Theorem natpropd 16636

Description: If two categories have the same set of objects, morphisms, and compositions, then they have the same natural transformations. (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
natpropd	|- ( ph -> ( A Nat C ) = ( B Nat D ) )

Proof of Theorem natpropd

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

Step	Hyp	Ref	Expression
1		fucpropd.1	. . . 4 |- ( ph -> ( Hom f ` A ) = ( Hom f ` B ) )
2		fucpropd.2	. . . 4 |- ( ph -> (compf ` A ) = (compf ` B ) )
3		fucpropd.3	. . . 4 |- ( ph -> ( Hom f ` C ) = ( Hom f ` D ) )
4		fucpropd.4	. . . 4 |- ( ph -> (compf ` C ) = (compf ` D ) )
5		fucpropd.a	. . . 4 |- ( ph -> A e. Cat )
6		fucpropd.b	. . . 4 |- ( ph -> B e. Cat )
7		fucpropd.c	. . . 4 |- ( ph -> C e. Cat )
8		fucpropd.d	. . . 4 |- ( ph -> D e. Cat )
9	1, 2, 3, 4, 5, 6, 7, 8	funcpropd 16560	. . 3 |- ( ph -> ( A Func C ) = ( B Func D ) )
10	9	adantr 481	. . 3 |- ( ( ph /\ f e. ( A Func C ) ) -> ( A Func C ) = ( B Func D ) )
11		nfv 1843	. . . 4 |- F/ r ( ph /\ ( f e. ( A Func C ) /\ g e. ( A Func C ) ) )
12		nfcsb1v 3549	. . . . 5 |- F/_ r [_ ( 1st ` f ) / r ]_ [_ ( 1st ` g ) / s ]_ { a e. X_ x e. ( Base ` B ) ( ( r ` x ) ( Hom ` D ) ( s ` x ) ) | A. x e. ( Base ` B ) A. y e. ( Base ` B ) A. h e. ( x ( Hom ` B ) y ) ( ( a ` y ) ( <. ( r ` x ) , ( r ` y ) >. (comp ` D ) ( s ` y ) ) ( ( x ( 2nd ` f ) y ) ` h ) ) = ( ( ( x ( 2nd ` g ) y ) ` h ) ( <. ( r ` x ) , ( s ` x ) >. (comp ` D ) ( s ` y ) ) ( a ` x ) ) }
13	12	a1i 11	. . . 4 |- ( ( ph /\ ( f e. ( A Func C ) /\ g e. ( A Func C ) ) ) -> F/_ r [_ ( 1st ` f ) / r ]_ [_ ( 1st ` g ) / s ]_ { a e. X_ x e. ( Base ` B ) ( ( r ` x ) ( Hom ` D ) ( s ` x ) ) | A. x e. ( Base ` B ) A. y e. ( Base ` B ) A. h e. ( x ( Hom ` B ) y ) ( ( a ` y ) ( <. ( r ` x ) , ( r ` y ) >. (comp ` D ) ( s ` y ) ) ( ( x ( 2nd ` f ) y ) ` h ) ) = ( ( ( x ( 2nd ` g ) y ) ` h ) ( <. ( r ` x ) , ( s ` x ) >. (comp ` D ) ( s ` y ) ) ( a ` x ) ) } )
14		fvexd 6203	. . . 4 |- ( ( ph /\ ( f e. ( A Func C ) /\ g e. ( A Func C ) ) ) -> ( 1st ` f ) e. _V )
15		nfv 1843	. . . . . 6 |- F/ s ( ( ph /\ ( f e. ( A Func C ) /\ g e. ( A Func C ) ) ) /\ r = ( 1st ` f ) )
16		nfcsb1v 3549	. . . . . . 7 |- F/_ s [_ ( 1st ` g ) / s ]_ { a e. X_ x e. ( Base ` B ) ( ( r ` x ) ( Hom ` D ) ( s ` x ) ) | A. x e. ( Base ` B ) A. y e. ( Base ` B ) A. h e. ( x ( Hom ` B ) y ) ( ( a ` y ) ( <. ( r ` x ) , ( r ` y ) >. (comp ` D ) ( s ` y ) ) ( ( x ( 2nd ` f ) y ) ` h ) ) = ( ( ( x ( 2nd ` g ) y ) ` h ) ( <. ( r ` x ) , ( s ` x ) >. (comp ` D ) ( s ` y ) ) ( a ` x ) ) }
