Theorem fucpropd 16637

Description: If two categories have the same set of objects, morphisms, and compositions, then they have the same functor categories. (Contributed by Mario Carneiro, 26-Jan-2017.)

Hypotheses

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

Assertion

Ref	Expression
fucpropd	|- ( ph -> ( A FuncCat C ) = ( B FuncCat D ) )

Proof of Theorem fucpropd

Dummy variables a b f g h v x are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fucpropd.1	. . . . 5 |- ( ph -> ( Hom f ` A ) = ( Hom f ` B ) )
2		fucpropd.2	. . . . 5 |- ( ph -> (compf ` A ) = (compf ` B ) )
3		fucpropd.3	. . . . 5 |- ( ph -> ( Hom f ` C ) = ( Hom f ` D ) )
4		fucpropd.4	. . . . 5 |- ( ph -> (compf ` C ) = (compf ` D ) )
5		fucpropd.a	. . . . 5 |- ( ph -> A e. Cat )
6		fucpropd.b	. . . . 5 |- ( ph -> B e. Cat )
7		fucpropd.c	. . . . 5 |- ( ph -> C e. Cat )
8		fucpropd.d	. . . . 5 |- ( ph -> D e. Cat )
9	1, 2, 3, 4, 5, 6, 7, 8	funcpropd 16560	. . . 4 |- ( ph -> ( A Func C ) = ( B Func D ) )
10	9	opeq2d 4409	. . 3 |- ( ph -> <. ( Base ` ndx ) , ( A Func C ) >. = <. ( Base ` ndx ) , ( B Func D ) >. )
11	1, 2, 3, 4, 5, 6, 7, 8	natpropd 16636	. . . 4 |- ( ph -> ( A Nat C ) = ( B Nat D ) )
12	11	opeq2d 4409	. . 3 |- ( ph -> <. ( Hom ` ndx ) , ( A Nat C ) >. = <. ( Hom ` ndx ) , ( B Nat D ) >. )
13	9	sqxpeqd 5141	. . . . 5 |- ( ph -> ( ( A Func C ) X. ( A Func C ) ) = ( ( B Func D ) X. ( B Func D ) ) )
14	9	adantr 481	. . . . 5 |- ( ( ph /\ v e. ( ( A Func C ) X. ( A Func C ) ) ) -> ( A Func C ) = ( B Func D ) )
15		nfv 1843	. . . . . 6 |- F/ f ( ph /\ ( v e. ( ( A Func C ) X. ( A Func C ) ) /\ h e. ( A Func C ) ) )
16		nfcsb1v 3549	. . . . . . 7 |- F/_ f [_ ( 1st ` v ) / f ]_ [_ ( 2nd ` v ) / g ]_ ( b e. ( g ( B Nat D ) h ) , a e. ( f ( B Nat D ) g ) |-> ( x e. ( Base ` B ) |-> ( ( b ` x ) ( <. ( ( 1st ` f ) ` x ) , ( ( 1st ` g ) ` x ) >. (comp ` D ) ( ( 1st ` h ) ` x ) ) ( a ` x ) ) ) )
17	16	a1i 11	. . . . . 6 |- ( ( ph /\ ( v e. ( ( A Func C ) X. ( A Func C ) ) /\ h e. ( A Func C ) ) ) -> F/_ f [_ ( 1st ` v ) / f ]_ [_ ( 2nd ` v ) / g ]_ ( b e. ( g ( B Nat D ) h ) , a e. ( f ( B Nat D ) g ) |-> ( x e. ( Base ` B ) |-> ( ( b ` x ) ( <. ( ( 1st ` f ) ` x ) , ( ( 1st ` g ) ` x ) >. (comp ` D ) ( ( 1st ` h ) ` x ) ) ( a ` x ) ) ) ) )
18		fvexd 6203	. . . . . 6 |- ( ( ph /\ ( v e. ( ( A Func C ) X. ( A Func C ) ) /\ h e. ( A Func C ) ) ) -> ( 1st ` v ) e. _V )
19		nfv 1843	. . . . . . . 8 |- F/ g ( ( ph /\ ( v e. ( ( A Func C ) X. ( A Func C ) ) /\ h e. ( A Func C ) ) ) /\ f = ( 1st ` v ) )
20		nfcsb1v 3549	. . . . . . . . 9 |- F/_ g [_ ( 2nd ` v ) / g ]_ ( b e. ( g ( B Nat D ) h ) , a e. ( f ( B Nat D ) g ) |-> ( x e. ( Base ` B ) |-> ( ( b ` x ) ( <. ( ( 1st ` f ) ` x ) , ( ( 1st ` g ) ` x ) >. (comp ` D ) ( ( 1st ` h ) ` x ) ) ( a ` x ) ) ) )
21	20	a1i 11	. . . . . . . 8 |- ( ( ( ph /\ ( v e. ( ( A Func C ) X. ( A Func C ) ) /\ h e. ( A Func C ) ) ) /\ f = ( 1st ` v ) ) -> F/_ g [_ ( 2nd ` v ) / g ]_ ( b e. ( g ( B Nat D ) h ) , a e. ( f ( B Nat D ) g ) |-> ( x e. ( Base ` B ) |-> ( ( b ` x ) ( <. ( ( 1st ` f ) ` x ) , ( ( 1st ` g ) ` x ) >. (comp ` D ) ( ( 1st ` h ) ` x ) ) ( a ` x ) ) ) ) )
