Theorem xpcpropd 16848

Description: If two categories have the same set of objects, morphisms, and compositions, then they have the same product category. (Contributed by Mario Carneiro, 17-Jan-2017.)

Hypotheses

Ref	Expression
xpcpropd.1	|- ( ph -> ( Hom f ` A ) = ( Hom f ` B ) )
xpcpropd.2	|- ( ph -> (compf ` A ) = (compf ` B ) )
xpcpropd.3	|- ( ph -> ( Hom f ` C ) = ( Hom f ` D ) )
xpcpropd.4	|- ( ph -> (compf ` C ) = (compf ` D ) )
xpcpropd.a	|- ( ph -> A e. V )
xpcpropd.b	|- ( ph -> B e. V )
xpcpropd.c	|- ( ph -> C e. V )
xpcpropd.d	|- ( ph -> D e. V )

Assertion

Ref	Expression
xpcpropd	|- ( ph -> ( A X.c C ) = ( B X.c D ) )

Proof of Theorem xpcpropd

Dummy variables f g u v x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2622	. . 3 |- ( A X.c C ) = ( A X.c C )
2		eqid 2622	. . 3 |- ( Base ` A ) = ( Base ` A )
3		eqid 2622	. . 3 |- ( Base ` C ) = ( Base ` C )
4		eqid 2622	. . 3 |- ( Hom ` A ) = ( Hom ` A )
5		eqid 2622	. . 3 |- ( Hom ` C ) = ( Hom ` C )
6		eqid 2622	. . 3 |- (comp ` A ) = (comp ` A )
7		eqid 2622	. . 3 |- (comp ` C ) = (comp ` C )
8		xpcpropd.a	. . 3 |- ( ph -> A e. V )
9		xpcpropd.c	. . 3 |- ( ph -> C e. V )
10		eqidd 2623	. . 3 |- ( ph -> ( ( Base ` A ) X. ( Base ` C ) ) = ( ( Base ` A ) X. ( Base ` C ) ) )
11	1, 2, 3	xpcbas 16818	. . . . 5 |- ( ( Base ` A ) X. ( Base ` C ) ) = ( Base ` ( A X.c C ) )
12		eqid 2622	. . . . 5 |- ( Hom ` ( A X.c C ) ) = ( Hom ` ( A X.c C ) )
13	1, 11, 4, 5, 12	xpchomfval 16819	. . . 4 |- ( Hom ` ( A X.c C ) ) = ( u e. ( ( Base ` A ) X. ( Base ` C ) ) , v e. ( ( Base ` A ) X. ( Base ` C ) ) |-> ( ( ( 1st ` u ) ( Hom ` A ) ( 1st ` v ) ) X. ( ( 2nd ` u ) ( Hom ` C ) ( 2nd ` v ) ) ) )
14	13	a1i 11	. . 3 |- ( ph -> ( Hom ` ( A X.c C ) ) = ( u e. ( ( Base ` A ) X. ( Base ` C ) ) , v e. ( ( Base ` A ) X. ( Base ` C ) ) |-> ( ( ( 1st ` u ) ( Hom ` A ) ( 1st ` v ) ) X. ( ( 2nd ` u ) ( Hom ` C ) ( 2nd ` v ) ) ) ) )
15		eqidd 2623	. . 3 |- ( ph -> ( x e. ( ( ( Base ` A ) X. ( Base ` C ) ) X. ( ( Base ` A ) X. ( Base ` C ) ) ) , y e. ( ( Base ` A ) X. ( Base ` C ) ) |-> ( g e. ( ( 2nd ` x ) ( Hom ` ( A X.c C ) ) y ) , f e. ( ( Hom ` ( A X.c C ) ) ` x ) |-> <. ( ( 1st ` g ) ( <. ( 1st ` ( 1st ` x ) ) , ( 1st ` ( 2nd ` x ) ) >. (comp ` A ) ( 1st ` y ) ) ( 1st ` f ) ) , ( ( 2nd ` g ) ( <. ( 2nd ` ( 1st ` x ) ) , ( 2nd ` ( 2nd ` x ) ) >. (comp ` C ) ( 2nd ` y ) ) ( 2nd ` f ) ) >. ) ) = ( x e. ( ( ( Base ` A ) X. ( Base ` C ) ) X. ( ( Base ` A ) X. ( Base ` C ) ) ) , y e. ( ( Base ` A ) X. ( Base ` C ) ) |-> ( g e. ( ( 2nd ` x ) ( Hom ` ( A X.c C ) ) y ) , f e. ( ( Hom ` ( A X.c C ) ) ` x ) |-> <. ( ( 1st ` g ) ( <. ( 1st ` ( 1st ` x ) ) , ( 1st ` ( 2nd ` x ) ) >. (comp ` A ) ( 1st ` y ) ) ( 1st ` f ) ) , ( ( 2nd ` g ) ( <. ( 2nd ` ( 1st ` x ) ) , ( 2nd ` ( 2nd ` x ) ) >. (comp ` C ) ( 2nd ` y ) ) ( 2nd ` f ) ) >. ) ) )
16	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 14, 15	xpcval 16817	. 2 |- ( ph -> ( A X.c C ) = { <. ( Base ` ndx ) , ( ( Base ` A ) X. ( Base ` C ) ) >. , <. ( Hom ` ndx ) , ( Hom ` ( A X.c C ) ) >. , <. (comp ` ndx ) , ( x e. ( ( ( Base ` A ) X. ( Base ` C ) ) X. ( ( Base ` A ) X. ( Base ` C ) ) ) , y e. ( ( Base ` A ) X. ( Base ` C ) ) |-> ( g e. ( ( 2nd ` x ) ( Hom ` ( A X.c C ) ) y ) , f e. ( ( Hom ` ( A X.c C ) ) ` x ) |-> <. ( ( 1st ` g ) ( <. ( 1st ` ( 1st ` x ) ) , ( 1st ` ( 2nd ` x ) ) >. (comp ` A ) ( 1st ` y ) ) ( 1st ` f ) ) , ( ( 2nd ` g ) ( <. ( 2nd ` ( 1st ` x ) ) , ( 2nd ` ( 2nd ` x ) ) >. (comp ` C ) ( 2nd ` y ) ) ( 2nd ` f ) ) >. ) ) >. } )