17	16	a1i 11	. . . . . 6 |- ( ( ( ph /\ ( f e. ( A Func C ) /\ g e. ( A Func C ) ) ) /\ r = ( 1st ` f ) ) -> F/_ s [_ ( 1st ` g ) / s ]_ { a e. X_ x e. ( Base ` B ) ( ( r ` x ) ( Hom ` D ) ( s ` x ) ) | A. x e. ( Base ` B ) A. y e. ( Base ` B ) A. h e. ( x ( Hom ` B ) y ) ( ( a ` y ) ( <. ( r ` x ) , ( r ` y ) >. (comp ` D ) ( s ` y ) ) ( ( x ( 2nd ` f ) y ) ` h ) ) = ( ( ( x ( 2nd ` g ) y ) ` h ) ( <. ( r ` x ) , ( s ` x ) >. (comp ` D ) ( s ` y ) ) ( a ` x ) ) } )
18		fvexd 6203	. . . . . 6 |- ( ( ( ph /\ ( f e. ( A Func C ) /\ g e. ( A Func C ) ) ) /\ r = ( 1st ` f ) ) -> ( 1st ` g ) e. _V )
19		eqid 2622	. . . . . . . . . . 11 |- ( Base ` C ) = ( Base ` C )
20		eqid 2622	. . . . . . . . . . 11 |- ( Hom ` C ) = ( Hom ` C )
21		eqid 2622	. . . . . . . . . . 11 |- ( Hom ` D ) = ( Hom ` D )
22	3	ad4antr 768	. . . . . . . . . . 11 |- ( ( ( ( ( ph /\ ( f e. ( A Func C ) /\ g e. ( A Func C ) ) ) /\ r = ( 1st ` f ) ) /\ s = ( 1st ` g ) ) /\ x e. ( Base ` A ) ) -> ( Hom f ` C ) = ( Hom f ` D ) )
23		eqid 2622	. . . . . . . . . . . . 13 |- ( Base ` A ) = ( Base ` A )
24		simplr 792	. . . . . . . . . . . . . 14 |- ( ( ( ( ph /\ ( f e. ( A Func C ) /\ g e. ( A Func C ) ) ) /\ r = ( 1st ` f ) ) /\ s = ( 1st ` g ) ) -> r = ( 1st ` f ) )
25		relfunc 16522	. . . . . . . . . . . . . . 15 |- Rel ( A Func C )
26		simpllr 799	. . . . . . . . . . . . . . . 16 |- ( ( ( ( ph /\ ( f e. ( A Func C ) /\ g e. ( A Func C ) ) ) /\ r = ( 1st ` f ) ) /\ s = ( 1st ` g ) ) -> ( f e. ( A Func C ) /\ g e. ( A Func C ) ) )
27	26	simpld 475	. . . . . . . . . . . . . . 15 |- ( ( ( ( ph /\ ( f e. ( A Func C ) /\ g e. ( A Func C ) ) ) /\ r = ( 1st ` f ) ) /\ s = ( 1st ` g ) ) -> f e. ( A Func C ) )
28		1st2ndbr 7217	. . . . . . . . . . . . . . 15 |- ( ( Rel ( A Func C ) /\ f e. ( A Func C ) ) -> ( 1st ` f ) ( A Func C ) ( 2nd ` f ) )
29	25, 27, 28	sylancr 695	. . . . . . . . . . . . . 14 |- ( ( ( ( ph /\ ( f e. ( A Func C ) /\ g e. ( A Func C ) ) ) /\ r = ( 1st ` f ) ) /\ s = ( 1st ` g ) ) -> ( 1st ` f ) ( A Func C ) ( 2nd ` f ) )
30	24, 29	eqbrtrd 4675	. . . . . . . . . . . . 13 |- ( ( ( ( ph /\ ( f e. ( A Func C ) /\ g e. ( A Func C ) ) ) /\ r = ( 1st ` f ) ) /\ s = ( 1st ` g ) ) -> r ( A Func C ) ( 2nd ` f ) )
31	23, 19, 30	funcf1 16526	. . . . . . . . . . . 12 |- ( ( ( ( ph /\ ( f e. ( A Func C ) /\ g e. ( A Func C ) ) ) /\ r = ( 1st ` f ) ) /\ s = ( 1st ` g ) ) -> r : ( Base ` A ) --> ( Base ` C ) )
