Base ` C ) ) ) /\ f e. ( ( 1st ` x ) ( Hom ` C ) ( 2nd ` x ) ) ) -> ( f ( <. ( 1st ` x ) , ( 1st ` x ) >. (comp ` C ) ( 2nd ` x ) ) ( ( Id ` C ) ` ( 1st ` x ) ) ) = f )
135	133, 134	eqtrd 2656	. . . . . . 7 |- ( ( ( ph /\ x e. ( ( Base ` C ) X. ( Base ` C ) ) ) /\ f e. ( ( 1st ` x ) ( Hom ` C ) ( 2nd ` x ) ) ) -> ( ( ( ( Id ` C ) ` ( 2nd ` x ) ) ( <. ( 1st ` x ) , ( 2nd ` x ) >. (comp ` C ) ( 2nd ` x ) ) f ) ( <. ( 1st ` x ) , ( 1st ` x ) >. (comp ` C ) ( 2nd ` x ) ) ( ( Id ` C ) ` ( 1st ` x ) ) ) = f )
136	135	mpteq2dva 4744	. . . . . 6 |- ( ( ph /\ x e. ( ( Base ` C ) X. ( Base ` C ) ) ) -> ( f e. ( ( 1st ` x ) ( Hom ` C ) ( 2nd ` x ) ) |-> ( ( ( ( Id ` C ) ` ( 2nd ` x ) ) ( <. ( 1st ` x ) , ( 2nd ` x ) >. (comp ` C ) ( 2nd ` x ) ) f ) ( <. ( 1st ` x ) , ( 1st ` x ) >. (comp ` C ) ( 2nd ` x ) ) ( ( Id ` C ) ` ( 1st ` x ) ) ) ) = ( f e. ( ( 1st ` x ) ( Hom ` C ) ( 2nd ` x ) ) |-> f ) )
137		df-ov 6653	. . . . . . 7 |- ( ( ( Id ` C ) ` ( 1st ` x ) ) ( <. ( 1st ` x ) , ( 2nd ` x ) >. ( 2nd ` M ) <. ( 1st ` x ) , ( 2nd ` x ) >. ) ( ( Id ` C ) ` ( 2nd ` x ) ) ) = ( ( <. ( 1st ` x ) , ( 2nd ` x ) >. ( 2nd ` M ) <. ( 1st ` x ) , ( 2nd ` x ) >. ) ` <. ( ( Id ` C ) ` ( 1st ` x ) ) , ( ( Id ` C ) ` ( 2nd ` x ) ) >. )
138	2	adantr 481	. . . . . . . 8 |- ( ( ph /\ x e. ( ( Base ` C ) X. ( Base ` C ) ) ) -> C e. Cat )
139	3, 4, 125, 138, 127	catidcl 16343	. . . . . . . 8 |- ( ( ph /\ x e. ( ( Base ` C ) X. ( Base ` C ) ) ) -> ( ( Id ` C ) ` ( 1st ` x ) ) e. ( ( 1st ` x ) ( Hom ` C ) ( 1st ` x ) ) )
140	3, 4, 125, 138, 129	catidcl 16343	. . . . . . . 8 |- ( ( ph /\ x e. ( ( Base ` C ) X. ( Base ` C ) ) ) -> ( ( Id ` C ) ` ( 2nd ` x ) ) e. ( ( 2nd ` x ) ( Hom ` C ) ( 2nd ` x ) ) )
141	1, 138, 3, 4, 127, 129, 127, 129, 5, 139, 140	hof2val 16896	. . . . . . 7 |- ( ( ph /\ x e. ( ( Base ` C ) X. ( Base ` C ) ) ) -> ( ( ( Id ` C ) ` ( 1st ` x ) ) ( <. ( 1st ` x ) , ( 2nd ` x ) >. ( 2nd ` M ) <. ( 1st ` x ) , ( 2nd ` x ) >. ) ( ( Id ` C ) ` ( 2nd ` x ) ) ) = ( f e. ( ( 1st ` x ) ( Hom ` C ) ( 2nd ` x ) ) |-> ( ( ( ( Id ` C ) ` ( 2nd ` x ) ) ( <. ( 1st ` x ) , ( 2nd ` x ) >. (comp ` C ) ( 2nd ` x ) ) f ) ( <. ( 1st ` x ) , ( 1st ` x ) >. (comp ` C ) ( 2nd ` x ) ) ( ( Id ` C ) ` ( 1st ` x ) ) ) ) )
142	137, 141	syl5eqr 2670	. . . . . 6 |- ( ( ph /\ x e. ( ( Base ` C ) X. ( Base ` C ) ) ) -> ( ( <. ( 1st ` x ) , ( 2nd ` x ) >. ( 2nd ` M ) <. ( 1st ` x ) , ( 2nd ` x ) >. ) ` <. ( ( Id ` C ) ` ( 1st ` x ) ) , ( ( Id ` C ) ` ( 2nd ` x ) ) >. ) = ( f e. ( ( 1st ` x ) ( Hom ` C ) ( 2nd ` x ) ) |-> ( ( ( ( Id ` C ) ` ( 2nd ` x ) ) ( <. ( 1st ` x ) , ( 2nd ` x ) >. (comp ` C ) ( 2nd ` x ) ) f ) ( <. ( 1st ` x ) , ( 1st ` x ) >. (comp ` C ) ( 2nd ` x ) ) ( ( Id ` C ) ` ( 1st ` x ) ) ) ) )
143	62	adantl 482	. . . . . . . . . . 11 |- ( ( ph /\ x e. ( ( Base ` C ) X. ( Base ` C ) ) ) -> x = <. ( 1st ` x ) , ( 2nd ` x ) >. )
144	143	fveq2d 6195	. . . . . . . . . 10 |- ( ( ph /\ x e. ( ( Base ` C ) X. ( Base ` C ) ) ) -> ( ( Hom f ` C ) ` x ) = ( ( Hom f ` C ) ` <. ( 1st ` x ) , ( 2nd ` x ) >. ) )
145	144, 91	syl6eqr 2674	. . . . . . . . 9 |- ( ( ph /\ x e. ( ( Base ` C ) X. ( Base ` C ) ) ) -> ( ( Hom f ` C ) ` x ) = ( ( 1st ` x ) ( Hom f ` C ) ( 2nd ` x ) ) )
146	33, 3, 4, 127, 129	homfval 16352	. . . . . . . . 9 |- ( ( ph /\ x e. ( ( Base ` C ) X. ( Base ` C ) ) ) -> ( ( 1st ` x ) ( Hom f ` C ) ( 2nd ` x ) ) = ( ( 1st ` x ) ( Hom ` C ) ( 2nd ` x ) ) )
147	145, 146	eqtrd 2656	. . . . . . . 8 |- ( ( ph /\ x e. ( ( Base ` C ) X. ( Base ` C ) ) ) -> ( ( Hom f ` C ) ` x ) = ( ( 1st ` x ) ( Hom ` C ) ( 2nd ` x ) ) )
148	147	reseq2d 5396	. . . . . . 7 |- ( ( ph /\ x e. ( ( Base ` C ) X. ( Base ` C ) ) ) -> ( _I |` ( ( Hom f ` C ) ` x ) ) = ( _I |` ( ( 1st ` x ) ( Hom ` C ) ( 2nd ` x ) ) ) )
149		mptresid 5456	. . . . . . 7 |- ( f e. ( ( 1st ` x ) ( Hom ` C ) ( 2nd ` x ) ) |-> f ) = ( _I |` ( ( 1st ` x ) ( Hom ` C ) ( 2nd ` x ) ) )
150	148, 149	syl6eqr 2674	. . . . . 6 |- ( ( ph /\ x e. ( ( Base ` C ) X. ( Base ` C ) ) ) -> ( _I |` ( ( Hom f ` C ) ` x ) ) = ( f e. ( ( 1st ` x ) ( Hom ` C ) ( 2nd ` x ) ) |-> f ) )
151	136, 142, 150	3eqtr4d 2666	. . . . 5 |- ( ( ph /\ x e. ( ( Base ` C ) X. ( Base ` C ) ) ) -> ( ( <. ( 1st ` x ) , ( 2nd ` x ) >. ( 2nd ` M ) <. ( 1st ` x ) , ( 2nd ` x ) >. ) ` <. ( ( Id ` C ) ` ( 1st ` x ) ) , ( ( Id ` C ) ` ( 2nd ` x ) ) >. ) = ( _I |` ( ( Hom f ` C ) ` x ) ) )