17		eqid 2622	. . 3 |- ( B X.c D ) = ( B X.c D )
18		eqid 2622	. . 3 |- ( Base ` B ) = ( Base ` B )
19		eqid 2622	. . 3 |- ( Base ` D ) = ( Base ` D )
20		eqid 2622	. . 3 |- ( Hom ` B ) = ( Hom ` B )
21		eqid 2622	. . 3 |- ( Hom ` D ) = ( Hom ` D )
22		eqid 2622	. . 3 |- (comp ` B ) = (comp ` B )
23		eqid 2622	. . 3 |- (comp ` D ) = (comp ` D )
24		xpcpropd.b	. . 3 |- ( ph -> B e. V )
25		xpcpropd.d	. . 3 |- ( ph -> D e. V )
26		xpcpropd.1	. . . . 5 |- ( ph -> ( Hom f ` A ) = ( Hom f ` B ) )
27	26	homfeqbas 16356	. . . 4 |- ( ph -> ( Base ` A ) = ( Base ` B ) )
28		xpcpropd.3	. . . . 5 |- ( ph -> ( Hom f ` C ) = ( Hom f ` D ) )
29	28	homfeqbas 16356	. . . 4 |- ( ph -> ( Base ` C ) = ( Base ` D ) )
30	27, 29	xpeq12d 5140	. . 3 |- ( ph -> ( ( Base ` A ) X. ( Base ` C ) ) = ( ( Base ` B ) X. ( Base ` D ) ) )
31	26	3ad2ant1 1082	. . . . . . 7 |- ( ( ph /\ u e. ( ( Base ` A ) X. ( Base ` C ) ) /\ v e. ( ( Base ` A ) X. ( Base ` C ) ) ) -> ( Hom f ` A ) = ( Hom f ` B ) )
32		xp1st 7198	. . . . . . . 8 |- ( u e. ( ( Base ` A ) X. ( Base ` C ) ) -> ( 1st ` u ) e. ( Base ` A ) )
33	32	3ad2ant2 1083	. . . . . . 7 |- ( ( ph /\ u e. ( ( Base ` A ) X. ( Base ` C ) ) /\ v e. ( ( Base ` A ) X. ( Base ` C ) ) ) -> ( 1st ` u ) e. ( Base ` A ) )
34		xp1st 7198	. . . . . . . 8 |- ( v e. ( ( Base ` A ) X. ( Base ` C ) ) -> ( 1st ` v ) e. ( Base ` A ) )
35	34	3ad2ant3 1084	. . . . . . 7 |- ( ( ph /\ u e. ( ( Base ` A ) X. ( Base ` C ) ) /\ v e. ( ( Base ` A ) X. ( Base ` C ) ) ) -> ( 1st ` v ) e. ( Base ` A ) )
36	2, 4, 20, 31, 33, 35	homfeqval 16357	. . . . . 6 |- ( ( ph /\ u e. ( ( Base ` A ) X. ( Base ` C ) ) /\ v e. ( ( Base ` A ) X. ( Base ` C ) ) ) -> ( ( 1st ` u ) ( Hom ` A ) ( 1st ` v ) ) = ( ( 1st ` u ) ( Hom ` B ) ( 1st ` v ) ) )
37	28	3ad2ant1 1082	. . . . . . 7 |- ( ( ph /\ u e. ( ( Base ` A ) X. ( Base ` C ) ) /\ v e. ( ( Base ` A ) X. ( Base ` C ) ) ) -> ( Hom f ` C ) = ( Hom f ` D ) )
38		xp2nd 7199	. . . . . . . 8 |- ( u e. ( ( Base ` A ) X. ( Base ` C ) ) -> ( 2nd ` u ) e. ( Base ` C ) )
39	38	3ad2ant2 1083	. . . . . . 7 |- ( ( ph /\ u e. ( ( Base ` A ) X. ( Base ` C ) ) /\ v e. ( ( Base ` A ) X. ( Base ` C ) ) ) -> ( 2nd ` u ) e. ( Base ` C ) )
40		xp2nd 7199	. . . . . . . 8 |- ( v e. ( ( Base ` A ) X. ( Base ` C ) ) -> ( 2nd ` v ) e. ( Base ` C ) )
41	40	3ad2ant3 1084	. . . . . . 7 |- ( ( ph /\ u e. ( ( Base ` A ) X. ( Base ` C ) ) /\ v e. ( ( Base ` A ) X. ( Base ` C ) ) ) -> ( 2nd ` v ) e. ( Base ` C ) )
42	3, 5, 21, 37, 39, 41	homfeqval 16357	. . . . . 6 |- ( ( ph /\ u e. ( ( Base ` A ) X. ( Base ` C ) ) /\ v e. ( ( Base ` A ) X. ( Base ` C ) ) ) -> ( ( 2nd ` u ) ( Hom ` C ) ( 2nd ` v ) ) = ( ( 2nd ` u ) ( Hom ` D ) ( 2nd ` v ) ) )
43	36, 42	xpeq12d 5140	. . . . 5 |- ( ( ph /\ u e. ( ( Base ` A ) X. ( Base ` C ) ) /\ v e. ( ( Base ` A ) X. ( Base ` C ) ) ) -> ( ( ( 1st ` u ) ( Hom ` A ) ( 1st ` v ) ) X. ( ( 2nd ` u ) ( Hom ` C ) ( 2nd ` v ) ) ) = ( ( ( 1st ` u ) ( Hom ` B ) ( 1st ` v ) ) X. ( ( 2nd ` u ) ( Hom ` D ) ( 2nd ` v ) ) ) )