32	31	ffvelrnda 6359	. . . . . . . . . . 11 |- ( ( ( ( ( ph /\ ( f e. ( A Func C ) /\ g e. ( A Func C ) ) ) /\ r = ( 1st ` f ) ) /\ s = ( 1st ` g ) ) /\ x e. ( Base ` A ) ) -> ( r ` x ) e. ( Base ` C ) )
33		simpr 477	. . . . . . . . . . . . . 14 |- ( ( ( ( ph /\ ( f e. ( A Func C ) /\ g e. ( A Func C ) ) ) /\ r = ( 1st ` f ) ) /\ s = ( 1st ` g ) ) -> s = ( 1st ` g ) )
34	26	simprd 479	. . . . . . . . . . . . . . 15 |- ( ( ( ( ph /\ ( f e. ( A Func C ) /\ g e. ( A Func C ) ) ) /\ r = ( 1st ` f ) ) /\ s = ( 1st ` g ) ) -> g e. ( A Func C ) )
35		1st2ndbr 7217	. . . . . . . . . . . . . . 15 |- ( ( Rel ( A Func C ) /\ g e. ( A Func C ) ) -> ( 1st ` g ) ( A Func C ) ( 2nd ` g ) )
36	25, 34, 35	sylancr 695	. . . . . . . . . . . . . 14 |- ( ( ( ( ph /\ ( f e. ( A Func C ) /\ g e. ( A Func C ) ) ) /\ r = ( 1st ` f ) ) /\ s = ( 1st ` g ) ) -> ( 1st ` g ) ( A Func C ) ( 2nd ` g ) )
37	33, 36	eqbrtrd 4675	. . . . . . . . . . . . 13 |- ( ( ( ( ph /\ ( f e. ( A Func C ) /\ g e. ( A Func C ) ) ) /\ r = ( 1st ` f ) ) /\ s = ( 1st ` g ) ) -> s ( A Func C ) ( 2nd ` g ) )
38	23, 19, 37	funcf1 16526	. . . . . . . . . . . 12 |- ( ( ( ( ph /\ ( f e. ( A Func C ) /\ g e. ( A Func C ) ) ) /\ r = ( 1st ` f ) ) /\ s = ( 1st ` g ) ) -> s : ( Base ` A ) --> ( Base ` C ) )
39	38	ffvelrnda 6359	. . . . . . . . . . 11 |- ( ( ( ( ( ph /\ ( f e. ( A Func C ) /\ g e. ( A Func C ) ) ) /\ r = ( 1st ` f ) ) /\ s = ( 1st ` g ) ) /\ x e. ( Base ` A ) ) -> ( s ` x ) e. ( Base ` C ) )
40	19, 20, 21, 22, 32, 39	homfeqval 16357	. . . . . . . . . 10 |- ( ( ( ( ( ph /\ ( f e. ( A Func C ) /\ g e. ( A Func C ) ) ) /\ r = ( 1st ` f ) ) /\ s = ( 1st ` g ) ) /\ x e. ( Base ` A ) ) -> ( ( r ` x ) ( Hom ` C ) ( s ` x ) ) = ( ( r ` x ) ( Hom ` D ) ( s ` x ) ) )
41	40	ixpeq2dva 7923	. . . . . . . . 9 |- ( ( ( ( ph /\ ( f e. ( A Func C ) /\ g e. ( A Func C ) ) ) /\ r = ( 1st ` f ) ) /\ s = ( 1st ` g ) ) -> X_ x e. ( Base ` A ) ( ( r ` x ) ( Hom ` C ) ( s ` x ) ) = X_ x e. ( Base ` A ) ( ( r ` x ) ( Hom ` D ) ( s ` x ) ) )
42	1	homfeqbas 16356	. . . . . . . . . . 11 |- ( ph -> ( Base ` A ) = ( Base ` B ) )
43	42	ad3antrrr 766	. . . . . . . . . 10 |- ( ( ( ( ph /\ ( f e. ( A Func C ) /\ g e. ( A Func C ) ) ) /\ r = ( 1st ` f ) ) /\ s = ( 1st ` g ) ) -> ( Base ` A ) = ( Base ` B ) )
44	43	ixpeq1d 7920	. . . . . . . . 9 |- ( ( ( ( ph /\ ( f e. ( A Func C ) /\ g e. ( A Func C ) ) ) /\ r = ( 1st ` f ) ) /\ s = ( 1st ` g ) ) -> X_ x e. ( Base ` A ) ( ( r ` x ) ( Hom ` D ) ( s ` x ) ) = X_ x e. ( Base ` B ) ( ( r ` x ) ( Hom ` D ) ( s ` x ) ) )