22		fvexd 6203	. . . . . . . 8 |- ( ( ( ph /\ ( v e. ( ( A Func C ) X. ( A Func C ) ) /\ h e. ( A Func C ) ) ) /\ f = ( 1st ` v ) ) -> ( 2nd ` v ) e. _V )
23	11	ad3antrrr 766	. . . . . . . . . . 11 |- ( ( ( ( ph /\ ( v e. ( ( A Func C ) X. ( A Func C ) ) /\ h e. ( A Func C ) ) ) /\ f = ( 1st ` v ) ) /\ g = ( 2nd ` v ) ) -> ( A Nat C ) = ( B Nat D ) )
24	23	oveqd 6667	. . . . . . . . . 10 |- ( ( ( ( ph /\ ( v e. ( ( A Func C ) X. ( A Func C ) ) /\ h e. ( A Func C ) ) ) /\ f = ( 1st ` v ) ) /\ g = ( 2nd ` v ) ) -> ( g ( A Nat C ) h ) = ( g ( B Nat D ) h ) )
25	23	oveqdr 6674	. . . . . . . . . 10 |- ( ( ( ( ( ph /\ ( v e. ( ( A Func C ) X. ( A Func C ) ) /\ h e. ( A Func C ) ) ) /\ f = ( 1st ` v ) ) /\ g = ( 2nd ` v ) ) /\ b e. ( g ( A Nat C ) h ) ) -> ( f ( A Nat C ) g ) = ( f ( B Nat D ) g ) )
26	1	homfeqbas 16356	. . . . . . . . . . . 12 |- ( ph -> ( Base ` A ) = ( Base ` B ) )
27	26	ad4antr 768	. . . . . . . . . . 11 |- ( ( ( ( ( ph /\ ( v e. ( ( A Func C ) X. ( A Func C ) ) /\ h e. ( A Func C ) ) ) /\ f = ( 1st ` v ) ) /\ g = ( 2nd ` v ) ) /\ ( b e. ( g ( A Nat C ) h ) /\ a e. ( f ( A Nat C ) g ) ) ) -> ( Base ` A ) = ( Base ` B ) )
28		eqid 2622	. . . . . . . . . . . 12 |- ( Base ` C ) = ( Base ` C )
29		eqid 2622	. . . . . . . . . . . 12 |- ( Hom ` C ) = ( Hom ` C )
30		eqid 2622	. . . . . . . . . . . 12 |- (comp ` C ) = (comp ` C )
31		eqid 2622	. . . . . . . . . . . 12 |- (comp ` D ) = (comp ` D )
32	3	ad5antr 770	. . . . . . . . . . . 12 |- ( ( ( ( ( ( ph /\ ( v e. ( ( A Func C ) X. ( A Func C ) ) /\ h e. ( A Func C ) ) ) /\ f = ( 1st ` v ) ) /\ g = ( 2nd ` v ) ) /\ ( b e. ( g ( A Nat C ) h ) /\ a e. ( f ( A Nat C ) g ) ) ) /\ x e. ( Base ` A ) ) -> ( Hom f ` C ) = ( Hom f ` D ) )
33	4	ad5antr 770	. . . . . . . . . . . 12 |- ( ( ( ( ( ( ph /\ ( v e. ( ( A Func C ) X. ( A Func C ) ) /\ h e. ( A Func C ) ) ) /\ f = ( 1st ` v ) ) /\ g = ( 2nd ` v ) ) /\ ( b e. ( g ( A Nat C ) h ) /\ a e. ( f ( A Nat C ) g ) ) ) /\ x e. ( Base ` A ) ) -> (compf ` C ) = (compf ` D ) )
34		eqid 2622	. . . . . . . . . . . . . 14 |- ( Base ` A ) = ( Base ` A )
35		relfunc 16522	. . . . . . . . . . . . . . 15 |- Rel ( A Func C )
36		simpllr 799	. . . . . . . . . . . . . . . 16 |- ( ( ( ( ( ph /\ ( v e. ( ( A Func C ) X. ( A Func C ) ) /\ h e. ( A Func C ) ) ) /\ f = ( 1st ` v ) ) /\ g = ( 2nd ` v ) ) /\ ( b e. ( g ( A Nat C ) h ) /\ a e. ( f ( A Nat C ) g ) ) ) -> f = ( 1st ` v ) )
37		simp-4r 807	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ( ( ph /\ ( v e. ( ( A Func C ) X. ( A Func C ) ) /\ h e. ( A Func C ) ) ) /\ f = ( 1st ` v ) ) /\ g = ( 2nd ` v ) ) /\ ( b e. ( g ( A Nat C ) h ) /\ a e. ( f ( A Nat C ) g ) ) ) -> ( v e. ( ( A Func C ) X. ( A Func C ) ) /\ h e. ( A Func C ) ) )
38	37	simpld 475	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( ( ph /\ ( v e. ( ( A Func C ) X. ( A Func C ) ) /\ h e. ( A Func C ) ) ) /\ f = ( 1st ` v ) ) /\ g = ( 2nd ` v ) ) /\ ( b e. ( g ( A Nat C ) h ) /\ a e. ( f ( A Nat C ) g ) ) ) -> v e. ( ( A Func C ) X. ( A Func C ) ) )
39		xp1st 7198	. . . . . . . . . . . . . . . . 17 |- ( v e. ( ( A Func C ) X. ( A Func C ) ) -> ( 1st ` v ) e. ( A Func C ) )