44	43	mpt2eq3dva 6719	. . . 4 |- ( ph -> ( u e. ( ( Base ` A ) X. ( Base ` C ) ) , v e. ( ( Base ` A ) X. ( Base ` C ) ) |-> ( ( ( 1st ` u ) ( Hom ` A ) ( 1st ` v ) ) X. ( ( 2nd ` u ) ( Hom ` C ) ( 2nd ` v ) ) ) ) = ( u e. ( ( Base ` A ) X. ( Base ` C ) ) , v e. ( ( Base ` A ) X. ( Base ` C ) ) |-> ( ( ( 1st ` u ) ( Hom ` B ) ( 1st ` v ) ) X. ( ( 2nd ` u ) ( Hom ` D ) ( 2nd ` v ) ) ) ) )
45	13, 44	syl5eq 2668	. . 3 |- ( ph -> ( Hom ` ( A X.c C ) ) = ( u e. ( ( Base ` A ) X. ( Base ` C ) ) , v e. ( ( Base ` A ) X. ( Base ` C ) ) |-> ( ( ( 1st ` u ) ( Hom ` B ) ( 1st ` v ) ) X. ( ( 2nd ` u ) ( Hom ` D ) ( 2nd ` v ) ) ) ) )
46	26	ad4antr 768	. . . . . . . . 9 |- ( ( ( ( ( ph /\ x e. ( ( ( Base ` A ) X. ( Base ` C ) ) X. ( ( Base ` A ) X. ( Base ` C ) ) ) ) /\ y e. ( ( Base ` A ) X. ( Base ` C ) ) ) /\ g e. ( ( 2nd ` x ) ( Hom ` ( A X.c C ) ) y ) ) /\ f e. ( ( Hom ` ( A X.c C ) ) ` x ) ) -> ( Hom f ` A ) = ( Hom f ` B ) )
47		xpcpropd.2	. . . . . . . . . 10 |- ( ph -> (compf ` A ) = (compf ` B ) )
48	47	ad4antr 768	. . . . . . . . 9 |- ( ( ( ( ( ph /\ x e. ( ( ( Base ` A ) X. ( Base ` C ) ) X. ( ( Base ` A ) X. ( Base ` C ) ) ) ) /\ y e. ( ( Base ` A ) X. ( Base ` C ) ) ) /\ g e. ( ( 2nd ` x ) ( Hom ` ( A X.c C ) ) y ) ) /\ f e. ( ( Hom ` ( A X.c C ) ) ` x ) ) -> (compf ` A ) = (compf ` B ) )
49		simp-4r 807	. . . . . . . . . . 11 |- ( ( ( ( ( ph /\ x e. ( ( ( Base ` A ) X. ( Base ` C ) ) X. ( ( Base ` A ) X. ( Base ` C ) ) ) ) /\ y e. ( ( Base ` A ) X. ( Base ` C ) ) ) /\ g e. ( ( 2nd ` x ) ( Hom ` ( A X.c C ) ) y ) ) /\ f e. ( ( Hom ` ( A X.c C ) ) ` x ) ) -> x e. ( ( ( Base ` A ) X. ( Base ` C ) ) X. ( ( Base ` A ) X. ( Base ` C ) ) ) )
50		xp1st 7198	. . . . . . . . . . 11 |- ( x e. ( ( ( Base ` A ) X. ( Base ` C ) ) X. ( ( Base ` A ) X. ( Base ` C ) ) ) -> ( 1st ` x ) e. ( ( Base ` A ) X. ( Base ` C ) ) )
51	49, 50	syl 17	. . . . . . . . . 10 |- ( ( ( ( ( ph /\ x e. ( ( ( Base ` A ) X. ( Base ` C ) ) X. ( ( Base ` A ) X. ( Base ` C ) ) ) ) /\ y e. ( ( Base ` A ) X. ( Base ` C ) ) ) /\ g e. ( ( 2nd ` x ) ( Hom ` ( A X.c C ) ) y ) ) /\ f e. ( ( Hom ` ( A X.c C ) ) ` x ) ) -> ( 1st ` x ) e. ( ( Base ` A ) X. ( Base ` C ) ) )
52		xp1st 7198	. . . . . . . . . 10 |- ( ( 1st ` x ) e. ( ( Base ` A ) X. ( Base ` C ) ) -> ( 1st ` ( 1st ` x ) ) e. ( Base ` A ) )
53	51, 52	syl 17	. . . . . . . . 9 |- ( ( ( ( ( ph /\ x e. ( ( ( Base ` A ) X. ( Base ` C ) ) X. ( ( Base ` A ) X. ( Base ` C ) ) ) ) /\ y e. ( ( Base ` A ) X. ( Base ` C ) ) ) /\ g e. ( ( 2nd ` x ) ( Hom ` ( A X.c C ) ) y ) ) /\ f e. ( ( Hom ` ( A X.c C ) ) ` x ) ) -> ( 1st ` ( 1st ` x ) ) e. ( Base ` A ) )
54		xp2nd 7199	. . . . . . . . . . 11 |- ( x e. ( ( ( Base ` A ) X. ( Base ` C ) ) X. ( ( Base ` A ) X. ( Base ` C ) ) ) -> ( 2nd ` x ) e. ( ( Base ` A ) X. ( Base ` C ) ) )
55	49, 54	syl 17	. . . . . . . . . 10 |- ( ( ( ( ( ph /\ x e. ( ( ( Base ` A ) X. ( Base ` C ) ) X. ( ( Base ` A ) X. ( Base ` C ) ) ) ) /\ y e. ( ( Base ` A ) X. ( Base ` C ) ) ) /\ g e. ( ( 2nd ` x ) ( Hom ` ( A X.c C ) ) y ) ) /\ f e. ( ( Hom ` ( A X.c C ) ) ` x ) ) -> ( 2nd ` x ) e. ( ( Base ` A ) X. ( Base ` C ) ) )