45	41, 44	eqtrd 2656	. . . . . . . 8 |- ( ( ( ( ph /\ ( f e. ( A Func C ) /\ g e. ( A Func C ) ) ) /\ r = ( 1st ` f ) ) /\ s = ( 1st ` g ) ) -> X_ x e. ( Base ` A ) ( ( r ` x ) ( Hom ` C ) ( s ` x ) ) = X_ x e. ( Base ` B ) ( ( r ` x ) ( Hom ` D ) ( s ` x ) ) )
46		fveq2 6191	. . . . . . . . . . . 12 |- ( x = z -> ( r ` x ) = ( r ` z ) )
47		fveq2 6191	. . . . . . . . . . . 12 |- ( x = z -> ( s ` x ) = ( s ` z ) )
48	46, 47	oveq12d 6668	. . . . . . . . . . 11 |- ( x = z -> ( ( r ` x ) ( Hom ` C ) ( s ` x ) ) = ( ( r ` z ) ( Hom ` C ) ( s ` z ) ) )
49	48	cbvixpv 7926	. . . . . . . . . 10 |- X_ x e. ( Base ` A ) ( ( r ` x ) ( Hom ` C ) ( s ` x ) ) = X_ z e. ( Base ` A ) ( ( r ` z ) ( Hom ` C ) ( s ` z ) )
50	49	eleq2i 2693	. . . . . . . . 9 |- ( a e. X_ x e. ( Base ` A ) ( ( r ` x ) ( Hom ` C ) ( s ` x ) ) <-> a e. X_ z e. ( Base ` A ) ( ( r ` z ) ( Hom ` C ) ( s ` z ) ) )
51	43	adantr 481	. . . . . . . . . 10 |- ( ( ( ( ( ph /\ ( f e. ( A Func C ) /\ g e. ( A Func C ) ) ) /\ r = ( 1st ` f ) ) /\ s = ( 1st ` g ) ) /\ a e. X_ z e. ( Base ` A ) ( ( r ` z ) ( Hom ` C ) ( s ` z ) ) ) -> ( Base ` A ) = ( Base ` B ) )
52	51	adantr 481	. . . . . . . . . . 11 |- ( ( ( ( ( ( ph /\ ( f e. ( A Func C ) /\ g e. ( A Func C ) ) ) /\ r = ( 1st ` f ) ) /\ s = ( 1st ` g ) ) /\ a e. X_ z e. ( Base ` A ) ( ( r ` z ) ( Hom ` C ) ( s ` z ) ) ) /\ x e. ( Base ` A ) ) -> ( Base ` A ) = ( Base ` B ) )
53		eqid 2622	. . . . . . . . . . . . 13 |- ( Hom ` A ) = ( Hom ` A )
54		eqid 2622	. . . . . . . . . . . . 13 |- ( Hom ` B ) = ( Hom ` B )
55	1	ad6antr 772	. . . . . . . . . . . . 13 |- ( ( ( ( ( ( ( ph /\ ( f e. ( A Func C ) /\ g e. ( A Func C ) ) ) /\ r = ( 1st ` f ) ) /\ s = ( 1st ` g ) ) /\ a e. X_ z e. ( Base ` A ) ( ( r ` z ) ( Hom ` C ) ( s ` z ) ) ) /\ x e. ( Base ` A ) ) /\ y e. ( Base ` A ) ) -> ( Hom f ` A ) = ( Hom f ` B ) )
56		simplr 792	. . . . . . . . . . . . 13 |- ( ( ( ( ( ( ( ph /\ ( f e. ( A Func C ) /\ g e. ( A Func C ) ) ) /\ r = ( 1st ` f ) ) /\ s = ( 1st ` g ) ) /\ a e. X_ z e. ( Base ` A ) ( ( r ` z ) ( Hom ` C ) ( s ` z ) ) ) /\ x e. ( Base ` A ) ) /\ y e. ( Base ` A ) ) -> x e. ( Base ` A ) )
57		simpr 477	. . . . . . . . . . . . 13 |- ( ( ( ( ( ( ( ph /\ ( f e. ( A Func C ) /\ g e. ( A Func C ) ) ) /\ r = ( 1st ` f ) ) /\ s = ( 1st ` g ) ) /\ a e. X_ z e. ( Base ` A ) ( ( r ` z ) ( Hom ` C ) ( s ` z ) ) ) /\ x e. ( Base ` A ) ) /\ y e. ( Base ` A ) ) -> y e. ( Base ` A ) )