152	143, 143	oveq12d 6668	. . . . . 6 |- ( ( ph /\ x e. ( ( Base ` C ) X. ( Base ` C ) ) ) -> ( x ( 2nd ` M ) x ) = ( <. ( 1st ` x ) , ( 2nd ` x ) >. ( 2nd ` M ) <. ( 1st ` x ) , ( 2nd ` x ) >. ) )
153	143	fveq2d 6195	. . . . . . 7 |- ( ( ph /\ x e. ( ( Base ` C ) X. ( Base ` C ) ) ) -> ( ( Id ` ( O X.c C ) ) ` x ) = ( ( Id ` ( O X.c C ) ) ` <. ( 1st ` x ) , ( 2nd ` x ) >. ) )
154	27	adantr 481	. . . . . . . 8 |- ( ( ph /\ x e. ( ( Base ` C ) X. ( Base ` C ) ) ) -> O e. Cat )
155		eqid 2622	. . . . . . . 8 |- ( Id ` O ) = ( Id ` O )
156	15, 154, 138, 17, 3, 155, 125, 22, 127, 129	xpcid 16829	. . . . . . 7 |- ( ( ph /\ x e. ( ( Base ` C ) X. ( Base ` C ) ) ) -> ( ( Id ` ( O X.c C ) ) ` <. ( 1st ` x ) , ( 2nd ` x ) >. ) = <. ( ( Id ` O ) ` ( 1st ` x ) ) , ( ( Id ` C ) ` ( 2nd ` x ) ) >. )
157	16, 125	oppcid 16381	. . . . . . . . . 10 |- ( C e. Cat -> ( Id ` O ) = ( Id ` C ) )
158	138, 157	syl 17	. . . . . . . . 9 |- ( ( ph /\ x e. ( ( Base ` C ) X. ( Base ` C ) ) ) -> ( Id ` O ) = ( Id ` C ) )
159	158	fveq1d 6193	. . . . . . . 8 |- ( ( ph /\ x e. ( ( Base ` C ) X. ( Base ` C ) ) ) -> ( ( Id ` O ) ` ( 1st ` x ) ) = ( ( Id ` C ) ` ( 1st ` x ) ) )
160	159	opeq1d 4408	. . . . . . 7 |- ( ( ph /\ x e. ( ( Base ` C ) X. ( Base ` C ) ) ) -> <. ( ( Id ` O ) ` ( 1st ` x ) ) , ( ( Id ` C ) ` ( 2nd ` x ) ) >. = <. ( ( Id ` C ) ` ( 1st ` x ) ) , ( ( Id ` C ) ` ( 2nd ` x ) ) >. )
161	153, 156, 160	3eqtrd 2660	. . . . . 6 |- ( ( ph /\ x e. ( ( Base ` C ) X. ( Base ` C ) ) ) -> ( ( Id ` ( O X.c C ) ) ` x ) = <. ( ( Id ` C ) ` ( 1st ` x ) ) , ( ( Id ` C ) ` ( 2nd ` x ) ) >. )
162	152, 161	fveq12d 6197	. . . . 5 |- ( ( ph /\ x e. ( ( Base ` C ) X. ( Base ` C ) ) ) -> ( ( x ( 2nd ` M ) x ) ` ( ( Id ` ( O X.c C ) ) ` x ) ) = ( ( <. ( 1st ` x ) , ( 2nd ` x ) >. ( 2nd ` M ) <. ( 1st ` x ) , ( 2nd ` x ) >. ) ` <. ( ( Id ` C ) ` ( 1st ` x ) ) , ( ( Id ` C ) ` ( 2nd ` x ) ) >. ) )
163	29	adantr 481	. . . . . 6 |- ( ( ph /\ x e. ( ( Base ` C ) X. ( Base ` C ) ) ) -> U e. V )
164	38	ffvelrnda 6359	. . . . . 6 |- ( ( ph /\ x e. ( ( Base ` C ) X. ( Base ` C ) ) ) -> ( ( Hom f ` C ) ` x ) e. U )
165	30, 23, 163, 164	setcid 16736	. . . . 5 |- ( ( ph /\ x e. ( ( Base ` C ) X. ( Base ` C ) ) ) -> ( ( Id ` D ) ` ( ( Hom f ` C ) ` x ) ) = ( _I |` ( ( Hom f ` C ) ` x ) ) )
166	151, 162, 165	3eqtr4d 2666	. . . 4 |- ( ( ph /\ x e. ( ( Base ` C ) X. ( Base ` C ) ) ) -> ( ( x ( 2nd ` M ) x ) ` ( ( Id ` ( O X.c C ) ) ` x ) ) = ( ( Id ` D ) ` ( ( Hom f ` C ) ` x ) ) )
167	2	3ad2ant1 1082	. . . . . 6 |- ( ( ph /\ ( x e. ( ( Base ` C ) X. ( Base ` C ) ) /\ y e. ( ( Base ` C ) X. ( Base ` C ) ) /\ z e. ( ( Base ` C ) X. ( Base ` C ) ) ) /\ ( f e. ( x ( Hom ` ( O X.c C ) ) y ) /\ g e. ( y ( Hom ` ( O X.c C ) ) z ) ) ) -> C e. Cat )
168	29	3ad2ant1 1082	. . . . . 6 |- ( ( ph /\ ( x e. ( ( Base ` C ) X. ( Base ` C ) ) /\ y e. ( ( Base ` C ) X. ( Base ` C ) ) /\ z e. ( ( Base ` C ) X. ( Base ` C ) ) ) /\ ( f e. ( x ( Hom ` ( O X.c C ) ) y ) /\ g e. ( y ( Hom ` ( O X.c C ) ) z ) ) ) -> U e. V )
169	36	3ad2ant1 1082	. . . . . 6 |- ( ( ph /\ ( x e. ( ( Base ` C ) X. ( Base ` C ) ) /\ y e. ( ( Base ` C ) X. ( Base ` C ) ) /\ z e. ( ( Base ` C ) X. ( Base ` C ) ) ) /\ ( f e. ( x ( Hom ` ( O X.c C ) ) y ) /\ g e. ( y ( Hom ` ( O X.c C ) ) z ) ) ) -> ran ( Hom f ` C ) C_ U )
170		simp21 1094	. . . . . . 7 |- ( ( ph /\ ( x e. ( ( Base ` C ) X. ( Base ` C ) ) /\ y e. ( ( Base ` C ) X. ( Base ` C ) ) /\ z e. ( ( Base ` C ) X. ( Base ` C ) ) ) /\ ( f e. ( x ( Hom ` ( O X.c C ) ) y ) /\ g e. ( y ( Hom ` ( O X.c C ) ) z ) ) ) -> x e. ( ( Base ` C ) X. ( Base ` C ) ) )
171	170, 55	syl 17	. . . . . 6 |- ( ( ph /\ ( x e. ( ( Base ` C ) X. ( Base ` C ) ) /\ y e. ( ( Base ` C ) X. ( Base ` C ) ) /\ z e. ( ( Base ` C ) X. ( Base ` C ) ) ) /\ ( f e. ( x ( Hom ` ( O X.c C ) ) y ) /\ g e. ( y ( Hom ` ( O X.c C ) ) z ) ) ) -> ( 1st ` x ) e. ( Base ` C ) )
172	170, 67	syl 17	. . . . . 6 |- ( ( ph /\ ( x e. ( ( Base ` C ) X. ( Base ` C ) ) /\ y e. ( ( Base ` C ) X. ( Base ` C ) ) /\ z e. ( ( Base ` C ) X. ( Base ` C ) ) ) /\ ( f e. ( x ( Hom ` ( O X.c C ) ) y ) /\ g e. ( y ( Hom ` ( O X.c C ) ) z ) ) ) -> ( 2nd ` x ) e. ( Base ` C ) )
173		simp22 1095	. . . . . . 7 |- ( ( ph /\ ( x e. ( ( Base ` C ) X. ( Base ` C ) ) /\ y e. ( ( Base ` C ) X. ( Base ` C ) ) /\ z e. ( ( Base ` C ) X. ( Base ` C ) ) ) /\ ( f e. ( x ( Hom ` ( O X.c C ) ) y ) /\ g e. ( y ( Hom ` ( O X.c C ) ) z ) ) ) -> y e. ( ( Base ` C ) X. ( Base ` C ) ) )