56		xp1st 7198	. . . . . . . . . 10 |- ( ( 2nd ` x ) e. ( ( Base ` A ) X. ( Base ` C ) ) -> ( 1st ` ( 2nd ` x ) ) e. ( Base ` A ) )
57	55, 56	syl 17	. . . . . . . . 9 |- ( ( ( ( ( ph /\ x e. ( ( ( Base ` A ) X. ( Base ` C ) ) X. ( ( Base ` A ) X. ( Base ` C ) ) ) ) /\ y e. ( ( Base ` A ) X. ( Base ` C ) ) ) /\ g e. ( ( 2nd ` x ) ( Hom ` ( A X.c C ) ) y ) ) /\ f e. ( ( Hom ` ( A X.c C ) ) ` x ) ) -> ( 1st ` ( 2nd ` x ) ) e. ( Base ` A ) )
58		simpllr 799	. . . . . . . . . 10 |- ( ( ( ( ( ph /\ x e. ( ( ( Base ` A ) X. ( Base ` C ) ) X. ( ( Base ` A ) X. ( Base ` C ) ) ) ) /\ y e. ( ( Base ` A ) X. ( Base ` C ) ) ) /\ g e. ( ( 2nd ` x ) ( Hom ` ( A X.c C ) ) y ) ) /\ f e. ( ( Hom ` ( A X.c C ) ) ` x ) ) -> y e. ( ( Base ` A ) X. ( Base ` C ) ) )
59		xp1st 7198	. . . . . . . . . 10 |- ( y e. ( ( Base ` A ) X. ( Base ` C ) ) -> ( 1st ` y ) e. ( Base ` A ) )
60	58, 59	syl 17	. . . . . . . . 9 |- ( ( ( ( ( ph /\ x e. ( ( ( Base ` A ) X. ( Base ` C ) ) X. ( ( Base ` A ) X. ( Base ` C ) ) ) ) /\ y e. ( ( Base ` A ) X. ( Base ` C ) ) ) /\ g e. ( ( 2nd ` x ) ( Hom ` ( A X.c C ) ) y ) ) /\ f e. ( ( Hom ` ( A X.c C ) ) ` x ) ) -> ( 1st ` y ) e. ( Base ` A ) )
61		simpr 477	. . . . . . . . . . 11 |- ( ( ( ( ( ph /\ x e. ( ( ( Base ` A ) X. ( Base ` C ) ) X. ( ( Base ` A ) X. ( Base ` C ) ) ) ) /\ y e. ( ( Base ` A ) X. ( Base ` C ) ) ) /\ g e. ( ( 2nd ` x ) ( Hom ` ( A X.c C ) ) y ) ) /\ f e. ( ( Hom ` ( A X.c C ) ) ` x ) ) -> f e. ( ( Hom ` ( A X.c C ) ) ` x ) )
62		1st2nd2 7205	. . . . . . . . . . . . . . 15 |- ( x e. ( ( ( Base ` A ) X. ( Base ` C ) ) X. ( ( Base ` A ) X. ( Base ` C ) ) ) -> x = <. ( 1st ` x ) , ( 2nd ` x ) >. )
63	49, 62	syl 17	. . . . . . . . . . . . . 14 |- ( ( ( ( ( ph /\ x e. ( ( ( Base ` A ) X. ( Base ` C ) ) X. ( ( Base ` A ) X. ( Base ` C ) ) ) ) /\ y e. ( ( Base ` A ) X. ( Base ` C ) ) ) /\ g e. ( ( 2nd ` x ) ( Hom ` ( A X.c C ) ) y ) ) /\ f e. ( ( Hom ` ( A X.c C ) ) ` x ) ) -> x = <. ( 1st ` x ) , ( 2nd ` x ) >. )
64	63	fveq2d 6195	. . . . . . . . . . . . 13 |- ( ( ( ( ( ph /\ x e. ( ( ( Base ` A ) X. ( Base ` C ) ) X. ( ( Base ` A ) X. ( Base ` C ) ) ) ) /\ y e. ( ( Base ` A ) X. ( Base ` C ) ) ) /\ g e. ( ( 2nd ` x ) ( Hom ` ( A X.c C ) ) y ) ) /\ f e. ( ( Hom ` ( A X.c C ) ) ` x ) ) -> ( ( Hom ` ( A X.c C ) ) ` x ) = ( ( Hom ` ( A X.c C ) ) ` <. ( 1st ` x ) , ( 2nd ` x ) >. ) )
65		df-ov 6653	. . . . . . . . . . . . 13 |- ( ( 1st ` x ) ( Hom ` ( A X.c C ) ) ( 2nd ` x ) ) = ( ( Hom ` ( A X.c C ) ) ` <. ( 1st ` x ) , ( 2nd ` x ) >. )
66	64, 65	syl6eqr 2674	. . . . . . . . . . . 12 |- ( ( ( ( ( ph /\ x e. ( ( ( Base ` A ) X. ( Base ` C ) ) X. ( ( Base ` A ) X. ( Base ` C ) ) ) ) /\ y e. ( ( Base ` A ) X. ( Base ` C ) ) ) /\ g e. ( ( 2nd ` x ) ( Hom ` ( A X.c C ) ) y ) ) /\ f e. ( ( Hom ` ( A X.c C ) ) ` x ) ) -> ( ( Hom ` ( A X.c C ) ) ` x ) = ( ( 1st ` x ) ( Hom ` ( A X.c C ) ) ( 2nd ` x ) ) )
67	1, 11, 4, 5, 12, 51, 55	xpchom 16820	. . . . . . . . . . . 12 |- ( ( ( ( ( ph /\ x e. ( ( ( Base ` A ) X. ( Base ` C ) ) X. ( ( Base ` A ) X. ( Base ` C ) ) ) ) /\ y e. ( ( Base ` A ) X. ( Base ` C ) ) ) /\ g e. ( ( 2nd ` x ) ( Hom ` ( A X.c C ) ) y ) ) /\ f e. ( ( Hom ` ( A X.c C ) ) ` x ) ) -> ( ( 1st ` x ) ( Hom ` ( A X.c C ) ) ( 2nd ` x ) ) = ( ( ( 1st ` ( 1st ` x ) ) ( Hom ` A ) ( 1st ` ( 2nd ` x ) ) ) X. ( ( 2nd ` ( 1st ` x ) ) ( Hom ` C ) ( 2nd ` ( 2nd ` x ) ) ) ) )