58	23, 53, 54, 55, 56, 57	homfeqval 16357	. . . . . . . . . . . 12 |- ( ( ( ( ( ( ( ph /\ ( f e. ( A Func C ) /\ g e. ( A Func C ) ) ) /\ r = ( 1st ` f ) ) /\ s = ( 1st ` g ) ) /\ a e. X_ z e. ( Base ` A ) ( ( r ` z ) ( Hom ` C ) ( s ` z ) ) ) /\ x e. ( Base ` A ) ) /\ y e. ( Base ` A ) ) -> ( x ( Hom ` A ) y ) = ( x ( Hom ` B ) y ) )
59		eqid 2622	. . . . . . . . . . . . . 14 |- (comp ` C ) = (comp ` C )
60		eqid 2622	. . . . . . . . . . . . . 14 |- (comp ` D ) = (comp ` D )
61	3	ad7antr 774	. . . . . . . . . . . . . 14 |- ( ( ( ( ( ( ( ( ph /\ ( f e. ( A Func C ) /\ g e. ( A Func C ) ) ) /\ r = ( 1st ` f ) ) /\ s = ( 1st ` g ) ) /\ a e. X_ z e. ( Base ` A ) ( ( r ` z ) ( Hom ` C ) ( s ` z ) ) ) /\ x e. ( Base ` A ) ) /\ y e. ( Base ` A ) ) /\ h e. ( x ( Hom ` A ) y ) ) -> ( Hom f ` C ) = ( Hom f ` D ) )
62	4	ad7antr 774	. . . . . . . . . . . . . 14 |- ( ( ( ( ( ( ( ( ph /\ ( f e. ( A Func C ) /\ g e. ( A Func C ) ) ) /\ r = ( 1st ` f ) ) /\ s = ( 1st ` g ) ) /\ a e. X_ z e. ( Base ` A ) ( ( r ` z ) ( Hom ` C ) ( s ` z ) ) ) /\ x e. ( Base ` A ) ) /\ y e. ( Base ` A ) ) /\ h e. ( x ( Hom ` A ) y ) ) -> (compf ` C ) = (compf ` D ) )
63	32	adantlr 751	. . . . . . . . . . . . . . 15 |- ( ( ( ( ( ( ph /\ ( f e. ( A Func C ) /\ g e. ( A Func C ) ) ) /\ r = ( 1st ` f ) ) /\ s = ( 1st ` g ) ) /\ a e. X_ z e. ( Base ` A ) ( ( r ` z ) ( Hom ` C ) ( s ` z ) ) ) /\ x e. ( Base ` A ) ) -> ( r ` x ) e. ( Base ` C ) )
64	63	ad2antrr 762	. . . . . . . . . . . . . 14 |- ( ( ( ( ( ( ( ( ph /\ ( f e. ( A Func C ) /\ g e. ( A Func C ) ) ) /\ r = ( 1st ` f ) ) /\ s = ( 1st ` g ) ) /\ a e. X_ z e. ( Base ` A ) ( ( r ` z ) ( Hom ` C ) ( s ` z ) ) ) /\ x e. ( Base ` A ) ) /\ y e. ( Base ` A ) ) /\ h e. ( x ( Hom ` A ) y ) ) -> ( r ` x ) e. ( Base ` C ) )
65	31	ad2antrr 762	. . . . . . . . . . . . . . . 16 |- ( ( ( ( ( ( ph /\ ( f e. ( A Func C ) /\ g e. ( A Func C ) ) ) /\ r = ( 1st ` f ) ) /\ s = ( 1st ` g ) ) /\ a e. X_ z e. ( Base ` A ) ( ( r ` z ) ( Hom ` C ) ( s ` z ) ) ) /\ x e. ( Base ` A ) ) -> r : ( Base ` A ) --> ( Base ` C ) )
66	65	ffvelrnda 6359	. . . . . . . . . . . . . . 15 |- ( ( ( ( ( ( ( ph /\ ( f e. ( A Func C ) /\ g e. ( A Func C ) ) ) /\ r = ( 1st ` f ) ) /\ s = ( 1st ` g ) ) /\ a e. X_ z e. ( Base ` A ) ( ( r ` z ) ( Hom ` C ) ( s ` z ) ) ) /\ x e. ( Base ` A ) ) /\ y e. ( Base ` A ) ) -> ( r ` y ) e. ( Base ` C ) )