174	173, 51	syl 17	. . . . . 6 |- ( ( ph /\ ( x e. ( ( Base ` C ) X. ( Base ` C ) ) /\ y e. ( ( Base ` C ) X. ( Base ` C ) ) /\ z e. ( ( Base ` C ) X. ( Base ` C ) ) ) /\ ( f e. ( x ( Hom ` ( O X.c C ) ) y ) /\ g e. ( y ( Hom ` ( O X.c C ) ) z ) ) ) -> ( 1st ` y ) e. ( Base ` C ) )
175	173, 58	syl 17	. . . . . 6 |- ( ( ph /\ ( x e. ( ( Base ` C ) X. ( Base ` C ) ) /\ y e. ( ( Base ` C ) X. ( Base ` C ) ) /\ z e. ( ( Base ` C ) X. ( Base ` C ) ) ) /\ ( f e. ( x ( Hom ` ( O X.c C ) ) y ) /\ g e. ( y ( Hom ` ( O X.c C ) ) z ) ) ) -> ( 2nd ` y ) e. ( Base ` C ) )
176		simp23 1096	. . . . . . 7 |- ( ( ph /\ ( x e. ( ( Base ` C ) X. ( Base ` C ) ) /\ y e. ( ( Base ` C ) X. ( Base ` C ) ) /\ z e. ( ( Base ` C ) X. ( Base ` C ) ) ) /\ ( f e. ( x ( Hom ` ( O X.c C ) ) y ) /\ g e. ( y ( Hom ` ( O X.c C ) ) z ) ) ) -> z e. ( ( Base ` C ) X. ( Base ` C ) ) )
177		xp1st 7198	. . . . . . 7 |- ( z e. ( ( Base ` C ) X. ( Base ` C ) ) -> ( 1st ` z ) e. ( Base ` C ) )
178	176, 177	syl 17	. . . . . 6 |- ( ( ph /\ ( x e. ( ( Base ` C ) X. ( Base ` C ) ) /\ y e. ( ( Base ` C ) X. ( Base ` C ) ) /\ z e. ( ( Base ` C ) X. ( Base ` C ) ) ) /\ ( f e. ( x ( Hom ` ( O X.c C ) ) y ) /\ g e. ( y ( Hom ` ( O X.c C ) ) z ) ) ) -> ( 1st ` z ) e. ( Base ` C ) )
179		xp2nd 7199	. . . . . . 7 |- ( z e. ( ( Base ` C ) X. ( Base ` C ) ) -> ( 2nd ` z ) e. ( Base ` C ) )
180	176, 179	syl 17	. . . . . 6 |- ( ( ph /\ ( x e. ( ( Base ` C ) X. ( Base ` C ) ) /\ y e. ( ( Base ` C ) X. ( Base ` C ) ) /\ z e. ( ( Base ` C ) X. ( Base ` C ) ) ) /\ ( f e. ( x ( Hom ` ( O X.c C ) ) y ) /\ g e. ( y ( Hom ` ( O X.c C ) ) z ) ) ) -> ( 2nd ` z ) e. ( Base ` C ) )
181		simp3l 1089	. . . . . . . . 9 |- ( ( ph /\ ( x e. ( ( Base ` C ) X. ( Base ` C ) ) /\ y e. ( ( Base ` C ) X. ( Base ` C ) ) /\ z e. ( ( Base ` C ) X. ( Base ` C ) ) ) /\ ( f e. ( x ( Hom ` ( O X.c C ) ) y ) /\ g e. ( y ( Hom ` ( O X.c C ) ) z ) ) ) -> f e. ( x ( Hom ` ( O X.c C ) ) y ) )
182	15, 18, 116, 4, 20, 170, 173	xpchom 16820	. . . . . . . . 9 |- ( ( ph /\ ( x e. ( ( Base ` C ) X. ( Base ` C ) ) /\ y e. ( ( Base ` C ) X. ( Base ` C ) ) /\ z e. ( ( Base ` C ) X. ( Base ` C ) ) ) /\ ( f e. ( x ( Hom ` ( O X.c C ) ) y ) /\ g e. ( y ( Hom ` ( O X.c C ) ) z ) ) ) -> ( x ( Hom ` ( O X.c C ) ) y ) = ( ( ( 1st ` x ) ( Hom ` O ) ( 1st ` y ) ) X. ( ( 2nd ` x ) ( Hom ` C ) ( 2nd ` y ) ) ) )
183	181, 182	eleqtrd 2703	. . . . . . . 8 |- ( ( ph /\ ( x e. ( ( Base ` C ) X. ( Base ` C ) ) /\ y e. ( ( Base ` C ) X. ( Base ` C ) ) /\ z e. ( ( Base ` C ) X. ( Base ` C ) ) ) /\ ( f e. ( x ( Hom ` ( O X.c C ) ) y ) /\ g e. ( y ( Hom ` ( O X.c C ) ) z ) ) ) -> f e. ( ( ( 1st ` x ) ( Hom ` O ) ( 1st ` y ) ) X. ( ( 2nd ` x ) ( Hom ` C ) ( 2nd ` y ) ) ) )
184		xp1st 7198	. . . . . . . 8 |- ( f e. ( ( ( 1st ` x ) ( Hom ` O ) ( 1st ` y ) ) X. ( ( 2nd ` x ) ( Hom ` C ) ( 2nd ` y ) ) ) -> ( 1st ` f ) e. ( ( 1st ` x ) ( Hom ` O ) ( 1st ` y ) ) )
185	183, 184	syl 17	. . . . . . 7 |- ( ( ph /\ ( x e. ( ( Base ` C ) X. ( Base ` C ) ) /\ y e. ( ( Base ` C ) X. ( Base ` C ) ) /\ z e. ( ( Base ` C ) X. ( Base ` C ) ) ) /\ ( f e. ( x ( Hom ` ( O X.c C ) ) y ) /\ g e. ( y ( Hom ` ( O X.c C ) ) z ) ) ) -> ( 1st ` f ) e. ( ( 1st ` x ) ( Hom ` O ) ( 1st ` y ) ) )
186	185, 120	syl6eleq 2711	. . . . . 6 |- ( ( ph /\ ( x e. ( ( Base ` C ) X. ( Base ` C ) ) /\ y e. ( ( Base ` C ) X. ( Base ` C ) ) /\ z e. ( ( Base ` C ) X. ( Base ` C ) ) ) /\ ( f e. ( x ( Hom ` ( O X.c C ) ) y ) /\ g e. ( y ( Hom ` ( O X.c C ) ) z ) ) ) -> ( 1st ` f ) e. ( ( 1st ` y ) ( Hom ` C ) ( 1st ` x ) ) )
187		xp2nd 7199	. . . . . . 7 |- ( f e. ( ( ( 1st ` x ) ( Hom ` O ) ( 1st ` y ) ) X. ( ( 2nd ` x ) ( Hom ` C ) ( 2nd ` y ) ) ) -> ( 2nd ` f ) e. ( ( 2nd ` x ) ( Hom ` C ) ( 2nd ` y ) ) )
188	183, 187	syl 17	. . . . . 6 |- ( ( ph /\ ( x e. ( ( Base ` C ) X. ( Base ` C ) ) /\ y e. ( ( Base ` C ) X. ( Base ` C ) ) /\ z e. ( ( Base ` C ) X. ( Base ` C ) ) ) /\ ( f e. ( x ( Hom ` ( O X.c C ) ) y ) /\ g e. ( y ( Hom ` ( O X.c C ) ) z ) ) ) -> ( 2nd ` f ) e. ( ( 2nd ` x ) ( Hom ` C ) ( 2nd ` y ) ) )
189		simp3r 1090	. . . . . . . . 9 |- ( ( ph /\ ( x e. ( ( Base ` C ) X. ( Base ` C ) ) /\ y e. ( ( Base ` C ) X. ( Base ` C ) ) /\ z e. ( ( Base ` C ) X. ( Base ` C ) ) ) /\ ( f e. ( x ( Hom ` ( O X.c C ) ) y ) /\ g e. ( y ( Hom ` ( O X.c C ) ) z ) ) ) -> g e. ( y ( Hom ` ( O X.c C ) ) z ) )