68	66, 67	eqtrd 2656	. . . . . . . . . . 11 |- ( ( ( ( ( ph /\ x e. ( ( ( Base ` A ) X. ( Base ` C ) ) X. ( ( Base ` A ) X. ( Base ` C ) ) ) ) /\ y e. ( ( Base ` A ) X. ( Base ` C ) ) ) /\ g e. ( ( 2nd ` x ) ( Hom ` ( A X.c C ) ) y ) ) /\ f e. ( ( Hom ` ( A X.c C ) ) ` x ) ) -> ( ( Hom ` ( A X.c C ) ) ` x ) = ( ( ( 1st ` ( 1st ` x ) ) ( Hom ` A ) ( 1st ` ( 2nd ` x ) ) ) X. ( ( 2nd ` ( 1st ` x ) ) ( Hom ` C ) ( 2nd ` ( 2nd ` x ) ) ) ) )
69	61, 68	eleqtrd 2703	. . . . . . . . . 10 |- ( ( ( ( ( ph /\ x e. ( ( ( Base ` A ) X. ( Base ` C ) ) X. ( ( Base ` A ) X. ( Base ` C ) ) ) ) /\ y e. ( ( Base ` A ) X. ( Base ` C ) ) ) /\ g e. ( ( 2nd ` x ) ( Hom ` ( A X.c C ) ) y ) ) /\ f e. ( ( Hom ` ( A X.c C ) ) ` x ) ) -> f e. ( ( ( 1st ` ( 1st ` x ) ) ( Hom ` A ) ( 1st ` ( 2nd ` x ) ) ) X. ( ( 2nd ` ( 1st ` x ) ) ( Hom ` C ) ( 2nd ` ( 2nd ` x ) ) ) ) )
70		xp1st 7198	. . . . . . . . . 10 |- ( f e. ( ( ( 1st ` ( 1st ` x ) ) ( Hom ` A ) ( 1st ` ( 2nd ` x ) ) ) X. ( ( 2nd ` ( 1st ` x ) ) ( Hom ` C ) ( 2nd ` ( 2nd ` x ) ) ) ) -> ( 1st ` f ) e. ( ( 1st ` ( 1st ` x ) ) ( Hom ` A ) ( 1st ` ( 2nd ` x ) ) ) )
71	69, 70	syl 17	. . . . . . . . 9 |- ( ( ( ( ( ph /\ x e. ( ( ( Base ` A ) X. ( Base ` C ) ) X. ( ( Base ` A ) X. ( Base ` C ) ) ) ) /\ y e. ( ( Base ` A ) X. ( Base ` C ) ) ) /\ g e. ( ( 2nd ` x ) ( Hom ` ( A X.c C ) ) y ) ) /\ f e. ( ( Hom ` ( A X.c C ) ) ` x ) ) -> ( 1st ` f ) e. ( ( 1st ` ( 1st ` x ) ) ( Hom ` A ) ( 1st ` ( 2nd ` x ) ) ) )
72		simplr 792	. . . . . . . . . . 11 |- ( ( ( ( ( ph /\ x e. ( ( ( Base ` A ) X. ( Base ` C ) ) X. ( ( Base ` A ) X. ( Base ` C ) ) ) ) /\ y e. ( ( Base ` A ) X. ( Base ` C ) ) ) /\ g e. ( ( 2nd ` x ) ( Hom ` ( A X.c C ) ) y ) ) /\ f e. ( ( Hom ` ( A X.c C ) ) ` x ) ) -> g e. ( ( 2nd ` x ) ( Hom ` ( A X.c C ) ) y ) )
73	1, 11, 4, 5, 12, 55, 58	xpchom 16820	. . . . . . . . . . 11 |- ( ( ( ( ( ph /\ x e. ( ( ( Base ` A ) X. ( Base ` C ) ) X. ( ( Base ` A ) X. ( Base ` C ) ) ) ) /\ y e. ( ( Base ` A ) X. ( Base ` C ) ) ) /\ g e. ( ( 2nd ` x ) ( Hom ` ( A X.c C ) ) y ) ) /\ f e. ( ( Hom ` ( A X.c C ) ) ` x ) ) -> ( ( 2nd ` x ) ( Hom ` ( A X.c C ) ) y ) = ( ( ( 1st ` ( 2nd ` x ) ) ( Hom ` A ) ( 1st ` y ) ) X. ( ( 2nd ` ( 2nd ` x ) ) ( Hom ` C ) ( 2nd ` y ) ) ) )
74	72, 73	eleqtrd 2703	. . . . . . . . . 10 |- ( ( ( ( ( ph /\ x e. ( ( ( Base ` A ) X. ( Base ` C ) ) X. ( ( Base ` A ) X. ( Base ` C ) ) ) ) /\ y e. ( ( Base ` A ) X. ( Base ` C ) ) ) /\ g e. ( ( 2nd ` x ) ( Hom ` ( A X.c C ) ) y ) ) /\ f e. ( ( Hom ` ( A X.c C ) ) ` x ) ) -> g e. ( ( ( 1st ` ( 2nd ` x ) ) ( Hom ` A ) ( 1st ` y ) ) X. ( ( 2nd ` ( 2nd ` x ) ) ( Hom ` C ) ( 2nd ` y ) ) ) )
75		xp1st 7198	. . . . . . . . . 10 |- ( g e. ( ( ( 1st ` ( 2nd ` x ) ) ( Hom ` A ) ( 1st ` y ) ) X. ( ( 2nd ` ( 2nd ` x ) ) ( Hom ` C ) ( 2nd ` y ) ) ) -> ( 1st ` g ) e. ( ( 1st ` ( 2nd ` x ) ) ( Hom ` A ) ( 1st ` y ) ) )
76	74, 75	syl 17	. . . . . . . . 9 |- ( ( ( ( ( ph /\ x e. ( ( ( Base ` A ) X. ( Base ` C ) ) X. ( ( Base ` A ) X. ( Base ` C ) ) ) ) /\ y e. ( ( Base ` A ) X. ( Base ` C ) ) ) /\ g e. ( ( 2nd ` x ) ( Hom ` ( A X.c C ) ) y ) ) /\ f e. ( ( Hom ` ( A X.c C ) ) ` x ) ) -> ( 1st ` g ) e. ( ( 1st ` ( 2nd ` x ) ) ( Hom ` A ) ( 1st ` y ) ) )
