Theorem fucpropd 14872

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 14795	. . . 4 |- ( ph -> ( A Func C ) = ( B Func D ) )
10	9	opeq2d 4056	. . 3 |- ( ph -> <. ( Base ` ndx ) , ( A Func C ) >. = <. ( Base ` ndx ) , ( B Func D ) >. )
11	1, 2, 3, 4, 5, 6, 7, 8	natpropd 14871	. . . 4 |- ( ph -> ( A Nat C ) = ( B Nat D ) )
12	11	opeq2d 4056	. . 3 |- ( ph -> <. ( Hom ` ndx ) , ( A Nat C ) >. = <. ( Hom ` ndx ) , ( B Nat D ) >. )
13	9, 9	xpeq12d 4854	. . . . 5 |- ( ph -> ( ( A Func C ) X. ( A Func C ) ) = ( ( B Func D ) X. ( B Func D ) ) )
14	9	adantr 462	. . . . 5 |- ( ( ph /\ v e. ( ( A Func C ) X. ( A Func C ) ) ) -> ( A Func C ) = ( B Func D ) )
15		nfv 1674	. . . . . 6 |- F/ f ( ph /\ ( v e. ( ( A Func C ) X. ( A Func C ) ) /\ h e. ( A Func C ) ) )
16		nfcsb1v 3294	. . . . . . 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		fvex 5691	. . . . . . 7 |- ( 1st ` v ) e. _V
19	18	a1i 11	. . . . . 6 |- ( ( ph /\ ( v e. ( ( A Func C ) X. ( A Func C ) ) /\ h e. ( A Func C ) ) ) -> ( 1st ` v ) e. _V )
20		nfv 1674	. . . . . . . 8 |- F/ g ( ( ph /\ ( v e. ( ( A Func C ) X. ( A Func C ) ) /\ h e. ( A Func C ) ) ) /\ f = ( 1st ` v ) )
21		nfcsb1v 3294	. . . . . . . . 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 ) ) ) )
22	21	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 ) ) ) ) )
23		fvex 5691	. . . . . . . . 9 |- ( 2nd ` v ) e. _V
24	23	a1i 11	. . . . . . . 8 |- ( ( ( ph /\ ( v e. ( ( A Func C ) X. ( A Func C ) ) /\ h e. ( A Func C ) ) ) /\ f = ( 1st ` v ) ) -> ( 2nd ` v ) e. _V )
25	11	ad3antrrr 724	. . . . . . . . . . 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 ) )
26	25	oveqd 6099	. . . . . . . . . 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 ) )
27	25	proplem3 14614	. . . . . . . . . 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 ) )
28	1	homfeqbas 14620	. . . . . . . . . . . 12 |- ( ph -> ( Base ` A ) = ( Base ` B ) )
29	28	ad4antr 726	. . . . . . . . . . 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 ) )
30		eqid 2435	. . . . . . . . . . . 12 |- ( Base ` C ) = ( Base ` C )
31		eqid 2435	. . . . . . . . . . . 12 |- ( Hom ` C ) = ( Hom ` C )
32		eqid 2435	. . . . . . . . . . . 12 |- (comp ` C ) = (comp ` C )
33		eqid 2435	. . . . . . . . . . . 12 |- (comp ` D ) = (comp ` D )
34	3	ad5antr 728	. . . . . . . . . . . 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 ) )
35	4	ad5antr 728	. . . . . . . . . . . 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 ) )
36		eqid 2435	. . . . . . . . . . . . . 14 |- ( Base ` A ) = ( Base ` A )
37		relfunc 14757	. . . . . . . . . . . . . . 15 |- Rel ( A Func C )
38		simpllr 753	. . . . . . . . . . . . . . . 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 ) )
39		simp-4r 761	. . . . . . . . . . . . . . . . . 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 ) ) )
40	39	simpld 456	. . . . . . . . . . . . . . . . 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 ) ) )
41		xp1st 6597	. . . . . . . . . . . . . . . . 17 |- ( v e. ( ( A Func C ) X. ( A Func C ) ) -> ( 1st ` v ) e. ( A Func C ) )
42	40, 41	syl 16	. . . . . . . . . . . . . . . 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 ) )
43	38, 42	eqeltrd 2509	. . . . . . . . . . . . . . 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 ) )
44		1st2ndbr 6614	. . . . . . . . . . . . . . 15 |- ( ( Rel ( A Func C ) /\ f e. ( A Func C ) ) -> ( 1st ` f ) ( A Func C ) ( 2nd ` f ) )
45	37, 43, 44	sylancr 658	. . . . . . . . . . . . . 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 ) )
46	36, 30, 45	funcf1 14761	. . . . . . . . . . . . 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 ) )
47	46	ffvelrnda 5833	. . . . . . . . . . . 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 ) )
48		simplr 749	. . . . . . . . . . . . . . . 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 ) )
49		xp2nd 6598	. . . . . . . . . . . . . . . . 17 |- ( v e. ( ( A Func C ) X. ( A Func C ) ) -> ( 2nd ` v ) e. ( A Func C ) )
50	40, 49	syl 16	. . . . . . . . . . . . . . . 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 ) )
51	48, 50	eqeltrd 2509	. . . . . . . . . . . . . . 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 ) )