190	15, 18, 116, 4, 20, 173, 176	xpchom 16820	. . . . . . . . 9 |- ( ( ph /\ ( x e. ( ( Base ` C ) X. ( Base ` C ) ) /\ y e. ( ( Base ` C ) X. ( Base ` C ) ) /\ z e. ( ( Base ` C ) X. ( Base ` C ) ) ) /\ ( f e. ( x ( Hom ` ( O X.c C ) ) y ) /\ g e. ( y ( Hom ` ( O X.c C ) ) z ) ) ) -> ( y ( Hom ` ( O X.c C ) ) z ) = ( ( ( 1st ` y ) ( Hom ` O ) ( 1st ` z ) ) X. ( ( 2nd ` y ) ( Hom ` C ) ( 2nd ` z ) ) ) )
191	189, 190	eleqtrd 2703	. . . . . . . 8 |- ( ( ph /\ ( x e. ( ( Base ` C ) X. ( Base ` C ) ) /\ y e. ( ( Base ` C ) X. ( Base ` C ) ) /\ z e. ( ( Base ` C ) X. ( Base ` C ) ) ) /\ ( f e. ( x ( Hom ` ( O X.c C ) ) y ) /\ g e. ( y ( Hom ` ( O X.c C ) ) z ) ) ) -> g e. ( ( ( 1st ` y ) ( Hom ` O ) ( 1st ` z ) ) X. ( ( 2nd ` y ) ( Hom ` C ) ( 2nd ` z ) ) ) )
192		xp1st 7198	. . . . . . . 8 |- ( g e. ( ( ( 1st ` y ) ( Hom ` O ) ( 1st ` z ) ) X. ( ( 2nd ` y ) ( Hom ` C ) ( 2nd ` z ) ) ) -> ( 1st ` g ) e. ( ( 1st ` y ) ( Hom ` O ) ( 1st ` z ) ) )
193	191, 192	syl 17	. . . . . . 7 |- ( ( ph /\ ( x e. ( ( Base ` C ) X. ( Base ` C ) ) /\ y e. ( ( Base ` C ) X. ( Base ` C ) ) /\ z e. ( ( Base ` C ) X. ( Base ` C ) ) ) /\ ( f e. ( x ( Hom ` ( O X.c C ) ) y ) /\ g e. ( y ( Hom ` ( O X.c C ) ) z ) ) ) -> ( 1st ` g ) e. ( ( 1st ` y ) ( Hom ` O ) ( 1st ` z ) ) )
194	4, 16	oppchom 16375	. . . . . . 7 |- ( ( 1st ` y ) ( Hom ` O ) ( 1st ` z ) ) = ( ( 1st ` z ) ( Hom ` C ) ( 1st ` y ) )
195	193, 194	syl6eleq 2711	. . . . . 6 |- ( ( ph /\ ( x e. ( ( Base ` C ) X. ( Base ` C ) ) /\ y e. ( ( Base ` C ) X. ( Base ` C ) ) /\ z e. ( ( Base ` C ) X. ( Base ` C ) ) ) /\ ( f e. ( x ( Hom ` ( O X.c C ) ) y ) /\ g e. ( y ( Hom ` ( O X.c C ) ) z ) ) ) -> ( 1st ` g ) e. ( ( 1st ` z ) ( Hom ` C ) ( 1st ` y ) ) )
196		xp2nd 7199	. . . . . . 7 |- ( g e. ( ( ( 1st ` y ) ( Hom ` O ) ( 1st ` z ) ) X. ( ( 2nd ` y ) ( Hom ` C ) ( 2nd ` z ) ) ) -> ( 2nd ` g ) e. ( ( 2nd ` y ) ( Hom ` C ) ( 2nd ` z ) ) )
197	191, 196	syl 17	. . . . . 6 |- ( ( ph /\ ( x e. ( ( Base ` C ) X. ( Base ` C ) ) /\ y e. ( ( Base ` C ) X. ( Base ` C ) ) /\ z e. ( ( Base ` C ) X. ( Base ` C ) ) ) /\ ( f e. ( x ( Hom ` ( O X.c C ) ) y ) /\ g e. ( y ( Hom ` ( O X.c C ) ) z ) ) ) -> ( 2nd ` g ) e. ( ( 2nd ` y ) ( Hom ` C ) ( 2nd ` z ) ) )
198	1, 16, 30, 167, 168, 169, 3, 4, 171, 172, 174, 175, 178, 180, 186, 188, 195, 197	hofcllem 16898	. . . . 5 |- ( ( ph /\ ( x e. ( ( Base ` C ) X. ( Base ` C ) ) /\ y e. ( ( Base ` C ) X. ( Base ` C ) ) /\ z e. ( ( Base ` C ) X. ( Base ` C ) ) ) /\ ( f e. ( x ( Hom ` ( O X.c C ) ) y ) /\ g e. ( y ( Hom ` ( O X.c C ) ) z ) ) ) -> ( ( ( 1st ` f ) ( <. ( 1st ` z ) , ( 1st ` y ) >. (comp ` C ) ( 1st ` x ) ) ( 1st ` g ) ) ( <. ( 1st ` x ) , ( 2nd ` x ) >. ( 2nd ` M ) <. ( 1st ` z ) , ( 2nd ` z ) >. ) ( ( 2nd ` g ) ( <. ( 2nd ` x ) , ( 2nd ` y ) >. (comp ` C ) ( 2nd ` z ) ) ( 2nd ` f ) ) ) = ( ( ( 1st ` g ) ( <. ( 1st ` y ) , ( 2nd ` y ) >. ( 2nd ` M ) <. ( 1st ` z ) , ( 2nd ` z ) >. ) ( 2nd ` g ) ) ( <. ( ( 1st ` x ) ( Hom ` C ) ( 2nd ` x ) ) , ( ( 1st ` y ) ( Hom ` C ) ( 2nd ` y ) ) >. (comp ` D ) ( ( 1st ` z ) ( Hom ` C ) ( 2nd ` z ) ) ) ( ( 1st ` f ) ( <. ( 1st ` x ) , ( 2nd ` x ) >. ( 2nd ` M ) <. ( 1st ` y ) , ( 2nd ` y ) >. ) ( 2nd ` f ) ) ) )
199	170, 62	syl 17	. . . . . . . 8 |- ( ( ph /\ ( x e. ( ( Base ` C ) X. ( Base ` C ) ) /\ y e. ( ( Base ` C ) X. ( Base ` C ) ) /\ z e. ( ( Base ` C ) X. ( Base ` C ) ) ) /\ ( f e. ( x ( Hom ` ( O X.c C ) ) y ) /\ g e. ( y ( Hom ` ( O X.c C ) ) z ) ) ) -> x = <. ( 1st ` x ) , ( 2nd ` x ) >. )
200		1st2nd2 7205	. . . . . . . . 9 |- ( z e. ( ( Base ` C ) X. ( Base ` C ) ) -> z = <. ( 1st ` z ) , ( 2nd ` z ) >. )
201	176, 200	syl 17	. . . . . . . 8 |- ( ( ph /\ ( x e. ( ( Base ` C ) X. ( Base ` C ) ) /\ y e. ( ( Base ` C ) X. ( Base ` C ) ) /\ z e. ( ( Base ` C ) X. ( Base ` C ) ) ) /\ ( f e. ( x ( Hom ` ( O X.c C ) ) y ) /\ g e. ( y ( Hom ` ( O X.c C ) ) z ) ) ) -> z = <. ( 1st ` z ) , ( 2nd ` z ) >. )
202	199, 201	oveq12d 6668	. . . . . . 7 |- ( ( ph /\ ( x e. ( ( Base ` C ) X. ( Base ` C ) ) /\ y e. ( ( Base ` C ) X. ( Base ` C ) ) /\ z e. ( ( Base ` C ) X. ( Base ` C ) ) ) /\ ( f e. ( x ( Hom ` ( O X.c C ) ) y ) /\ g e. ( y ( Hom ` ( O X.c C ) ) z ) ) ) -> ( x ( 2nd ` M ) z ) = ( <. ( 1st ` x ) , ( 2nd ` x ) >. ( 2nd ` M ) <. ( 1st ` z ) , ( 2nd ` z ) >. ) )
203	173, 79	syl 17	. . . . . . . . . . 11 |- ( ( ph /\ ( x e. ( ( Base ` C ) X. ( Base ` C ) ) /\ y e. ( ( Base ` C ) X. ( Base ` C ) ) /\ z e. ( ( Base ` C ) X. ( Base ` C ) ) ) /\ ( f e. ( x ( Hom ` ( O X.c C ) ) y ) /\ g e. ( y ( Hom ` ( O X.c C ) ) z ) ) ) -> y = <. ( 1st ` y ) , ( 2nd ` y ) >. )