77	2, 4, 6, 22, 46, 48, 53, 57, 60, 71, 76	comfeqval 16368	. . . . . . . 8 |- ( ( ( ( ( ph /\ x e. ( ( ( Base ` A ) X. ( Base ` C ) ) X. ( ( Base ` A ) X. ( Base ` C ) ) ) ) /\ y e. ( ( Base ` A ) X. ( Base ` C ) ) ) /\ g e. ( ( 2nd ` x ) ( Hom ` ( A X.c C ) ) y ) ) /\ f e. ( ( Hom ` ( A X.c C ) ) ` x ) ) -> ( ( 1st ` g ) ( <. ( 1st ` ( 1st ` x ) ) , ( 1st ` ( 2nd ` x ) ) >. (comp ` A ) ( 1st ` y ) ) ( 1st ` f ) ) = ( ( 1st ` g ) ( <. ( 1st ` ( 1st ` x ) ) , ( 1st ` ( 2nd ` x ) ) >. (comp ` B ) ( 1st ` y ) ) ( 1st ` f ) ) )
78	28	ad4antr 768	. . . . . . . . 9 |- ( ( ( ( ( ph /\ x e. ( ( ( Base ` A ) X. ( Base ` C ) ) X. ( ( Base ` A ) X. ( Base ` C ) ) ) ) /\ y e. ( ( Base ` A ) X. ( Base ` C ) ) ) /\ g e. ( ( 2nd ` x ) ( Hom ` ( A X.c C ) ) y ) ) /\ f e. ( ( Hom ` ( A X.c C ) ) ` x ) ) -> ( Hom f ` C ) = ( Hom f ` D ) )
79		xpcpropd.4	. . . . . . . . . 10 |- ( ph -> (compf ` C ) = (compf ` D ) )
80	79	ad4antr 768	. . . . . . . . 9 |- ( ( ( ( ( ph /\ x e. ( ( ( Base ` A ) X. ( Base ` C ) ) X. ( ( Base ` A ) X. ( Base ` C ) ) ) ) /\ y e. ( ( Base ` A ) X. ( Base ` C ) ) ) /\ g e. ( ( 2nd ` x ) ( Hom ` ( A X.c C ) ) y ) ) /\ f e. ( ( Hom ` ( A X.c C ) ) ` x ) ) -> (compf ` C ) = (compf ` D ) )
81		xp2nd 7199	. . . . . . . . . 10 |- ( ( 1st ` x ) e. ( ( Base ` A ) X. ( Base ` C ) ) -> ( 2nd ` ( 1st ` x ) ) e. ( Base ` C ) )
82	51, 81	syl 17	. . . . . . . . 9 |- ( ( ( ( ( ph /\ x e. ( ( ( Base ` A ) X. ( Base ` C ) ) X. ( ( Base ` A ) X. ( Base ` C ) ) ) ) /\ y e. ( ( Base ` A ) X. ( Base ` C ) ) ) /\ g e. ( ( 2nd ` x ) ( Hom ` ( A X.c C ) ) y ) ) /\ f e. ( ( Hom ` ( A X.c C ) ) ` x ) ) -> ( 2nd ` ( 1st ` x ) ) e. ( Base ` C ) )
83		xp2nd 7199	. . . . . . . . . 10 |- ( ( 2nd ` x ) e. ( ( Base ` A ) X. ( Base ` C ) ) -> ( 2nd ` ( 2nd ` x ) ) e. ( Base ` C ) )
84	55, 83	syl 17	. . . . . . . . 9 |- ( ( ( ( ( ph /\ x e. ( ( ( Base ` A ) X. ( Base ` C ) ) X. ( ( Base ` A ) X. ( Base ` C ) ) ) ) /\ y e. ( ( Base ` A ) X. ( Base ` C ) ) ) /\ g e. ( ( 2nd ` x ) ( Hom ` ( A X.c C ) ) y ) ) /\ f e. ( ( Hom ` ( A X.c C ) ) ` x ) ) -> ( 2nd ` ( 2nd ` x ) ) e. ( Base ` C ) )
85		xp2nd 7199	. . . . . . . . . 10 |- ( y e. ( ( Base ` A ) X. ( Base ` C ) ) -> ( 2nd ` y ) e. ( Base ` C ) )
86	58, 85	syl 17	. . . . . . . . 9 |- ( ( ( ( ( ph /\ x e. ( ( ( Base ` A ) X. ( Base ` C ) ) X. ( ( Base ` A ) X. ( Base ` C ) ) ) ) /\ y e. ( ( Base ` A ) X. ( Base ` C ) ) ) /\ g e. ( ( 2nd ` x ) ( Hom ` ( A X.c C ) ) y ) ) /\ f e. ( ( Hom ` ( A X.c C ) ) ` x ) ) -> ( 2nd ` y ) e. ( Base ` C ) )
87		xp2nd 7199	. . . . . . . . . 10 |- ( f e. ( ( ( 1st ` ( 1st ` x ) ) ( Hom ` A ) ( 1st ` ( 2nd ` x ) ) ) X. ( ( 2nd ` ( 1st ` x ) ) ( Hom ` C ) ( 2nd ` ( 2nd ` x ) ) ) ) -> ( 2nd ` f ) e. ( ( 2nd ` ( 1st ` x ) ) ( Hom ` C ) ( 2nd ` ( 2nd ` x ) ) ) )
88	69, 87	syl 17	. . . . . . . . 9 |- ( ( ( ( ( ph /\ x e. ( ( ( Base ` A ) X. ( Base ` C ) ) X. ( ( Base ` A ) X. ( Base ` C ) ) ) ) /\ y e. ( ( Base ` A ) X. ( Base ` C ) ) ) /\ g e. ( ( 2nd ` x ) ( Hom ` ( A X.c C ) ) y ) ) /\ f e. ( ( Hom ` ( A X.c C ) ) ` x ) ) -> ( 2nd ` f ) e. ( ( 2nd ` ( 1st ` x ) ) ( Hom ` C ) ( 2nd ` ( 2nd ` x ) ) ) )