40	38, 39	syl 17	. . . . . . . . . . . . . . . 16 |- ( ( ( ( ( ph /\ ( v e. ( ( A Func C ) X. ( A Func C ) ) /\ h e. ( A Func C ) ) ) /\ f = ( 1st ` v ) ) /\ g = ( 2nd ` v ) ) /\ ( b e. ( g ( A Nat C ) h ) /\ a e. ( f ( A Nat C ) g ) ) ) -> ( 1st ` v ) e. ( A Func C ) )
41	36, 40	eqeltrd 2701	. . . . . . . . . . . . . . 15 |- ( ( ( ( ( ph /\ ( v e. ( ( A Func C ) X. ( A Func C ) ) /\ h e. ( A Func C ) ) ) /\ f = ( 1st ` v ) ) /\ g = ( 2nd ` v ) ) /\ ( b e. ( g ( A Nat C ) h ) /\ a e. ( f ( A Nat C ) g ) ) ) -> f e. ( A Func C ) )
42		1st2ndbr 7217	. . . . . . . . . . . . . . 15 |- ( ( Rel ( A Func C ) /\ f e. ( A Func C ) ) -> ( 1st ` f ) ( A Func C ) ( 2nd ` f ) )
43	35, 41, 42	sylancr 695	. . . . . . . . . . . . . 14 |- ( ( ( ( ( ph /\ ( v e. ( ( A Func C ) X. ( A Func C ) ) /\ h e. ( A Func C ) ) ) /\ f = ( 1st ` v ) ) /\ g = ( 2nd ` v ) ) /\ ( b e. ( g ( A Nat C ) h ) /\ a e. ( f ( A Nat C ) g ) ) ) -> ( 1st ` f ) ( A Func C ) ( 2nd ` f ) )
44	34, 28, 43	funcf1 16526	. . . . . . . . . . . . 13 |- ( ( ( ( ( ph /\ ( v e. ( ( A Func C ) X. ( A Func C ) ) /\ h e. ( A Func C ) ) ) /\ f = ( 1st ` v ) ) /\ g = ( 2nd ` v ) ) /\ ( b e. ( g ( A Nat C ) h ) /\ a e. ( f ( A Nat C ) g ) ) ) -> ( 1st ` f ) : ( Base ` A ) --> ( Base ` C ) )
45	44	ffvelrnda 6359	. . . . . . . . . . . 12 |- ( ( ( ( ( ( ph /\ ( v e. ( ( A Func C ) X. ( A Func C ) ) /\ h e. ( A Func C ) ) ) /\ f = ( 1st ` v ) ) /\ g = ( 2nd ` v ) ) /\ ( b e. ( g ( A Nat C ) h ) /\ a e. ( f ( A Nat C ) g ) ) ) /\ x e. ( Base ` A ) ) -> ( ( 1st ` f ) ` x ) e. ( Base ` C ) )
46		simplr 792	. . . . . . . . . . . . . . . 16 |- ( ( ( ( ( ph /\ ( v e. ( ( A Func C ) X. ( A Func C ) ) /\ h e. ( A Func C ) ) ) /\ f = ( 1st ` v ) ) /\ g = ( 2nd ` v ) ) /\ ( b e. ( g ( A Nat C ) h ) /\ a e. ( f ( A Nat C ) g ) ) ) -> g = ( 2nd ` v ) )
47		xp2nd 7199	. . . . . . . . . . . . . . . . 17 |- ( v e. ( ( A Func C ) X. ( A Func C ) ) -> ( 2nd ` v ) e. ( A Func C ) )
48	38, 47	syl 17	. . . . . . . . . . . . . . . 16 |- ( ( ( ( ( ph /\ ( v e. ( ( A Func C ) X. ( A Func C ) ) /\ h e. ( A Func C ) ) ) /\ f = ( 1st ` v ) ) /\ g = ( 2nd ` v ) ) /\ ( b e. ( g ( A Nat C ) h ) /\ a e. ( f ( A Nat C ) g ) ) ) -> ( 2nd ` v ) e. ( A Func C ) )
49	46, 48	eqeltrd 2701	. . . . . . . . . . . . . . 15 |- ( ( ( ( ( ph /\ ( v e. ( ( A Func C ) X. ( A Func C ) ) /\ h e. ( A Func C ) ) ) /\ f = ( 1st ` v ) ) /\ g = ( 2nd ` v ) ) /\ ( b e. ( g ( A Nat C ) h ) /\ a e. ( f ( A Nat C ) g ) ) ) -> g e. ( A Func C ) )
50		1st2ndbr 7217	. . . . . . . . . . . . . . 15 |- ( ( Rel ( A Func C ) /\ g e. ( A Func C ) ) -> ( 1st ` g ) ( A Func C ) ( 2nd ` g ) )
51	35, 49, 50	sylancr 695	. . . . . . . . . . . . . 14 |- ( ( ( ( ( ph /\ ( v e. ( ( A Func C ) X. ( A Func C ) ) /\ h e. ( A Func C ) ) ) /\ f = ( 1st ` v ) ) /\ g = ( 2nd ` v ) ) /\ ( b e. ( g ( A Nat C ) h ) /\ a e. ( f ( A Nat C ) g ) ) ) -> ( 1st ` g ) ( A Func C ) ( 2nd ` g ) )
52	34, 28, 51	funcf1 16526	. . . . . . . . . . . . 13 |- ( ( ( ( ( ph /\ ( v e. ( ( A Func C ) X. ( A Func C ) ) /\ h e. ( A Func C ) ) ) /\ f = ( 1st ` v ) ) /\ g = ( 2nd ` v ) ) /\ ( b e. ( g ( A Nat C ) h ) /\ a e. ( f ( A Nat C ) g ) ) ) -> ( 1st ` g ) : ( Base ` A ) --> ( Base ` C ) )