204	199, 203	opeq12d 4410	. . . . . . . . . 10 |- ( ( ph /\ ( x e. ( ( Base ` C ) X. ( Base ` C ) ) /\ y e. ( ( Base ` C ) X. ( Base ` C ) ) /\ z e. ( ( Base ` C ) X. ( Base ` C ) ) ) /\ ( f e. ( x ( Hom ` ( O X.c C ) ) y ) /\ g e. ( y ( Hom ` ( O X.c C ) ) z ) ) ) -> <. x , y >. = <. <. ( 1st ` x ) , ( 2nd ` x ) >. , <. ( 1st ` y ) , ( 2nd ` y ) >. >. )
205	204, 201	oveq12d 6668	. . . . . . . . 9 |- ( ( ph /\ ( x e. ( ( Base ` C ) X. ( Base ` C ) ) /\ y e. ( ( Base ` C ) X. ( Base ` C ) ) /\ z e. ( ( Base ` C ) X. ( Base ` C ) ) ) /\ ( f e. ( x ( Hom ` ( O X.c C ) ) y ) /\ g e. ( y ( Hom ` ( O X.c C ) ) z ) ) ) -> ( <. x , y >. (comp ` ( O X.c C ) ) z ) = ( <. <. ( 1st ` x ) , ( 2nd ` x ) >. , <. ( 1st ` y ) , ( 2nd ` y ) >. >. (comp ` ( O X.c C ) ) <. ( 1st ` z ) , ( 2nd ` z ) >. ) )
206		1st2nd2 7205	. . . . . . . . . 10 |- ( g e. ( ( ( 1st ` y ) ( Hom ` O ) ( 1st ` z ) ) X. ( ( 2nd ` y ) ( Hom ` C ) ( 2nd ` z ) ) ) -> g = <. ( 1st ` g ) , ( 2nd ` g ) >. )
207	191, 206	syl 17	. . . . . . . . 9 |- ( ( ph /\ ( x e. ( ( Base ` C ) X. ( Base ` C ) ) /\ y e. ( ( Base ` C ) X. ( Base ` C ) ) /\ z e. ( ( Base ` C ) X. ( Base ` C ) ) ) /\ ( f e. ( x ( Hom ` ( O X.c C ) ) y ) /\ g e. ( y ( Hom ` ( O X.c C ) ) z ) ) ) -> g = <. ( 1st ` g ) , ( 2nd ` g ) >. )
208		1st2nd2 7205	. . . . . . . . . 10 |- ( f e. ( ( ( 1st ` x ) ( Hom ` O ) ( 1st ` y ) ) X. ( ( 2nd ` x ) ( Hom ` C ) ( 2nd ` y ) ) ) -> f = <. ( 1st ` f ) , ( 2nd ` f ) >. )
209	183, 208	syl 17	. . . . . . . . 9 |- ( ( ph /\ ( x e. ( ( Base ` C ) X. ( Base ` C ) ) /\ y e. ( ( Base ` C ) X. ( Base ` C ) ) /\ z e. ( ( Base ` C ) X. ( Base ` C ) ) ) /\ ( f e. ( x ( Hom ` ( O X.c C ) ) y ) /\ g e. ( y ( Hom ` ( O X.c C ) ) z ) ) ) -> f = <. ( 1st ` f ) , ( 2nd ` f ) >. )
210	205, 207, 209	oveq123d 6671	. . . . . . . 8 |- ( ( ph /\ ( x e. ( ( Base ` C ) X. ( Base ` C ) ) /\ y e. ( ( Base ` C ) X. ( Base ` C ) ) /\ z e. ( ( Base ` C ) X. ( Base ` C ) ) ) /\ ( f e. ( x ( Hom ` ( O X.c C ) ) y ) /\ g e. ( y ( Hom ` ( O X.c C ) ) z ) ) ) -> ( g ( <. x , y >. (comp ` ( O X.c C ) ) z ) f ) = ( <. ( 1st ` g ) , ( 2nd ` g ) >. ( <. <. ( 1st ` x ) , ( 2nd ` x ) >. , <. ( 1st ` y ) , ( 2nd ` y ) >. >. (comp ` ( O X.c C ) ) <. ( 1st ` z ) , ( 2nd ` z ) >. ) <. ( 1st ` f ) , ( 2nd ` f ) >. ) )
211		eqid 2622	. . . . . . . . 9 |- (comp ` O ) = (comp ` O )
212	15, 17, 3, 116, 4, 171, 172, 174, 175, 211, 5, 24, 178, 180, 185, 188, 193, 197	xpcco2 16827	. . . . . . . 8 |- ( ( ph /\ ( x e. ( ( Base ` C ) X. ( Base ` C ) ) /\ y e. ( ( Base ` C ) X. ( Base ` C ) ) /\ z e. ( ( Base ` C ) X. ( Base ` C ) ) ) /\ ( f e. ( x ( Hom ` ( O X.c C ) ) y ) /\ g e. ( y ( Hom ` ( O X.c C ) ) z ) ) ) -> ( <. ( 1st ` g ) , ( 2nd ` g ) >. ( <. <. ( 1st ` x ) , ( 2nd ` x ) >. , <. ( 1st ` y ) , ( 2nd ` y ) >. >. (comp ` ( O X.c C ) ) <. ( 1st ` z ) , ( 2nd ` z ) >. ) <. ( 1st ` f ) , ( 2nd ` f ) >. ) = <. ( ( 1st ` g ) ( <. ( 1st ` x ) , ( 1st ` y ) >. (comp ` O ) ( 1st ` z ) ) ( 1st ` f ) ) , ( ( 2nd ` g ) ( <. ( 2nd ` x ) , ( 2nd ` y ) >. (comp ` C ) ( 2nd ` z ) ) ( 2nd ` f ) ) >. )
213	3, 5, 16, 171, 174, 178	oppcco 16377	. . . . . . . . 9 |- ( ( ph /\ ( x e. ( ( Base ` C ) X. ( Base ` C ) ) /\ y e. ( ( Base ` C ) X. ( Base ` C ) ) /\ z e. ( ( Base ` C ) X. ( Base ` C ) ) ) /\ ( f e. ( x ( Hom ` ( O X.c C ) ) y ) /\ g e. ( y ( Hom ` ( O X.c C ) ) z ) ) ) -> ( ( 1st ` g ) ( <. ( 1st ` x ) , ( 1st ` y ) >. (comp ` O ) ( 1st ` z ) ) ( 1st ` f ) ) = ( ( 1st ` f ) ( <. ( 1st ` z ) , ( 1st ` y ) >. (comp ` C ) ( 1st ` x ) ) ( 1st ` g ) ) )
214	213	opeq1d 4408	. . . . . . . 8 |- ( ( ph /\ ( x e. ( ( Base ` C ) X. ( Base ` C ) ) /\ y e. ( ( Base ` C ) X. ( Base ` C ) ) /\ z e. ( ( Base ` C ) X. ( Base ` C ) ) ) /\ ( f e. ( x ( Hom ` ( O X.c C ) ) y ) /\ g e. ( y ( Hom ` ( O X.c C ) ) z ) ) ) -> <. ( ( 1st ` g ) ( <. ( 1st ` x ) , ( 1st ` y ) >. (comp ` O ) ( 1st ` z ) ) ( 1st ` f ) ) , ( ( 2nd ` g ) ( <. ( 2nd ` x ) , ( 2nd ` y ) >. (comp ` C ) ( 2nd ` z ) ) ( 2nd ` f ) ) >. = <. ( ( 1st ` f ) ( <. ( 1st ` z ) , ( 1st ` y ) >. (comp ` C ) ( 1st ` x ) ) ( 1st ` g ) ) , ( ( 2nd ` g ) ( <. ( 2nd ` x ) , ( 2nd ` y ) >. (comp ` C ) ( 2nd ` z ) ) ( 2nd ` f ) ) >. )
215	210, 212, 214	3eqtrd 2660	. . . . . . 7 |- ( ( ph /\ ( x e. ( ( Base ` C ) X. ( Base ` C ) ) /\ y e. ( ( Base ` C ) X. ( Base ` C ) ) /\ z e. ( ( Base ` C ) X. ( Base ` C ) ) ) /\ ( f e. ( x ( Hom ` ( O X.c C ) ) y ) /\ g e. ( y ( Hom ` ( O X.c C ) ) z ) ) ) -> ( g ( <. x , y >. (comp ` ( O X.c C ) ) z ) f ) = <. ( ( 1st ` f ) ( <. ( 1st ` z ) , ( 1st ` y ) >. (comp ` C ) ( 1st ` x ) ) ( 1st ` g ) ) , ( ( 2nd ` g ) ( <. ( 2nd ` x ) , ( 2nd ` y ) >. (comp ` C ) ( 2nd ` z ) ) ( 2nd ` f ) ) >. )