67	66	adantr 481	. . . . . . . . . . . . . 14 |- ( ( ( ( ( ( ( ( ph /\ ( f e. ( A Func C ) /\ g e. ( A Func C ) ) ) /\ r = ( 1st ` f ) ) /\ s = ( 1st ` g ) ) /\ a e. X_ z e. ( Base ` A ) ( ( r ` z ) ( Hom ` C ) ( s ` z ) ) ) /\ x e. ( Base ` A ) ) /\ y e. ( Base ` A ) ) /\ h e. ( x ( Hom ` A ) y ) ) -> ( r ` y ) e. ( Base ` C ) )
68	38	ad2antrr 762	. . . . . . . . . . . . . . . 16 |- ( ( ( ( ( ( ph /\ ( f e. ( A Func C ) /\ g e. ( A Func C ) ) ) /\ r = ( 1st ` f ) ) /\ s = ( 1st ` g ) ) /\ a e. X_ z e. ( Base ` A ) ( ( r ` z ) ( Hom ` C ) ( s ` z ) ) ) /\ x e. ( Base ` A ) ) -> s : ( Base ` A ) --> ( Base ` C ) )
69	68	ffvelrnda 6359	. . . . . . . . . . . . . . 15 |- ( ( ( ( ( ( ( ph /\ ( f e. ( A Func C ) /\ g e. ( A Func C ) ) ) /\ r = ( 1st ` f ) ) /\ s = ( 1st ` g ) ) /\ a e. X_ z e. ( Base ` A ) ( ( r ` z ) ( Hom ` C ) ( s ` z ) ) ) /\ x e. ( Base ` A ) ) /\ y e. ( Base ` A ) ) -> ( s ` y ) e. ( Base ` C ) )
70	69	adantr 481	. . . . . . . . . . . . . 14 |- ( ( ( ( ( ( ( ( ph /\ ( f e. ( A Func C ) /\ g e. ( A Func C ) ) ) /\ r = ( 1st ` f ) ) /\ s = ( 1st ` g ) ) /\ a e. X_ z e. ( Base ` A ) ( ( r ` z ) ( Hom ` C ) ( s ` z ) ) ) /\ x e. ( Base ` A ) ) /\ y e. ( Base ` A ) ) /\ h e. ( x ( Hom ` A ) y ) ) -> ( s ` y ) e. ( Base ` C ) )
71	30	ad3antrrr 766	. . . . . . . . . . . . . . . 16 |- ( ( ( ( ( ( ( ph /\ ( f e. ( A Func C ) /\ g e. ( A Func C ) ) ) /\ r = ( 1st ` f ) ) /\ s = ( 1st ` g ) ) /\ a e. X_ z e. ( Base ` A ) ( ( r ` z ) ( Hom ` C ) ( s ` z ) ) ) /\ x e. ( Base ` A ) ) /\ y e. ( Base ` A ) ) -> r ( A Func C ) ( 2nd ` f ) )
72	23, 53, 20, 71, 56, 57	funcf2 16528	. . . . . . . . . . . . . . 15 |- ( ( ( ( ( ( ( ph /\ ( f e. ( A Func C ) /\ g e. ( A Func C ) ) ) /\ r = ( 1st ` f ) ) /\ s = ( 1st ` g ) ) /\ a e. X_ z e. ( Base ` A ) ( ( r ` z ) ( Hom ` C ) ( s ` z ) ) ) /\ x e. ( Base ` A ) ) /\ y e. ( Base ` A ) ) -> ( x ( 2nd ` f ) y ) : ( x ( Hom ` A ) y ) --> ( ( r ` x ) ( Hom ` C ) ( r ` y ) ) )
73	72	ffvelrnda 6359	. . . . . . . . . . . . . 14 |- ( ( ( ( ( ( ( ( ph /\ ( f e. ( A Func C ) /\ g e. ( A Func C ) ) ) /\ r = ( 1st ` f ) ) /\ s = ( 1st ` g ) ) /\ a e. X_ z e. ( Base ` A ) ( ( r ` z ) ( Hom ` C ) ( s ` z ) ) ) /\ x e. ( Base ` A ) ) /\ y e. ( Base ` A ) ) /\ h e. ( x ( Hom ` A ) y ) ) -> ( ( x ( 2nd ` f ) y ) ` h ) e. ( ( r ` x ) ( Hom ` C ) ( r ` y ) ) )
74		simplr 792	. . . . . . . . . . . . . . . 16 |- ( ( ( ( ( ( ph /\ ( f e. ( A Func C ) /\ g e. ( A Func C ) ) ) /\ r = ( 1st ` f ) ) /\ s = ( 1st ` g ) ) /\ a e. X_ z e. ( Base ` A ) ( ( r ` z ) ( Hom ` C ) ( s ` z ) ) ) /\ x e. ( Base ` A ) ) -> a e. X_ z e. ( Base ` A ) ( ( r ` z ) ( Hom ` C ) ( s ` z ) ) )