52		1st2ndbr 6614	. . . . . . . . . . . . . . 15 |- ( ( Rel ( A Func C ) /\ g e. ( A Func C ) ) -> ( 1st ` g ) ( A Func C ) ( 2nd ` g ) )
53	37, 51, 52	sylancr 658	. . . . . . . . . . . . . 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 ) )
54	36, 30, 53	funcf1 14761	. . . . . . . . . . . . 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 ) )
55	54	ffvelrnda 5833	. . . . . . . . . . . 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 ) )
56	39	simprd 460	. . . . . . . . . . . . . . 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 ) )
57		1st2ndbr 6614	. . . . . . . . . . . . . . 15 |- ( ( Rel ( A Func C ) /\ h e. ( A Func C ) ) -> ( 1st ` h ) ( A Func C ) ( 2nd ` h ) )
58	37, 56, 57	sylancr 658	. . . . . . . . . . . . . 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 ) )
59	36, 30, 58	funcf1 14761	. . . . . . . . . . . . 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 ) )
60	59	ffvelrnda 5833	. . . . . . . . . . . 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 ) )
61		eqid 2435	. . . . . . . . . . . . 13 |- ( A Nat C ) = ( A Nat C )
62		simplrr 755	. . . . . . . . . . . . . 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 ) )
63	61, 62	nat1st2nd 14846	. . . . . . . . . . . . 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 ) >. ) )
64		simpr 458	. . . . . . . . . . . . 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 ) )
65	61, 63, 36, 31, 64	natcl 14848	. . . . . . . . . . . 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 ) ) )
66		simplrl 754	. . . . . . . . . . . . . 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 ) )
67	61, 66	nat1st2nd 14846	. . . . . . . . . . . . 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 ) >. ) )
68	61, 67, 36, 31, 64	natcl 14848	. . . . . . . . . . . 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 ) ) )
69	30, 31, 32, 33, 34, 35, 47, 55, 60, 65, 68	comfeqval 14632	. . . . . . . . . . 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 ) ) )
70	29, 69	mpteq12dva 4359	. . . . . . . . . 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 ) ) ) )
71	26, 27, 70	mpt2eq123dva 6138	. . . . . . . . 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 ) ) ) ) )
72		csbeq1a 3287	. . . . . . . . . 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 ) ) ) ) )
73	72	adantl 463	. . . . . . . . 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 ) ) ) ) )
74	71, 73	eqtrd 2467	. . . . . . . 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 ) ) ) ) )
75	20, 22, 24, 74	csbiedf 3299	. . . . . . 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 ) ) ) ) )
76		csbeq1a 3287	. . . . . . . 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 ) ) ) ) )
77	76	adantl 463	. . . . . . 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 ) ) ) ) )
78	75, 77	eqtrd 2467	. . . . . 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 ) ) ) ) )
79	15, 17, 19, 78	csbiedf 3299	. . . . 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 ) ) ) ) )
80	13, 14, 79	mpt2eq123dva 6138	. . . 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 ) ) ) ) ) )
81	80	opeq2d 4056	. . 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 ) ) ) ) ) >. )
82	10, 12, 81	tpeq123d 3959	. 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 ) ) ) ) ) >. } )
83		eqid 2435	. . 3 |- ( A FuncCat C ) = ( A FuncCat C )
84		eqid 2435	. . 3 |- ( A Func C ) = ( A Func C )
85		eqidd 2436	. . 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 ) ) ) ) ) )
86	83, 84, 61, 36, 32, 5, 7, 85	fucval 14853	. 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 ) ) ) ) ) >. } )
87		eqid 2435	. . 3 |- ( B FuncCat D ) = ( B FuncCat D )
88		eqid 2435	. . 3 |- ( B Func D ) = ( B Func D )
89		eqid 2435	. . 3 |- ( B Nat D ) = ( B Nat D )
90		eqid 2435	. . 3 |- ( Base ` B ) = ( Base ` B )
91		eqidd 2436	. . 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 ) ) ) ) ) )
92	87, 88, 89, 90, 33, 6, 8, 91	fucval 14853	. 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 ) ) ) ) ) >. } )
93	82, 86, 92	3eqtr4d 2477	1 |- ( ph -> ( A FuncCat C ) = ( B FuncCat D ) )