53	52	ffvelrnda 6359	. . . . . . . . . . . 12 |- ( ( ( ( ( ( ph /\ ( v e. ( ( A Func C ) X. ( A Func C ) ) /\ h e. ( A Func C ) ) ) /\ f = ( 1st ` v ) ) /\ g = ( 2nd ` v ) ) /\ ( b e. ( g ( A Nat C ) h ) /\ a e. ( f ( A Nat C ) g ) ) ) /\ x e. ( Base ` A ) ) -> ( ( 1st ` g ) ` x ) e. ( Base ` C ) )
54	37	simprd 479	. . . . . . . . . . . . . . 15 |- ( ( ( ( ( ph /\ ( v e. ( ( A Func C ) X. ( A Func C ) ) /\ h e. ( A Func C ) ) ) /\ f = ( 1st ` v ) ) /\ g = ( 2nd ` v ) ) /\ ( b e. ( g ( A Nat C ) h ) /\ a e. ( f ( A Nat C ) g ) ) ) -> h e. ( A Func C ) )
55		1st2ndbr 7217	. . . . . . . . . . . . . . 15 |- ( ( Rel ( A Func C ) /\ h e. ( A Func C ) ) -> ( 1st ` h ) ( A Func C ) ( 2nd ` h ) )
56	35, 54, 55	sylancr 695	. . . . . . . . . . . . . 14 |- ( ( ( ( ( ph /\ ( v e. ( ( A Func C ) X. ( A Func C ) ) /\ h e. ( A Func C ) ) ) /\ f = ( 1st ` v ) ) /\ g = ( 2nd ` v ) ) /\ ( b e. ( g ( A Nat C ) h ) /\ a e. ( f ( A Nat C ) g ) ) ) -> ( 1st ` h ) ( A Func C ) ( 2nd ` h ) )
57	34, 28, 56	funcf1 16526	. . . . . . . . . . . . 13 |- ( ( ( ( ( ph /\ ( v e. ( ( A Func C ) X. ( A Func C ) ) /\ h e. ( A Func C ) ) ) /\ f = ( 1st ` v ) ) /\ g = ( 2nd ` v ) ) /\ ( b e. ( g ( A Nat C ) h ) /\ a e. ( f ( A Nat C ) g ) ) ) -> ( 1st ` h ) : ( Base ` A ) --> ( Base ` C ) )
58	57	ffvelrnda 6359	. . . . . . . . . . . 12 |- ( ( ( ( ( ( ph /\ ( v e. ( ( A Func C ) X. ( A Func C ) ) /\ h e. ( A Func C ) ) ) /\ f = ( 1st ` v ) ) /\ g = ( 2nd ` v ) ) /\ ( b e. ( g ( A Nat C ) h ) /\ a e. ( f ( A Nat C ) g ) ) ) /\ x e. ( Base ` A ) ) -> ( ( 1st ` h ) ` x ) e. ( Base ` C ) )
59		eqid 2622	. . . . . . . . . . . . 13 |- ( A Nat C ) = ( A Nat C )
60		simplrr 801	. . . . . . . . . . . . . 14 |- ( ( ( ( ( ( ph /\ ( v e. ( ( A Func C ) X. ( A Func C ) ) /\ h e. ( A Func C ) ) ) /\ f = ( 1st ` v ) ) /\ g = ( 2nd ` v ) ) /\ ( b e. ( g ( A Nat C ) h ) /\ a e. ( f ( A Nat C ) g ) ) ) /\ x e. ( Base ` A ) ) -> a e. ( f ( A Nat C ) g ) )
61	59, 60	nat1st2nd 16611	. . . . . . . . . . . . 13 |- ( ( ( ( ( ( ph /\ ( v e. ( ( A Func C ) X. ( A Func C ) ) /\ h e. ( A Func C ) ) ) /\ f = ( 1st ` v ) ) /\ g = ( 2nd ` v ) ) /\ ( b e. ( g ( A Nat C ) h ) /\ a e. ( f ( A Nat C ) g ) ) ) /\ x e. ( Base ` A ) ) -> a e. ( <. ( 1st ` f ) , ( 2nd ` f ) >. ( A Nat C ) <. ( 1st ` g ) , ( 2nd ` g ) >. ) )
62		simpr 477	. . . . . . . . . . . . 13 |- ( ( ( ( ( ( ph /\ ( v e. ( ( A Func C ) X. ( A Func C ) ) /\ h e. ( A Func C ) ) ) /\ f = ( 1st ` v ) ) /\ g = ( 2nd ` v ) ) /\ ( b e. ( g ( A Nat C ) h ) /\ a e. ( f ( A Nat C ) g ) ) ) /\ x e. ( Base ` A ) ) -> x e. ( Base ` A ) )
63	59, 61, 34, 29, 62	natcl 16613	. . . . . . . . . . . 12 |- ( ( ( ( ( ( ph /\ ( v e. ( ( A Func C ) X. ( A Func C ) ) /\ h e. ( A Func C ) ) ) /\ f = ( 1st ` v ) ) /\ g = ( 2nd ` v ) ) /\ ( b e. ( g ( A Nat C ) h ) /\ a e. ( f ( A Nat C ) g ) ) ) /\ x e. ( Base ` A ) ) -> ( a ` x ) e. ( ( ( 1st ` f ) ` x ) ( Hom ` C ) ( ( 1st ` g ) ` x ) ) )
64		simplrl 800	. . . . . . . . . . . . . 14 |- ( ( ( ( ( ( ph /\ ( v e. ( ( A Func C ) X. ( A Func C ) ) /\ h e. ( A Func C ) ) ) /\ f = ( 1st ` v ) ) /\ g = ( 2nd ` v ) ) /\ ( b e. ( g ( A Nat C ) h ) /\ a e. ( f ( A Nat C ) g ) ) ) /\ x e. ( Base ` A ) ) -> b e. ( g ( A Nat C ) h ) )
