Theorem fucpropd 15010

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 14933	. . . 4 |- ( ph -> ( A Func C ) = ( B Func D ) )
10	9	opeq2d 4177	. . 3 |- ( ph -> <. ( Base ` ndx ) , ( A Func C ) >. = <. ( Base ` ndx ) , ( B Func D ) >. )
11	1, 2, 3, 4, 5, 6, 7, 8	natpropd 15009	. . . 4 |- ( ph -> ( A Nat C ) = ( B Nat D ) )
12	11	opeq2d 4177	. . 3 |- ( ph -> <. ( Hom ` ndx ) , ( A Nat C ) >. = <. ( Hom ` ndx ) , ( B Nat D ) >. )
13	9, 9	xpeq12d 4976	. . . . 5 |- ( ph -> ( ( A Func C ) X. ( A Func C ) ) = ( ( B Func D ) X. ( B Func D ) ) )
14	9	adantr 465	. . . . 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 3414	. . . . . . 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 5812	. . . . . . 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 3414	. . . . . . . . 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 5812	. . . . . . . . 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 729	. . . . . . . . . . 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 6220	. . . . . . . . . 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 14752	. . . . . . . . . 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 14758	. . . . . . . . . . . 12 |- ( ph -> ( Base ` A ) = ( Base ` B ) )
29	28	ad4antr 731	. . . . . . . . . . 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 2454	. . . . . . . . . . . 12 |- ( Base ` C ) = ( Base ` C )
31		eqid 2454	. . . . . . . . . . . 12 |- ( Hom ` C ) = ( Hom ` C )
32		eqid 2454	. . . . . . . . . . . 12 |- (comp ` C ) = (comp ` C )
33		eqid 2454	. . . . . . . . . . . 12 |- (comp ` D ) = (comp ` D )
34	3	ad5antr 733	. . . . . . . . . . . 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 733	. . . . . . . . . . . 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 2454	. . . . . . . . . . . . . 14 |- ( Base ` A ) = ( Base ` A )
37		relfunc 14895	. . . . . . . . . . . . . . 15 |- Rel ( A Func C )
38		simpllr 758	. . . . . . . . . . . . . . . 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 766	. . . . . . . . . . . . . . . . . 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 459	. . . . . . . . . . . . . . . . 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 6719	. . . . . . . . . . . . . . . . 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 2542	. . . . . . . . . . . . . . 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 6736	. . . . . . . . . . . . . . 15 |- ( ( Rel ( A Func C ) /\ f e. ( A Func C ) ) -> ( 1st ` f ) ( A Func C ) ( 2nd ` f ) )
45	37, 43, 44	sylancr 663	. . . . . . . . . . . . . 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 14899	. . . . . . . . . . . . 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 5955	. . . . . . . . . . . 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 754	. . . . . . . . . . . . . . . 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 6720	. . . . . . . . . . . . . . . . 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 2542	. . . . . . . . . . . . . . 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 6736	. . . . . . . . . . . . . . 15 |- ( ( Rel ( A Func C ) /\ g e. ( A Func C ) ) -> ( 1st ` g ) ( A Func C ) ( 2nd ` g ) )
53	37, 51, 52	sylancr 663	. . . . . . . . . . . . . 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 14899	. . . . . . . . . . . . 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 5955	. . . . . . . . . . . 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 463	. . . . . . . . . . . . . . 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 6736	. . . . . . . . . . . . . . 15 |- ( ( Rel ( A Func C ) /\ h e. ( A Func C ) ) -> ( 1st ` h ) ( A Func C ) ( 2nd ` h ) )
58	37, 56, 57	sylancr 663	. . . . . . . . . . . . . 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 14899	. . . . . . . . . . . . 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 5955	. . . . . . . . . . . 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 2454	. . . . . . . . . . . . 13 |- ( A Nat C ) = ( A Nat C )
62		simplrr 760	. . . . . . . . . . . . . 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 14984	. . . . . . . . . . . . 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 461	. . . . . . . . . . . . 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 14986	. . . . . . . . . . . 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 759	. . . . . . . . . . . . . 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 14984	. . . . . . . . . . . . 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 14986	. . . . . . . . . . . 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 14770	. . . . . . . . . . 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 4480	. . . . . . . . . 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 6259	. . . . . . . . 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 3407	. . . . . . . . . 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 466	. . . . . . . . 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 2495	. . . . . . . 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 3419	. . . . . . 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 3407	. . . . . . . 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 466	. . . . . . 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 2495	. . . . . 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 3419	. . . . 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 6259	. . . 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 4177	. . 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 4080	. 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 2454	. . 3 |- ( A FuncCat C ) = ( A FuncCat C )
84		eqid 2454	. . 3 |- ( A Func C ) = ( A Func C )
85		eqidd 2455	. . 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 14991	. 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 2454	. . 3 |- ( B FuncCat D ) = ( B FuncCat D )
88		eqid 2454	. . 3 |- ( B Func D ) = ( B Func D )
89		eqid 2454	. . 3 |- ( B Nat D ) = ( B Nat D )
90		eqid 2454	. . 3 |- ( Base ` B ) = ( Base ` B )
91		eqidd 2455	. . 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 14991	. 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 2505	1 |- ( ph -> ( A FuncCat C ) = ( B FuncCat D ) )

Syntax hints:   -> wi 4   /\ wa 369   = wceq 1370   e. wcel 1758   F/_wnfc 2602   _Vcvv 3078   [_csb 3398   {ctp 3992   <.cop 3994   class class class wbr 4403   |-> cmpt 4461   X. cxp 4949   Rel wrel 4956   ` cfv 5529  (class class class)co 6203   |-> cmpt2 6205   1stc1st 6688   2ndc2nd 6689   ndxcnx 14293   Basecbs 14296   Hom chom 14372  compcco 14373   Catccat 14725   Hom _f chomf 14727  comp_fccomf 14728   Func cfunc 14887   Nat cnat 14974   FuncCat cfuc 14975

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4514  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-fal 1376  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-reu 2806  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-tp 3993  df-op 3995  df-uni 4203  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-id 4747  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-riota 6164  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-1st 6690  df-2nd 6691  df-map 7329  df-ixp 7377  df-cat 14729  df-cid 14730  df-homf 14731  df-comf 14732  df-func 14891  df-nat 14976  df-fuc 14977

This theorem is referenced by:  oyoncl  15203