Syntax hints:   -> wi 4   /\ wa 369   = wceq 1364   e. wcel 1757   F/_wnfc 2558   _Vcvv 2964   [_csb 3278   {ctp 3871   <.cop 3873   class class class wbr 4282   e. cmpt 4340   X. cxp 4827   Rel wrel 4834   ` cfv 5408  (class class class)co 6082   e. cmpt2 6084   1stc1st 6566   2ndc2nd 6567   ndxcnx 14156   Basecbs 14159   Hom chom 14234  compcco 14235   Catccat 14587   Hom _f chomf 14589  comp_fccomf 14590   Func cfunc 14749   Nat cnat 14836   FuncCat cfuc 14837

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1671  ax-6 1709  ax-7 1729  ax-8 1759  ax-9 1761  ax-10 1776  ax-11 1781  ax-12 1793  ax-13 1944  ax-ext 2416  ax-rep 4393  ax-sep 4403  ax-nul 4411  ax-pow 4460  ax-pr 4521  ax-un 6363

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 962  df-tru 1367  df-fal 1370  df-ex 1592  df-nf 1595  df-sb 1702  df-eu 2260  df-mo 2261  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2966  df-sbc 3178  df-csb 3279  df-dif 3321  df-un 3323  df-in 3325  df-ss 3332  df-nul 3628  df-if 3782  df-pw 3852  df-sn 3868  df-pr 3870  df-tp 3872  df-op 3874  df-uni 4082  df-iun 4163  df-br 4283  df-opab 4341  df-mpt 4342  df-id 4625  df-xp 4835  df-rel 4836  df-cnv 4837  df-co 4838  df-dm 4839  df-rn 4840  df-res 4841  df-ima 4842  df-iota 5371  df-fun 5410  df-fn 5411  df-f 5412  df-f1 5413  df-fo 5414  df-f1o 5415  df-fv 5416  df-riota 6041  df-ov 6085  df-oprab 6086  df-mpt2 6087  df-1st 6568  df-2nd 6569  df-map 7206  df-ixp 7254  df-cat 14591  df-cid 14592  df-homf 14593  df-comf 14594  df-func 14753  df-nat 14838  df-fuc 14839

This theorem is referenced by:  oyoncl  15065