65	59, 64	nat1st2nd 16611	. . . . . . . . . . . . 13 |- ( ( ( ( ( ( ph /\ ( v e. ( ( A Func C ) X. ( A Func C ) ) /\ h e. ( A Func C ) ) ) /\ f = ( 1st ` v ) ) /\ g = ( 2nd ` v ) ) /\ ( b e. ( g ( A Nat C ) h ) /\ a e. ( f ( A Nat C ) g ) ) ) /\ x e. ( Base ` A ) ) -> b e. ( <. ( 1st ` g ) , ( 2nd ` g ) >. ( A Nat C ) <. ( 1st ` h ) , ( 2nd ` h ) >. ) )
66	59, 65, 34, 29, 62	natcl 16613	. . . . . . . . . . . 12 |- ( ( ( ( ( ( ph /\ ( v e. ( ( A Func C ) X. ( A Func C ) ) /\ h e. ( A Func C ) ) ) /\ f = ( 1st ` v ) ) /\ g = ( 2nd ` v ) ) /\ ( b e. ( g ( A Nat C ) h ) /\ a e. ( f ( A Nat C ) g ) ) ) /\ x e. ( Base ` A ) ) -> ( b ` x ) e. ( ( ( 1st ` g ) ` x ) ( Hom ` C ) ( ( 1st ` h ) ` x ) ) )
67	28, 29, 30, 31, 32, 33, 45, 53, 58, 63, 66	comfeqval 16368	. . . . . . . . . . 11 |- ( ( ( ( ( ( ph /\ ( v e. ( ( A Func C ) X. ( A Func C ) ) /\ h e. ( A Func C ) ) ) /\ f = ( 1st ` v ) ) /\ g = ( 2nd ` v ) ) /\ ( b e. ( g ( A Nat C ) h ) /\ a e. ( f ( A Nat C ) g ) ) ) /\ x e. ( Base ` A ) ) -> ( ( b ` x ) ( <. ( ( 1st ` f ) ` x ) , ( ( 1st ` g ) ` x ) >. (comp ` C ) ( ( 1st ` h ) ` x ) ) ( a ` x ) ) = ( ( b ` x ) ( <. ( ( 1st ` f ) ` x ) , ( ( 1st ` g ) ` x ) >. (comp ` D ) ( ( 1st ` h ) ` x ) ) ( a ` x ) ) )
68	27, 67	mpteq12dva 4732	. . . . . . . . . 10 |- ( ( ( ( ( ph /\ ( v e. ( ( A Func C ) X. ( A Func C ) ) /\ h e. ( A Func C ) ) ) /\ f = ( 1st ` v ) ) /\ g = ( 2nd ` v ) ) /\ ( b e. ( g ( A Nat C ) h ) /\ a e. ( f ( A Nat C ) g ) ) ) -> ( x e. ( Base ` A ) |-> ( ( b ` x ) ( <. ( ( 1st ` f ) ` x ) , ( ( 1st ` g ) ` x ) >. (comp ` C ) ( ( 1st ` h ) ` x ) ) ( a ` x ) ) ) = ( x e. ( Base ` B ) |-> ( ( b ` x ) ( <. ( ( 1st ` f ) ` x ) , ( ( 1st ` g ) ` x ) >. (comp ` D ) ( ( 1st ` h ) ` x ) ) ( a ` x ) ) ) )
69	24, 25, 68	mpt2eq123dva 6716	. . . . . . . . 9 |- ( ( ( ( ph /\ ( v e. ( ( A Func C ) X. ( A Func C ) ) /\ h e. ( A Func C ) ) ) /\ f = ( 1st ` v ) ) /\ g = ( 2nd ` v ) ) -> ( b e. ( g ( A Nat C ) h ) , a e. ( f ( A Nat C ) g ) |-> ( x e. ( Base ` A ) |-> ( ( b ` x ) ( <. ( ( 1st ` f ) ` x ) , ( ( 1st ` g ) ` x ) >. (comp ` C ) ( ( 1st ` h ) ` x ) ) ( a ` x ) ) ) ) = ( b e. ( g ( B Nat D ) h ) , a e. ( f ( B Nat D ) g ) |-> ( x e. ( Base ` B ) |-> ( ( b ` x ) ( <. ( ( 1st ` f ) ` x ) , ( ( 1st ` g ) ` x ) >. (comp ` D ) ( ( 1st ` h ) ` x ) ) ( a ` x ) ) ) ) )
70		csbeq1a 3542	. . . . . . . . . 10 |- ( g = ( 2nd ` v ) -> ( b e. ( g ( B Nat D ) h ) , a e. ( f ( B Nat D ) g ) |-> ( x e. ( Base ` B ) |-> ( ( b ` x ) ( <. ( ( 1st ` f ) ` x ) , ( ( 1st ` g ) ` x ) >. (comp ` D ) ( ( 1st ` h ) ` x ) ) ( a ` x ) ) ) ) = [_ ( 2nd ` v ) / g ]_ ( b e. ( g ( B Nat D ) h ) , a e. ( f ( B Nat D ) g ) |-> ( x e. ( Base ` B ) |-> ( ( b ` x ) ( <. ( ( 1st ` f ) ` x ) , ( ( 1st ` g ) ` x ) >. (comp ` D ) ( ( 1st ` h ) ` x ) ) ( a ` x ) ) ) ) )