89		xp2nd 7199	. . . . . . . . . 10 |- ( g e. ( ( ( 1st ` ( 2nd ` x ) ) ( Hom ` A ) ( 1st ` y ) ) X. ( ( 2nd ` ( 2nd ` x ) ) ( Hom ` C ) ( 2nd ` y ) ) ) -> ( 2nd ` g ) e. ( ( 2nd ` ( 2nd ` x ) ) ( Hom ` C ) ( 2nd ` y ) ) )
90	74, 89	syl 17	. . . . . . . . 9 |- ( ( ( ( ( ph /\ x e. ( ( ( Base ` A ) X. ( Base ` C ) ) X. ( ( Base ` A ) X. ( Base ` C ) ) ) ) /\ y e. ( ( Base ` A ) X. ( Base ` C ) ) ) /\ g e. ( ( 2nd ` x ) ( Hom ` ( A X.c C ) ) y ) ) /\ f e. ( ( Hom ` ( A X.c C ) ) ` x ) ) -> ( 2nd ` g ) e. ( ( 2nd ` ( 2nd ` x ) ) ( Hom ` C ) ( 2nd ` y ) ) )
91	3, 5, 7, 23, 78, 80, 82, 84, 86, 88, 90	comfeqval 16368	. . . . . . . 8 |- ( ( ( ( ( ph /\ x e. ( ( ( Base ` A ) X. ( Base ` C ) ) X. ( ( Base ` A ) X. ( Base ` C ) ) ) ) /\ y e. ( ( Base ` A ) X. ( Base ` C ) ) ) /\ g e. ( ( 2nd ` x ) ( Hom ` ( A X.c C ) ) y ) ) /\ f e. ( ( Hom ` ( A X.c C ) ) ` x ) ) -> ( ( 2nd ` g ) ( <. ( 2nd ` ( 1st ` x ) ) , ( 2nd ` ( 2nd ` x ) ) >. (comp ` C ) ( 2nd ` y ) ) ( 2nd ` f ) ) = ( ( 2nd ` g ) ( <. ( 2nd ` ( 1st ` x ) ) , ( 2nd ` ( 2nd ` x ) ) >. (comp ` D ) ( 2nd ` y ) ) ( 2nd ` f ) ) )
92	77, 91	opeq12d 4410	. . . . . . 7 |- ( ( ( ( ( ph /\ x e. ( ( ( Base ` A ) X. ( Base ` C ) ) X. ( ( Base ` A ) X. ( Base ` C ) ) ) ) /\ y e. ( ( Base ` A ) X. ( Base ` C ) ) ) /\ g e. ( ( 2nd ` x ) ( Hom ` ( A X.c C ) ) y ) ) /\ f e. ( ( Hom ` ( A X.c C ) ) ` x ) ) -> <. ( ( 1st ` g ) ( <. ( 1st ` ( 1st ` x ) ) , ( 1st ` ( 2nd ` x ) ) >. (comp ` A ) ( 1st ` y ) ) ( 1st ` f ) ) , ( ( 2nd ` g ) ( <. ( 2nd ` ( 1st ` x ) ) , ( 2nd ` ( 2nd ` x ) ) >. (comp ` C ) ( 2nd ` y ) ) ( 2nd ` f ) ) >. = <. ( ( 1st ` g ) ( <. ( 1st ` ( 1st ` x ) ) , ( 1st ` ( 2nd ` x ) ) >. (comp ` B ) ( 1st ` y ) ) ( 1st ` f ) ) , ( ( 2nd ` g ) ( <. ( 2nd ` ( 1st ` x ) ) , ( 2nd ` ( 2nd ` x ) ) >. (comp ` D ) ( 2nd ` y ) ) ( 2nd ` f ) ) >. )
93	92	3impa 1259	. . . . . 6 |- ( ( ( ( ph /\ x e. ( ( ( Base ` A ) X. ( Base ` C ) ) X. ( ( Base ` A ) X. ( Base ` C ) ) ) ) /\ y e. ( ( Base ` A ) X. ( Base ` C ) ) ) /\ g e. ( ( 2nd ` x ) ( Hom ` ( A X.c C ) ) y ) /\ f e. ( ( Hom ` ( A X.c C ) ) ` x ) ) -> <. ( ( 1st ` g ) ( <. ( 1st ` ( 1st ` x ) ) , ( 1st ` ( 2nd ` x ) ) >. (comp ` A ) ( 1st ` y ) ) ( 1st ` f ) ) , ( ( 2nd ` g ) ( <. ( 2nd ` ( 1st ` x ) ) , ( 2nd ` ( 2nd ` x ) ) >. (comp ` C ) ( 2nd ` y ) ) ( 2nd ` f ) ) >. = <. ( ( 1st ` g ) ( <. ( 1st ` ( 1st ` x ) ) , ( 1st ` ( 2nd ` x ) ) >. (comp ` B ) ( 1st ` y ) ) ( 1st ` f ) ) , ( ( 2nd ` g ) ( <. ( 2nd ` ( 1st ` x ) ) , ( 2nd ` ( 2nd ` x ) ) >. (comp ` D ) ( 2nd ` y ) ) ( 2nd ` f ) ) >. )
94	93	mpt2eq3dva 6719	. . . . 5 |- ( ( ( ph /\ x e. ( ( ( Base ` A ) X. ( Base ` C ) ) X. ( ( Base ` A ) X. ( Base ` C ) ) ) ) /\ y e. ( ( Base ` A ) X. ( Base ` C ) ) ) -> ( g e. ( ( 2nd ` x ) ( Hom ` ( A X.c C ) ) y ) , f e. ( ( Hom ` ( A X.c C ) ) ` x ) |-> <. ( ( 1st ` g ) ( <. ( 1st ` ( 1st ` x ) ) , ( 1st ` ( 2nd ` x ) ) >. (comp ` A ) ( 1st ` y ) ) ( 1st ` f ) ) , ( ( 2nd ` g ) ( <. ( 2nd ` ( 1st ` x ) ) , ( 2nd ` ( 2nd ` x ) ) >. (comp ` C ) ( 2nd ` y ) ) ( 2nd ` f ) ) >. ) = ( g e. ( ( 2nd ` x ) ( Hom ` ( A X.c C ) ) y ) , f e. ( ( Hom ` ( A X.c C ) ) ` x ) |-> <. ( ( 1st ` g ) ( <. ( 1st ` ( 1st ` x ) ) , ( 1st ` ( 2nd ` x ) ) >. (comp ` B ) ( 1st ` y ) ) ( 1st ` f ) ) , ( ( 2nd ` g ) ( <. ( 2nd ` ( 1st ` x ) ) , ( 2nd ` ( 2nd ` x ) ) >. (comp ` D ) ( 2nd ` y ) ) ( 2nd ` f ) ) >. ) )