75		fveq2 6191	. . . . . . . . . . . . . . . . . 18 |- ( z = y -> ( r ` z ) = ( r ` y ) )
76		fveq2 6191	. . . . . . . . . . . . . . . . . 18 |- ( z = y -> ( s ` z ) = ( s ` y ) )
77	75, 76	oveq12d 6668	. . . . . . . . . . . . . . . . 17 |- ( z = y -> ( ( r ` z ) ( Hom ` C ) ( s ` z ) ) = ( ( r ` y ) ( Hom ` C ) ( s ` y ) ) )
78	77	fvixp 7913	. . . . . . . . . . . . . . . 16 |- ( ( a e. X_ z e. ( Base ` A ) ( ( r ` z ) ( Hom ` C ) ( s ` z ) ) /\ y e. ( Base ` A ) ) -> ( a ` y ) e. ( ( r ` y ) ( Hom ` C ) ( s ` y ) ) )
79	74, 78	sylan 488	. . . . . . . . . . . . . . 15 |- ( ( ( ( ( ( ( ph /\ ( f e. ( A Func C ) /\ g e. ( A Func C ) ) ) /\ r = ( 1st ` f ) ) /\ s = ( 1st ` g ) ) /\ a e. X_ z e. ( Base ` A ) ( ( r ` z ) ( Hom ` C ) ( s ` z ) ) ) /\ x e. ( Base ` A ) ) /\ y e. ( Base ` A ) ) -> ( a ` y ) e. ( ( r ` y ) ( Hom ` C ) ( s ` y ) ) )
80	79	adantr 481	. . . . . . . . . . . . . 14 |- ( ( ( ( ( ( ( ( ph /\ ( f e. ( A Func C ) /\ g e. ( A Func C ) ) ) /\ r = ( 1st ` f ) ) /\ s = ( 1st ` g ) ) /\ a e. X_ z e. ( Base ` A ) ( ( r ` z ) ( Hom ` C ) ( s ` z ) ) ) /\ x e. ( Base ` A ) ) /\ y e. ( Base ` A ) ) /\ h e. ( x ( Hom ` A ) y ) ) -> ( a ` y ) e. ( ( r ` y ) ( Hom ` C ) ( s ` y ) ) )
81	19, 20, 59, 60, 61, 62, 64, 67, 70, 73, 80	comfeqval 16368	. . . . . . . . . . . . 13 |- ( ( ( ( ( ( ( ( ph /\ ( f e. ( A Func C ) /\ g e. ( A Func C ) ) ) /\ r = ( 1st ` f ) ) /\ s = ( 1st ` g ) ) /\ a e. X_ z e. ( Base ` A ) ( ( r ` z ) ( Hom ` C ) ( s ` z ) ) ) /\ x e. ( Base ` A ) ) /\ y e. ( Base ` A ) ) /\ h e. ( x ( Hom ` A ) y ) ) -> ( ( a ` y ) ( <. ( r ` x ) , ( r ` y ) >. (comp ` C ) ( s ` y ) ) ( ( x ( 2nd ` f ) y ) ` h ) ) = ( ( a ` y ) ( <. ( r ` x ) , ( r ` y ) >. (comp ` D ) ( s ` y ) ) ( ( x ( 2nd ` f ) y ) ` h ) ) )
82	39	adantlr 751	. . . . . . . . . . . . . . 15 |- ( ( ( ( ( ( ph /\ ( f e. ( A Func C ) /\ g e. ( A Func C ) ) ) /\ r = ( 1st ` f ) ) /\ s = ( 1st ` g ) ) /\ a e. X_ z e. ( Base ` A ) ( ( r ` z ) ( Hom ` C ) ( s ` z ) ) ) /\ x e. ( Base ` A ) ) -> ( s ` x ) e. ( Base ` C ) )
83	82	ad2antrr 762	. . . . . . . . . . . . . 14 |- ( ( ( ( ( ( ( ( ph /\ ( f e. ( A Func C ) /\ g e. ( A Func C ) ) ) /\ r = ( 1st ` f ) ) /\ s = ( 1st ` g ) ) /\ a e. X_ z e. ( Base ` A ) ( ( r ` z ) ( Hom ` C ) ( s ` z ) ) ) /\ x e. ( Base ` A ) ) /\ y e. ( Base ` A ) ) /\ h e. ( x ( Hom ` A ) y ) ) -> ( s ` x ) e. ( Base ` C ) )
84		fveq2 6191	. . . . . . . . . . . . . . . . . 18 |- ( z = x -> ( r ` z ) = ( r ` x ) )