71	70	adantl 482	. . . . . . . . 9 |- ( ( ( ( ph /\ ( v e. ( ( A Func C ) X. ( A Func C ) ) /\ h e. ( A Func C ) ) ) /\ f = ( 1st ` v ) ) /\ g = ( 2nd ` v ) ) -> ( b e. ( g ( B Nat D ) h ) , a e. ( f ( B Nat D ) g ) |-> ( x e. ( Base ` B ) |-> ( ( b ` x ) ( <. ( ( 1st ` f ) ` x ) , ( ( 1st ` g ) ` x ) >. (comp ` D ) ( ( 1st ` h ) ` x ) ) ( a ` x ) ) ) ) = [_ ( 2nd ` v ) / g ]_ ( b e. ( g ( B Nat D ) h ) , a e. ( f ( B Nat D ) g ) |-> ( x e. ( Base ` B ) |-> ( ( b ` x ) ( <. ( ( 1st ` f ) ` x ) , ( ( 1st ` g ) ` x ) >. (comp ` D ) ( ( 1st ` h ) ` x ) ) ( a ` x ) ) ) ) )
72	69, 71	eqtrd 2656	. . . . . . . 8 |- ( ( ( ( ph /\ ( v e. ( ( A Func C ) X. ( A Func C ) ) /\ h e. ( A Func C ) ) ) /\ f = ( 1st ` v ) ) /\ g = ( 2nd ` v ) ) -> ( b e. ( g ( A Nat C ) h ) , a e. ( f ( A Nat C ) g ) |-> ( x e. ( Base ` A ) |-> ( ( b ` x ) ( <. ( ( 1st ` f ) ` x ) , ( ( 1st ` g ) ` x ) >. (comp ` C ) ( ( 1st ` h ) ` x ) ) ( a ` x ) ) ) ) = [_ ( 2nd ` v ) / g ]_ ( b e. ( g ( B Nat D ) h ) , a e. ( f ( B Nat D ) g ) |-> ( x e. ( Base ` B ) |-> ( ( b ` x ) ( <. ( ( 1st ` f ) ` x ) , ( ( 1st ` g ) ` x ) >. (comp ` D ) ( ( 1st ` h ) ` x ) ) ( a ` x ) ) ) ) )
73	19, 21, 22, 72	csbiedf 3554	. . . . . . 7 |- ( ( ( ph /\ ( v e. ( ( A Func C ) X. ( A Func C ) ) /\ h e. ( A Func C ) ) ) /\ f = ( 1st ` v ) ) -> [_ ( 2nd ` v ) / g ]_ ( b e. ( g ( A Nat C ) h ) , a e. ( f ( A Nat C ) g ) |-> ( x e. ( Base ` A ) |-> ( ( b ` x ) ( <. ( ( 1st ` f ) ` x ) , ( ( 1st ` g ) ` x ) >. (comp ` C ) ( ( 1st ` h ) ` x ) ) ( a ` x ) ) ) ) = [_ ( 2nd ` v ) / g ]_ ( b e. ( g ( B Nat D ) h ) , a e. ( f ( B Nat D ) g ) |-> ( x e. ( Base ` B ) |-> ( ( b ` x ) ( <. ( ( 1st ` f ) ` x ) , ( ( 1st ` g ) ` x ) >. (comp ` D ) ( ( 1st ` h ) ` x ) ) ( a ` x ) ) ) ) )
74		csbeq1a 3542	. . . . . . . 8 |- ( f = ( 1st ` v ) -> [_ ( 2nd ` v ) / g ]_ ( b e. ( g ( B Nat D ) h ) , a e. ( f ( B Nat D ) g ) |-> ( x e. ( Base ` B ) |-> ( ( b ` x ) ( <. ( ( 1st ` f ) ` x ) , ( ( 1st ` g ) ` x ) >. (comp ` D ) ( ( 1st ` h ) ` x ) ) ( a ` x ) ) ) ) = [_ ( 1st ` v ) / f ]_ [_ ( 2nd ` v ) / g ]_ ( b e. ( g ( B Nat D ) h ) , a e. ( f ( B Nat D ) g ) |-> ( x e. ( Base ` B ) |-> ( ( b ` x ) ( <. ( ( 1st ` f ) ` x ) , ( ( 1st ` g ) ` x ) >. (comp ` D ) ( ( 1st ` h ) ` x ) ) ( a ` x ) ) ) ) )
75	74	adantl 482	. . . . . . 7 |- ( ( ( ph /\ ( v e. ( ( A Func C ) X. ( A Func C ) ) /\ h e. ( A Func C ) ) ) /\ f = ( 1st ` v ) ) -> [_ ( 2nd ` v ) / g ]_ ( b e. ( g ( B Nat D ) h ) , a e. ( f ( B Nat D ) g ) |-> ( x e. ( Base ` B ) |-> ( ( b ` x ) ( <. ( ( 1st ` f ) ` x ) , ( ( 1st ` g ) ` x ) >. (comp ` D ) ( ( 1st ` h ) ` x ) ) ( a ` x ) ) ) ) = [_ ( 1st ` v ) / f ]_ [_ ( 2nd ` v ) / g ]_ ( b e. ( g ( B Nat D ) h ) , a e. ( f ( B Nat D ) g ) |-> ( x e. ( Base ` B ) |-> ( ( b ` x ) ( <. ( ( 1st ` f ) ` x ) , ( ( 1st ` g ) ` x ) >. (comp ` D ) ( ( 1st ` h ) ` x ) ) ( a ` x ) ) ) ) )
76	73, 75	eqtrd 2656	. . . . . 6 |- ( ( ( ph /\ ( v e. ( ( A Func C ) X. ( A Func C ) ) /\ h e. ( A Func C ) ) ) /\ f = ( 1st ` v ) ) -> [_ ( 2nd ` v ) / g ]_ ( b e. ( g ( A Nat C ) h ) , a e. ( f ( A Nat C ) g ) |-> ( x e. ( Base ` A ) |-> ( ( b ` x ) ( <. ( ( 1st ` f ) ` x ) , ( ( 1st ` g ) ` x ) >. (comp ` C ) ( ( 1st ` h ) ` x ) ) ( a ` x ) ) ) ) = [_ ( 1st ` v ) / f ]_ [_ ( 2nd ` v ) / g ]_ ( b e. ( g ( B Nat D ) h ) , a e. ( f ( B Nat D ) g ) |-> ( x e. ( Base ` B ) |-> ( ( b ` x ) ( <. ( ( 1st ` f ) ` x ) , ( ( 1st ` g ) ` x ) >. (comp ` D ) ( ( 1st ` h ) ` x ) ) ( a ` x ) ) ) ) )