216	202, 215	fveq12d 6197	. . . . . 6 |- ( ( ph /\ ( x e. ( ( Base ` C ) X. ( Base ` C ) ) /\ y e. ( ( Base ` C ) X. ( Base ` C ) ) /\ z e. ( ( Base ` C ) X. ( Base ` C ) ) ) /\ ( f e. ( x ( Hom ` ( O X.c C ) ) y ) /\ g e. ( y ( Hom ` ( O X.c C ) ) z ) ) ) -> ( ( x ( 2nd ` M ) z ) ` ( g ( <. x , y >. (comp ` ( O X.c C ) ) z ) f ) ) = ( ( <. ( 1st ` x ) , ( 2nd ` x ) >. ( 2nd ` M ) <. ( 1st ` z ) , ( 2nd ` z ) >. ) ` <. ( ( 1st ` f ) ( <. ( 1st ` z ) , ( 1st ` y ) >. (comp ` C ) ( 1st ` x ) ) ( 1st ` g ) ) , ( ( 2nd ` g ) ( <. ( 2nd ` x ) , ( 2nd ` y ) >. (comp ` C ) ( 2nd ` z ) ) ( 2nd ` f ) ) >. ) )
217		df-ov 6653	. . . . . 6 |- ( ( ( 1st ` f ) ( <. ( 1st ` z ) , ( 1st ` y ) >. (comp ` C ) ( 1st ` x ) ) ( 1st ` g ) ) ( <. ( 1st ` x ) , ( 2nd ` x ) >. ( 2nd ` M ) <. ( 1st ` z ) , ( 2nd ` z ) >. ) ( ( 2nd ` g ) ( <. ( 2nd ` x ) , ( 2nd ` y ) >. (comp ` C ) ( 2nd ` z ) ) ( 2nd ` f ) ) ) = ( ( <. ( 1st ` x ) , ( 2nd ` x ) >. ( 2nd ` M ) <. ( 1st ` z ) , ( 2nd ` z ) >. ) ` <. ( ( 1st ` f ) ( <. ( 1st ` z ) , ( 1st ` y ) >. (comp ` C ) ( 1st ` x ) ) ( 1st ` g ) ) , ( ( 2nd ` g ) ( <. ( 2nd ` x ) , ( 2nd ` y ) >. (comp ` C ) ( 2nd ` z ) ) ( 2nd ` f ) ) >. )
218	216, 217	syl6eqr 2674	. . . . 5 |- ( ( ph /\ ( x e. ( ( Base ` C ) X. ( Base ` C ) ) /\ y e. ( ( Base ` C ) X. ( Base ` C ) ) /\ z e. ( ( Base ` C ) X. ( Base ` C ) ) ) /\ ( f e. ( x ( Hom ` ( O X.c C ) ) y ) /\ g e. ( y ( Hom ` ( O X.c C ) ) z ) ) ) -> ( ( x ( 2nd ` M ) z ) ` ( g ( <. x , y >. (comp ` ( O X.c C ) ) z ) f ) ) = ( ( ( 1st ` f ) ( <. ( 1st ` z ) , ( 1st ` y ) >. (comp ` C ) ( 1st ` x ) ) ( 1st ` g ) ) ( <. ( 1st ` x ) , ( 2nd ` x ) >. ( 2nd ` M ) <. ( 1st ` z ) , ( 2nd ` z ) >. ) ( ( 2nd ` g ) ( <. ( 2nd ` x ) , ( 2nd ` y ) >. (comp ` C ) ( 2nd ` z ) ) ( 2nd ` f ) ) ) )
219	199	fveq2d 6195	. . . . . . . . . 10 |- ( ( ph /\ ( x e. ( ( Base ` C ) X. ( Base ` C ) ) /\ y e. ( ( Base ` C ) X. ( Base ` C ) ) /\ z e. ( ( Base ` C ) X. ( Base ` C ) ) ) /\ ( f e. ( x ( Hom ` ( O X.c C ) ) y ) /\ g e. ( y ( Hom ` ( O X.c C ) ) z ) ) ) -> ( ( Hom f ` C ) ` x ) = ( ( Hom f ` C ) ` <. ( 1st ` x ) , ( 2nd ` x ) >. ) )
220	219, 91	syl6eqr 2674	. . . . . . . . 9 |- ( ( ph /\ ( x e. ( ( Base ` C ) X. ( Base ` C ) ) /\ y e. ( ( Base ` C ) X. ( Base ` C ) ) /\ z e. ( ( Base ` C ) X. ( Base ` C ) ) ) /\ ( f e. ( x ( Hom ` ( O X.c C ) ) y ) /\ g e. ( y ( Hom ` ( O X.c C ) ) z ) ) ) -> ( ( Hom f ` C ) ` x ) = ( ( 1st ` x ) ( Hom f ` C ) ( 2nd ` x ) ) )
221	33, 3, 4, 171, 172	homfval 16352	. . . . . . . . 9 |- ( ( ph /\ ( x e. ( ( Base ` C ) X. ( Base ` C ) ) /\ y e. ( ( Base ` C ) X. ( Base ` C ) ) /\ z e. ( ( Base ` C ) X. ( Base ` C ) ) ) /\ ( f e. ( x ( Hom ` ( O X.c C ) ) y ) /\ g e. ( y ( Hom ` ( O X.c C ) ) z ) ) ) -> ( ( 1st ` x ) ( Hom f ` C ) ( 2nd ` x ) ) = ( ( 1st ` x ) ( Hom ` C ) ( 2nd ` x ) ) )
222	220, 221	eqtrd 2656	. . . . . . . 8 |- ( ( ph /\ ( x e. ( ( Base ` C ) X. ( Base ` C ) ) /\ y e. ( ( Base ` C ) X. ( Base ` C ) ) /\ z e. ( ( Base ` C ) X. ( Base ` C ) ) ) /\ ( f e. ( x ( Hom ` ( O X.c C ) ) y ) /\ g e. ( y ( Hom ` ( O X.c C ) ) z ) ) ) -> ( ( Hom f ` C ) ` x ) = ( ( 1st ` x ) ( Hom ` C ) ( 2nd ` x ) ) )
223	203	fveq2d 6195	. . . . . . . . . 10 |- ( ( ph /\ ( x e. ( ( Base ` C ) X. ( Base ` C ) ) /\ y e. ( ( Base ` C ) X. ( Base ` C ) ) /\ z e. ( ( Base ` C ) X. ( Base ` C ) ) ) /\ ( f e. ( x ( Hom ` ( O X.c C ) ) y ) /\ g e. ( y ( Hom ` ( O X.c C ) ) z ) ) ) -> ( ( Hom f ` C ) ` y ) = ( ( Hom f ` C ) ` <. ( 1st ` y ) , ( 2nd ` y ) >. ) )
224	223, 99	syl6eqr 2674	. . . . . . . . 9 |- ( ( ph /\ ( x e. ( ( Base ` C ) X. ( Base ` C ) ) /\ y e. ( ( Base ` C ) X. ( Base ` C ) ) /\ z e. ( ( Base ` C ) X. ( Base ` C ) ) ) /\ ( f e. ( x ( Hom ` ( O X.c C ) ) y ) /\ g e. ( y ( Hom ` ( O X.c C ) ) z ) ) ) -> ( ( Hom f ` C ) ` y ) = ( ( 1st ` y ) ( Hom f ` C ) ( 2nd ` y ) ) )
225	33, 3, 4, 174, 175	homfval 16352	. . . . . . . . 9 |- ( ( ph /\ ( x e. ( ( Base ` C ) X. ( Base ` C ) ) /\ y e. ( ( Base ` C ) X. ( Base ` C ) ) /\ z e. ( ( Base ` C ) X. ( Base ` C ) ) ) /\ ( f e. ( x ( Hom ` ( O X.c C ) ) y ) /\ g e. ( y ( Hom ` ( O X.c C ) ) z ) ) ) -> ( ( 1st ` y ) ( Hom f ` C ) ( 2nd ` y ) ) = ( ( 1st ` y ) ( Hom ` C ) ( 2nd ` y ) ) )