95	94	3impa 1259	. . . 4 |- ( ( ph /\ x e. ( ( ( Base ` A ) X. ( Base ` C ) ) X. ( ( Base ` A ) X. ( Base ` C ) ) ) /\ y e. ( ( Base ` A ) X. ( Base ` C ) ) ) -> ( g e. ( ( 2nd ` x ) ( Hom ` ( A X.c C ) ) y ) , f e. ( ( Hom ` ( A X.c C ) ) ` x ) |-> <. ( ( 1st ` g ) ( <. ( 1st ` ( 1st ` x ) ) , ( 1st ` ( 2nd ` x ) ) >. (comp ` A ) ( 1st ` y ) ) ( 1st ` f ) ) , ( ( 2nd ` g ) ( <. ( 2nd ` ( 1st ` x ) ) , ( 2nd ` ( 2nd ` x ) ) >. (comp ` C ) ( 2nd ` y ) ) ( 2nd ` f ) ) >. ) = ( g e. ( ( 2nd ` x ) ( Hom ` ( A X.c C ) ) y ) , f e. ( ( Hom ` ( A X.c C ) ) ` x ) |-> <. ( ( 1st ` g ) ( <. ( 1st ` ( 1st ` x ) ) , ( 1st ` ( 2nd ` x ) ) >. (comp ` B ) ( 1st ` y ) ) ( 1st ` f ) ) , ( ( 2nd ` g ) ( <. ( 2nd ` ( 1st ` x ) ) , ( 2nd ` ( 2nd ` x ) ) >. (comp ` D ) ( 2nd ` y ) ) ( 2nd ` f ) ) >. ) )
96	95	mpt2eq3dva 6719	. . 3 |- ( ph -> ( x e. ( ( ( Base ` A ) X. ( Base ` C ) ) X. ( ( Base ` A ) X. ( Base ` C ) ) ) , y e. ( ( Base ` A ) X. ( Base ` C ) ) |-> ( g e. ( ( 2nd ` x ) ( Hom ` ( A X.c C ) ) y ) , f e. ( ( Hom ` ( A X.c C ) ) ` x ) |-> <. ( ( 1st ` g ) ( <. ( 1st ` ( 1st ` x ) ) , ( 1st ` ( 2nd ` x ) ) >. (comp ` A ) ( 1st ` y ) ) ( 1st ` f ) ) , ( ( 2nd ` g ) ( <. ( 2nd ` ( 1st ` x ) ) , ( 2nd ` ( 2nd ` x ) ) >. (comp ` C ) ( 2nd ` y ) ) ( 2nd ` f ) ) >. ) ) = ( x e. ( ( ( Base ` A ) X. ( Base ` C ) ) X. ( ( Base ` A ) X. ( Base ` C ) ) ) , y e. ( ( Base ` A ) X. ( Base ` C ) ) |-> ( g e. ( ( 2nd ` x ) ( Hom ` ( A X.c C ) ) y ) , f e. ( ( Hom ` ( A X.c C ) ) ` x ) |-> <. ( ( 1st ` g ) ( <. ( 1st ` ( 1st ` x ) ) , ( 1st ` ( 2nd ` x ) ) >. (comp ` B ) ( 1st ` y ) ) ( 1st ` f ) ) , ( ( 2nd ` g ) ( <. ( 2nd ` ( 1st ` x ) ) , ( 2nd ` ( 2nd ` x ) ) >. (comp ` D ) ( 2nd ` y ) ) ( 2nd ` f ) ) >. ) ) )
97	17, 18, 19, 20, 21, 22, 23, 24, 25, 30, 45, 96	xpcval 16817	. 2 |- ( ph -> ( B X.c D ) = { <. ( Base ` ndx ) , ( ( Base ` A ) X. ( Base ` C ) ) >. , <. ( Hom ` ndx ) , ( Hom ` ( A X.c C ) ) >. , <. (comp ` ndx ) , ( x e. ( ( ( Base ` A ) X. ( Base ` C ) ) X. ( ( Base ` A ) X. ( Base ` C ) ) ) , y e. ( ( Base ` A ) X. ( Base ` C ) ) |-> ( g e. ( ( 2nd ` x ) ( Hom ` ( A X.c C ) ) y ) , f e. ( ( Hom ` ( A X.c C ) ) ` x ) |-> <. ( ( 1st ` g ) ( <. ( 1st ` ( 1st ` x ) ) , ( 1st ` ( 2nd ` x ) ) >. (comp ` A ) ( 1st ` y ) ) ( 1st ` f ) ) , ( ( 2nd ` g ) ( <. ( 2nd ` ( 1st ` x ) ) , ( 2nd ` ( 2nd ` x ) ) >. (comp ` C ) ( 2nd ` y ) ) ( 2nd ` f ) ) >. ) ) >. } )
98	16, 97	eqtr4d 2659	1 |- ( ph -> ( A X.c C ) = ( B X.c D ) )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   {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   Hom _f chomf 16327  comp_fccomf 16328   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  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-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-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  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-hom 15966  df-cco 15967  df-homf 16331  df-comf 16332  df-xpc 16812

This theorem is referenced by:  curfpropd  16873  oppchofcl  16900