77	15, 17, 18, 76	csbiedf 3554	. . . . 5 |- ( ( ph /\ ( v e. ( ( A Func C ) X. ( A Func C ) ) /\ h e. ( A Func C ) ) ) -> [_ ( 1st ` v ) / f ]_ [_ ( 2nd ` v ) / g ]_ ( b e. ( g ( A Nat C ) h ) , a e. ( f ( A Nat C ) g ) |-> ( x e. ( Base ` A ) |-> ( ( b ` x ) ( <. ( ( 1st ` f ) ` x ) , ( ( 1st ` g ) ` x ) >. (comp ` C ) ( ( 1st ` h ) ` x ) ) ( a ` x ) ) ) ) = [_ ( 1st ` v ) / f ]_ [_ ( 2nd ` v ) / g ]_ ( b e. ( g ( B Nat D ) h ) , a e. ( f ( B Nat D ) g ) |-> ( x e. ( Base ` B ) |-> ( ( b ` x ) ( <. ( ( 1st ` f ) ` x ) , ( ( 1st ` g ) ` x ) >. (comp ` D ) ( ( 1st ` h ) ` x ) ) ( a ` x ) ) ) ) )
78	13, 14, 77	mpt2eq123dva 6716	. . . 4 |- ( ph -> ( v e. ( ( A Func C ) X. ( A Func C ) ) , h e. ( A Func C ) |-> [_ ( 1st ` v ) / f ]_ [_ ( 2nd ` v ) / g ]_ ( b e. ( g ( A Nat C ) h ) , a e. ( f ( A Nat C ) g ) |-> ( x e. ( Base ` A ) |-> ( ( b ` x ) ( <. ( ( 1st ` f ) ` x ) , ( ( 1st ` g ) ` x ) >. (comp ` C ) ( ( 1st ` h ) ` x ) ) ( a ` x ) ) ) ) ) = ( v e. ( ( B Func D ) X. ( B Func D ) ) , h e. ( B Func D ) |-> [_ ( 1st ` v ) / f ]_ [_ ( 2nd ` v ) / g ]_ ( b e. ( g ( B Nat D ) h ) , a e. ( f ( B Nat D ) g ) |-> ( x e. ( Base ` B ) |-> ( ( b ` x ) ( <. ( ( 1st ` f ) ` x ) , ( ( 1st ` g ) ` x ) >. (comp ` D ) ( ( 1st ` h ) ` x ) ) ( a ` x ) ) ) ) ) )
79	78	opeq2d 4409	. . 3 |- ( ph -> <. (comp ` ndx ) , ( v e. ( ( A Func C ) X. ( A Func C ) ) , h e. ( A Func C ) |-> [_ ( 1st ` v ) / f ]_ [_ ( 2nd ` v ) / g ]_ ( b e. ( g ( A Nat C ) h ) , a e. ( f ( A Nat C ) g ) |-> ( x e. ( Base ` A ) |-> ( ( b ` x ) ( <. ( ( 1st ` f ) ` x ) , ( ( 1st ` g ) ` x ) >. (comp ` C ) ( ( 1st ` h ) ` x ) ) ( a ` x ) ) ) ) ) >. = <. (comp ` ndx ) , ( v e. ( ( B Func D ) X. ( B Func D ) ) , h e. ( B Func D ) |-> [_ ( 1st ` v ) / f ]_ [_ ( 2nd ` v ) / g ]_ ( b e. ( g ( B Nat D ) h ) , a e. ( f ( B Nat D ) g ) |-> ( x e. ( Base ` B ) |-> ( ( b ` x ) ( <. ( ( 1st ` f ) ` x ) , ( ( 1st ` g ) ` x ) >. (comp ` D ) ( ( 1st ` h ) ` x ) ) ( a ` x ) ) ) ) ) >. )
80	10, 12, 79	tpeq123d 4283	. 2 |- ( ph -> { <. ( Base ` ndx ) , ( A Func C ) >. , <. ( Hom ` ndx ) , ( A Nat C ) >. , <. (comp ` ndx ) , ( v e. ( ( A Func C ) X. ( A Func C ) ) , h e. ( A Func C ) |-> [_ ( 1st ` v ) / f ]_ [_ ( 2nd ` v ) / g ]_ ( b e. ( g ( A Nat C ) h ) , a e. ( f ( A Nat C ) g ) |-> ( x e. ( Base ` A ) |-> ( ( b ` x ) ( <. ( ( 1st ` f ) ` x ) , ( ( 1st ` g ) ` x ) >. (comp ` C ) ( ( 1st ` h ) ` x ) ) ( a ` x ) ) ) ) ) >. } = { <. ( Base ` ndx ) , ( B Func D ) >. , <. ( Hom ` ndx ) , ( B Nat D ) >. , <. (comp ` ndx ) , ( v e. ( ( B Func D ) X. ( B Func D ) ) , h e. ( B Func D ) |-> [_ ( 1st ` v ) / f ]_ [_ ( 2nd ` v ) / g ]_ ( b e. ( g ( B Nat D ) h ) , a e. ( f ( B Nat D ) g ) |-> ( x e. ( Base ` B ) |-> ( ( b ` x ) ( <. ( ( 1st ` f ) ` x ) , ( ( 1st ` g ) ` x ) >. (comp ` D ) ( ( 1st ` h ) ` x ) ) ( a ` x ) ) ) ) ) >. } )
81		eqid 2622	. . 3 |- ( A FuncCat C ) = ( A FuncCat C )
82		eqid 2622	. . 3 |- ( A Func C ) = ( A Func C )