226	224, 225	eqtrd 2656	. . . . . . . 8 |- ( ( ph /\ ( x e. ( ( Base ` C ) X. ( Base ` C ) ) /\ y e. ( ( Base ` C ) X. ( Base ` C ) ) /\ z e. ( ( Base ` C ) X. ( Base ` C ) ) ) /\ ( f e. ( x ( Hom ` ( O X.c C ) ) y ) /\ g e. ( y ( Hom ` ( O X.c C ) ) z ) ) ) -> ( ( Hom f ` C ) ` y ) = ( ( 1st ` y ) ( Hom ` C ) ( 2nd ` y ) ) )
227	222, 226	opeq12d 4410	. . . . . . 7 |- ( ( ph /\ ( x e. ( ( Base ` C ) X. ( Base ` C ) ) /\ y e. ( ( Base ` C ) X. ( Base ` C ) ) /\ z e. ( ( Base ` C ) X. ( Base ` C ) ) ) /\ ( f e. ( x ( Hom ` ( O X.c C ) ) y ) /\ g e. ( y ( Hom ` ( O X.c C ) ) z ) ) ) -> <. ( ( Hom f ` C ) ` x ) , ( ( Hom f ` C ) ` y ) >. = <. ( ( 1st ` x ) ( Hom ` C ) ( 2nd ` x ) ) , ( ( 1st ` y ) ( Hom ` C ) ( 2nd ` y ) ) >. )
228	201	fveq2d 6195	. . . . . . . . 9 |- ( ( ph /\ ( x e. ( ( Base ` C ) X. ( Base ` C ) ) /\ y e. ( ( Base ` C ) X. ( Base ` C ) ) /\ z e. ( ( Base ` C ) X. ( Base ` C ) ) ) /\ ( f e. ( x ( Hom ` ( O X.c C ) ) y ) /\ g e. ( y ( Hom ` ( O X.c C ) ) z ) ) ) -> ( ( Hom f ` C ) ` z ) = ( ( Hom f ` C ) ` <. ( 1st ` z ) , ( 2nd ` z ) >. ) )
229		df-ov 6653	. . . . . . . . 9 |- ( ( 1st ` z ) ( Hom f ` C ) ( 2nd ` z ) ) = ( ( Hom f ` C ) ` <. ( 1st ` z ) , ( 2nd ` z ) >. )
230	228, 229	syl6eqr 2674	. . . . . . . 8 |- ( ( ph /\ ( x e. ( ( Base ` C ) X. ( Base ` C ) ) /\ y e. ( ( Base ` C ) X. ( Base ` C ) ) /\ z e. ( ( Base ` C ) X. ( Base ` C ) ) ) /\ ( f e. ( x ( Hom ` ( O X.c C ) ) y ) /\ g e. ( y ( Hom ` ( O X.c C ) ) z ) ) ) -> ( ( Hom f ` C ) ` z ) = ( ( 1st ` z ) ( Hom f ` C ) ( 2nd ` z ) ) )
231	33, 3, 4, 178, 180	homfval 16352	. . . . . . . 8 |- ( ( ph /\ ( x e. ( ( Base ` C ) X. ( Base ` C ) ) /\ y e. ( ( Base ` C ) X. ( Base ` C ) ) /\ z e. ( ( Base ` C ) X. ( Base ` C ) ) ) /\ ( f e. ( x ( Hom ` ( O X.c C ) ) y ) /\ g e. ( y ( Hom ` ( O X.c C ) ) z ) ) ) -> ( ( 1st ` z ) ( Hom f ` C ) ( 2nd ` z ) ) = ( ( 1st ` z ) ( Hom ` C ) ( 2nd ` z ) ) )
232	230, 231	eqtrd 2656	. . . . . . 7 |- ( ( ph /\ ( x e. ( ( Base ` C ) X. ( Base ` C ) ) /\ y e. ( ( Base ` C ) X. ( Base ` C ) ) /\ z e. ( ( Base ` C ) X. ( Base ` C ) ) ) /\ ( f e. ( x ( Hom ` ( O X.c C ) ) y ) /\ g e. ( y ( Hom ` ( O X.c C ) ) z ) ) ) -> ( ( Hom f ` C ) ` z ) = ( ( 1st ` z ) ( Hom ` C ) ( 2nd ` z ) ) )
233	227, 232	oveq12d 6668	. . . . . 6 |- ( ( ph /\ ( x e. ( ( Base ` C ) X. ( Base ` C ) ) /\ y e. ( ( Base ` C ) X. ( Base ` C ) ) /\ z e. ( ( Base ` C ) X. ( Base ` C ) ) ) /\ ( f e. ( x ( Hom ` ( O X.c C ) ) y ) /\ g e. ( y ( Hom ` ( O X.c C ) ) z ) ) ) -> ( <. ( ( Hom f ` C ) ` x ) , ( ( Hom f ` C ) ` y ) >. (comp ` D ) ( ( Hom f ` C ) ` z ) ) = ( <. ( ( 1st ` x ) ( Hom ` C ) ( 2nd ` x ) ) , ( ( 1st ` y ) ( Hom ` C ) ( 2nd ` y ) ) >. (comp ` D ) ( ( 1st ` z ) ( Hom ` C ) ( 2nd ` z ) ) ) )
234	203, 201	oveq12d 6668	. . . . . . . 8 |- ( ( ph /\ ( x e. ( ( Base ` C ) X. ( Base ` C ) ) /\ y e. ( ( Base ` C ) X. ( Base ` C ) ) /\ z e. ( ( Base ` C ) X. ( Base ` C ) ) ) /\ ( f e. ( x ( Hom ` ( O X.c C ) ) y ) /\ g e. ( y ( Hom ` ( O X.c C ) ) z ) ) ) -> ( y ( 2nd ` M ) z ) = ( <. ( 1st ` y ) , ( 2nd ` y ) >. ( 2nd ` M ) <. ( 1st ` z ) , ( 2nd ` z ) >. ) )
235	234, 207	fveq12d 6197	. . . . . . 7 |- ( ( ph /\ ( x e. ( ( Base ` C ) X. ( Base ` C ) ) /\ y e. ( ( Base ` C ) X. ( Base ` C ) ) /\ z e. ( ( Base ` C ) X. ( Base ` C ) ) ) /\ ( f e. ( x ( Hom ` ( O X.c C ) ) y ) /\ g e. ( y ( Hom ` ( O X.c C ) ) z ) ) ) -> ( ( y ( 2nd ` M ) z ) ` g ) = ( ( <. ( 1st ` y ) , ( 2nd ` y ) >. ( 2nd ` M ) <. ( 1st ` z ) , ( 2nd ` z ) >. ) ` <. ( 1st ` g ) , ( 2nd ` g ) >. ) )
236		df-ov 6653	. . . . . . 7 |- ( ( 1st ` g ) ( <. ( 1st ` y ) , ( 2nd ` y ) >. ( 2nd ` M ) <. ( 1st ` z ) , ( 2nd ` z ) >. ) ( 2nd ` g ) ) = ( ( <. ( 1st ` y ) , ( 2nd ` y ) >. ( 2nd ` M ) <. ( 1st ` z ) , ( 2nd ` z ) >. ) ` <. ( 1st ` g ) , ( 2nd ` g ) >. )
237	235, 236	syl6eqr 2674	. . . . . 6 |- ( ( ph /\ ( x e. ( ( Base ` C ) X. ( Base ` C ) ) /\ y e. ( ( Base ` C ) X. ( Base ` C ) ) /\ z e. ( ( Base ` C ) X. ( Base ` C ) ) ) /\ ( f e. ( x ( Hom ` ( O X.c C ) ) y ) /\ g e. ( y ( Hom ` ( O X.c C ) ) z ) ) ) -> ( ( y ( 2nd ` M ) z ) ` g ) = ( ( 1st ` g ) ( <. ( 1st ` y ) , ( 2nd ` y ) >. ( 2nd ` M ) <. ( 1st ` z ) , ( 2nd ` z ) >. ) ( 2nd ` g ) ) )
238	199, 203	oveq12d 6668	. . . . . . . 8 |- ( ( ph /\ ( x e. ( ( Base ` C ) X. ( Base ` C ) ) /\ y e. ( ( Base ` C ) X. ( Base ` C ) ) /\ z e. ( ( Base ` C ) X. ( Base ` C ) ) ) /\ ( f e. ( x ( Hom ` ( O X.c C ) ) y ) /\ g e. ( y ( Hom ` ( O X.c C ) ) z ) ) ) -> ( x ( 2nd ` M ) y ) = ( <. ( 1st ` x ) , ( 2nd ` x ) >. ( 2nd ` M ) <. ( 1st ` y ) , ( 2nd ` y ) >. ) )
239	238, 209	fveq12d 6197	. . . . . . 7 |- ( ( ph /\ ( x e. ( ( Base ` C ) X. ( Base ` C ) ) /\ y e. ( ( Base ` C ) X. ( Base ` C ) ) /\ z e. ( ( Base ` C ) X. ( Base ` C ) ) ) /\ ( f e. ( x ( Hom ` ( O X.c C ) ) y ) /\ g e. ( y ( Hom ` ( O X.c C ) ) z ) ) ) -> ( ( x ( 2nd ` M ) y ) ` f ) = ( ( <. ( 1st ` x ) , ( 2nd ` x ) >. ( 2nd ` M ) <. ( 1st ` y ) , ( 2nd ` y ) >. ) ` <. ( 1st ` f ) , ( 2nd ` f ) >. ) )
240		df-ov 6653	. . . . . . 7 |- ( ( 1st ` f ) ( <. ( 1st ` x ) , ( 2nd ` x ) >. ( 2nd ` M ) <. ( 1st ` y ) , ( 2nd ` y ) >. ) ( 2nd ` f ) ) = ( ( <. ( 1st ` x ) , ( 2nd ` x ) >. ( 2nd ` M ) <. ( 1st ` y ) , ( 2nd ` y ) >. ) ` <. ( 1st ` f ) , ( 2nd ` f ) >. )
241	239, 240	syl6eqr 2674	. . . . . 6 |- ( ( ph /\ ( x e. ( ( Base ` C ) X. ( Base ` C ) ) /\ y e. ( ( Base ` C ) X. ( Base ` C ) ) /\ z e. ( ( Base ` C ) X. ( Base ` C ) ) ) /\ ( f e. ( x ( Hom ` ( O X.c C ) ) y ) /\ g e. ( y ( Hom ` ( O X.c C ) ) z ) ) ) -> ( ( x ( 2nd ` M ) y ) ` f ) = ( ( 1st ` f ) ( <. ( 1st ` x ) , ( 2nd ` x ) >. ( 2nd ` M ) <. ( 1st ` y ) , ( 2nd ` y ) >. ) ( 2nd ` f ) ) )
242	233, 237, 241	oveq123d 6671	. . . . 5 |- ( ( ph /\ ( x e. ( ( Base ` C ) X. ( Base ` C ) ) /\ y e. ( ( Base ` C ) X. ( Base ` C ) ) /\ z e. ( ( Base ` C ) X. ( Base ` C ) ) ) /\ ( f e. ( x ( Hom ` ( O X.c C ) ) y ) /\ g e. ( y ( Hom ` ( O X.c C ) ) z ) ) ) -> ( ( ( y ( 2nd ` M ) z ) ` g ) ( <. ( ( Hom f ` C ) ` x ) , ( ( Hom f ` C ) ` y ) >. (comp ` D ) ( ( Hom f ` C ) ` z ) ) ( ( x ( 2nd ` M ) y ) ` f ) ) = ( ( ( 1st ` g ) ( <. ( 1st ` y ) , ( 2nd ` y ) >. ( 2nd ` M ) <. ( 1st ` z ) , ( 2nd ` z ) >. ) ( 2nd ` g ) ) ( <. ( ( 1st ` x ) ( Hom ` C ) ( 2nd ` x ) ) , ( ( 1st ` y ) ( Hom ` C ) ( 2nd ` y ) ) >. (comp ` D ) ( ( 1st ` z ) ( Hom ` C ) ( 2nd ` z ) ) ) ( ( 1st ` f ) ( <. ( 1st ` x ) , ( 2nd ` x ) >. ( 2nd ` M ) <. ( 1st ` y ) , ( 2nd ` y ) >. ) ( 2nd ` f ) ) ) )
243	198, 218, 242	3eqtr4d 2666	. . . 4 |- ( ( ph /\ ( x e. ( ( Base ` C ) X. ( Base ` C ) ) /\ y e. ( ( Base ` C ) X. ( Base ` C ) ) /\ z e. ( ( Base ` C ) X. ( Base ` C ) ) ) /\ ( f e. ( x ( Hom ` ( O X.c C ) ) y ) /\ g e. ( y ( Hom ` ( O X.c C ) ) z ) ) ) -> ( ( x ( 2nd ` M ) z ) ` ( g ( <. x , y >. (comp ` ( O X.c C ) ) z ) f ) ) = ( ( ( y ( 2nd ` M ) z ) ` g ) ( <. ( ( Hom f ` C ) ` x ) , ( ( Hom f ` C ) ` y ) >. (comp ` D ) ( ( Hom f ` C ) ` z ) ) ( ( x ( 2nd ` M ) y ) ` f ) ) )
244	18, 19, 20, 21, 22, 23, 24, 25, 28, 32, 41, 48, 124, 166, 243	isfuncd 16525	. . 3 |- ( ph -> ( Hom f ` C ) ( ( O X.c C ) Func D ) ( 2nd ` M ) )
245		df-br 4654	. . 3 |- ( ( Hom f ` C ) ( ( O X.c C ) Func D ) ( 2nd ` M ) <-> <. ( Hom f ` C ) , ( 2nd ` M ) >. e. ( ( O X.c C ) Func D ) )
246	244, 245	sylib 208	. 2 |- ( ph -> <. ( Hom f ` C ) , ( 2nd ` M ) >. e. ( ( O X.c C ) Func D ) )
247	14, 246	eqeltrd 2701	1 |- ( ph -> M e. ( ( O X.c C ) Func D ) )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   _Vcvv 3200   C_ wss 3574   <.cop 4183   class class class wbr 4653   |-> cmpt 4729   _I cid 5023   X. cxp 5112   ran crn 5115   |` cres 5116   Fn wfn 5883   -->wf 5884   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   1stc1st 7166   2ndc2nd 7167   Basecbs 15857   Hom chom 15952  compcco 15953   Catccat 16325   Idccid 16326   Hom _f chomf 16327  oppCatcoppc 16371   Func cfunc 16514   SetCatcsetc 16725   X._c cxpc 16808  Hom_Fchof 16888

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  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  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-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  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-pss 3590  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-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  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-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  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-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-tpos 7352  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-map 7859  df-ixp 7909  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-z 11378  df-dec 11494  df-uz 11688  df-fz 12327  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-hom 15966  df-cco 15967  df-cat 16329  df-cid 16330  df-homf 16331  df-oppc 16372  df-func 16518  df-setc 16726  df-xpc 16812  df-hof 16890

This theorem is referenced by:  oppchofcl  16900  oppcyon  16909  yonedalem1  16912  yonedalem21  16913  yonedalem22  16918