83		eqidd 2623	. . 3 |- ( ph -> ( v e. ( ( A Func C ) X. ( A Func C ) ) , h e. ( A Func C ) |-> [_ ( 1st ` v ) / f ]_ [_ ( 2nd ` v ) / g ]_ ( b e. ( g ( A Nat C ) h ) , a e. ( f ( A Nat C ) g ) |-> ( x e. ( Base ` A ) |-> ( ( b ` x ) ( <. ( ( 1st ` f ) ` x ) , ( ( 1st ` g ) ` x ) >. (comp ` C ) ( ( 1st ` h ) ` x ) ) ( a ` x ) ) ) ) ) = ( v e. ( ( A Func C ) X. ( A Func C ) ) , h e. ( A Func C ) |-> [_ ( 1st ` v ) / f ]_ [_ ( 2nd ` v ) / g ]_ ( b e. ( g ( A Nat C ) h ) , a e. ( f ( A Nat C ) g ) |-> ( x e. ( Base ` A ) |-> ( ( b ` x ) ( <. ( ( 1st ` f ) ` x ) , ( ( 1st ` g ) ` x ) >. (comp ` C ) ( ( 1st ` h ) ` x ) ) ( a ` x ) ) ) ) ) )
84	81, 82, 59, 34, 30, 5, 7, 83	fucval 16618	. 2 |- ( ph -> ( A FuncCat C ) = { <. ( Base ` ndx ) , ( A Func C ) >. , <. ( Hom ` ndx ) , ( A Nat C ) >. , <. (comp ` ndx ) , ( v e. ( ( A Func C ) X. ( A Func C ) ) , h e. ( A Func C ) |-> [_ ( 1st ` v ) / f ]_ [_ ( 2nd ` v ) / g ]_ ( b e. ( g ( A Nat C ) h ) , a e. ( f ( A Nat C ) g ) |-> ( x e. ( Base ` A ) |-> ( ( b ` x ) ( <. ( ( 1st ` f ) ` x ) , ( ( 1st ` g ) ` x ) >. (comp ` C ) ( ( 1st ` h ) ` x ) ) ( a ` x ) ) ) ) ) >. } )
85		eqid 2622	. . 3 |- ( B FuncCat D ) = ( B FuncCat D )
86		eqid 2622	. . 3 |- ( B Func D ) = ( B Func D )
87		eqid 2622	. . 3 |- ( B Nat D ) = ( B Nat D )
88		eqid 2622	. . 3 |- ( Base ` B ) = ( Base ` B )
89		eqidd 2623	. . 3 |- ( ph -> ( v e. ( ( B Func D ) X. ( B Func D ) ) , h e. ( B Func D ) |-> [_ ( 1st ` v ) / f ]_ [_ ( 2nd ` v ) / g ]_ ( b e. ( g ( B Nat D ) h ) , a e. ( f ( B Nat D ) g ) |-> ( x e. ( Base ` B ) |-> ( ( b ` x ) ( <. ( ( 1st ` f ) ` x ) , ( ( 1st ` g ) ` x ) >. (comp ` D ) ( ( 1st ` h ) ` x ) ) ( a ` x ) ) ) ) ) = ( v e. ( ( B Func D ) X. ( B Func D ) ) , h e. ( B Func D ) |-> [_ ( 1st ` v ) / f ]_ [_ ( 2nd ` v ) / g ]_ ( b e. ( g ( B Nat D ) h ) , a e. ( f ( B Nat D ) g ) |-> ( x e. ( Base ` B ) |-> ( ( b ` x ) ( <. ( ( 1st ` f ) ` x ) , ( ( 1st ` g ) ` x ) >. (comp ` D ) ( ( 1st ` h ) ` x ) ) ( a ` x ) ) ) ) ) )
90	85, 86, 87, 88, 31, 6, 8, 89	fucval 16618	. 2 |- ( ph -> ( B FuncCat D ) = { <. ( Base ` ndx ) , ( B Func D ) >. , <. ( Hom ` ndx ) , ( B Nat D ) >. , <. (comp ` ndx ) , ( v e. ( ( B Func D ) X. ( B Func D ) ) , h e. ( B Func D ) |-> [_ ( 1st ` v ) / f ]_ [_ ( 2nd ` v ) / g ]_ ( b e. ( g ( B Nat D ) h ) , a e. ( f ( B Nat D ) g ) |-> ( x e. ( Base ` B ) |-> ( ( b ` x ) ( <. ( ( 1st ` f ) ` x ) , ( ( 1st ` g ) ` x ) >. (comp ` D ) ( ( 1st ` h ) ` x ) ) ( a ` x ) ) ) ) ) >. } )
91	80, 84, 90	3eqtr4d 2666	1 |- ( ph -> ( A FuncCat C ) = ( B FuncCat D ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   F/_wnfc 2751   _Vcvv 3200   [_csb 3533   {ctp 4181   <.cop 4183   class class class wbr 4653   |-> cmpt 4729   X. cxp 5112   Rel wrel 5119   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   1stc1st 7166   2ndc2nd 7167   ndxcnx 15854   Basecbs 15857   Hom chom 15952  compcco 15953   Catccat 16325   Hom _f chomf 16327  comp_fccomf 16328   Func cfunc 16514   Nat cnat 16601   FuncCat cfuc 16602

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-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-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-map 7859  df-ixp 7909  df-cat 16329  df-cid 16330  df-homf 16331  df-comf 16332  df-func 16518  df-nat 16603  df-fuc 16604

This theorem is referenced by:  oyoncl  16910