85		fveq2 6191	. . . . . . . . . . . . . . . . . 18 |- ( z = x -> ( s ` z ) = ( s ` x ) )
86	84, 85	oveq12d 6668	. . . . . . . . . . . . . . . . 17 |- ( z = x -> ( ( r ` z ) ( Hom ` C ) ( s ` z ) ) = ( ( r ` x ) ( Hom ` C ) ( s ` x ) ) )
87	86	fvixp 7913	. . . . . . . . . . . . . . . 16 |- ( ( a e. X_ z e. ( Base ` A ) ( ( r ` z ) ( Hom ` C ) ( s ` z ) ) /\ x e. ( Base ` A ) ) -> ( a ` x ) e. ( ( r ` x ) ( Hom ` C ) ( s ` x ) ) )
88	87	adantll 750	. . . . . . . . . . . . . . 15 |- ( ( ( ( ( ( ph /\ ( f e. ( A Func C ) /\ g e. ( A Func C ) ) ) /\ r = ( 1st ` f ) ) /\ s = ( 1st ` g ) ) /\ a e. X_ z e. ( Base ` A ) ( ( r ` z ) ( Hom ` C ) ( s ` z ) ) ) /\ x e. ( Base ` A ) ) -> ( a ` x ) e. ( ( r ` x ) ( Hom ` C ) ( s ` x ) ) )
89	88	ad2antrr 762	. . . . . . . . . . . . . 14 |- ( ( ( ( ( ( ( ( ph /\ ( f e. ( A Func C ) /\ g e. ( A Func C ) ) ) /\ r = ( 1st ` f ) ) /\ s = ( 1st ` g ) ) /\ a e. X_ z e. ( Base ` A ) ( ( r ` z ) ( Hom ` C ) ( s ` z ) ) ) /\ x e. ( Base ` A ) ) /\ y e. ( Base ` A ) ) /\ h e. ( x ( Hom ` A ) y ) ) -> ( a ` x ) e. ( ( r ` x ) ( Hom ` C ) ( s ` x ) ) )
90	37	ad3antrrr 766	. . . . . . . . . . . . . . . 16 |- ( ( ( ( ( ( ( ph /\ ( f e. ( A Func C ) /\ g e. ( A Func C ) ) ) /\ r = ( 1st ` f ) ) /\ s = ( 1st ` g ) ) /\ a e. X_ z e. ( Base ` A ) ( ( r ` z ) ( Hom ` C ) ( s ` z ) ) ) /\ x e. ( Base ` A ) ) /\ y e. ( Base ` A ) ) -> s ( A Func C ) ( 2nd ` g ) )
91	23, 53, 20, 90, 56, 57	funcf2 16528	. . . . . . . . . . . . . . 15 |- ( ( ( ( ( ( ( ph /\ ( f e. ( A Func C ) /\ g e. ( A Func C ) ) ) /\ r = ( 1st ` f ) ) /\ s = ( 1st ` g ) ) /\ a e. X_ z e. ( Base ` A ) ( ( r ` z ) ( Hom ` C ) ( s ` z ) ) ) /\ x e. ( Base ` A ) ) /\ y e. ( Base ` A ) ) -> ( x ( 2nd ` g ) y ) : ( x ( Hom ` A ) y ) --> ( ( s ` x ) ( Hom ` C ) ( s ` y ) ) )
92	91	ffvelrnda 6359	. . . . . . . . . . . . . 14 |- ( ( ( ( ( ( ( ( ph /\ ( f e. ( A Func C ) /\ g e. ( A Func C ) ) ) /\ r = ( 1st ` f ) ) /\ s = ( 1st ` g ) ) /\ a e. X_ z e. ( Base ` A ) ( ( r ` z ) ( Hom ` C ) ( s ` z ) ) ) /\ x e. ( Base ` A ) ) /\ y e. ( Base ` A ) ) /\ h e. ( x ( Hom ` A ) y ) ) -> ( ( x ( 2nd ` g ) y ) ` h ) e. ( ( s ` x ) ( Hom ` C ) ( s ` y ) ) )
93	19, 20, 59, 60, 61, 62, 64, 83, 70, 89, 92	comfeqval 16368	. . . . . . . . . . . . 13 |- ( ( ( ( ( ( ( ( ph /\ ( f e. ( A Func C ) /\ g e. ( A Func C ) ) ) /\ r = ( 1st ` f ) ) /\ s = ( 1st ` g ) ) /\ a e. X_ z e. ( Base ` A ) ( ( r ` z ) ( Hom ` C ) ( s ` z ) ) ) /\ x e. ( Base ` A ) ) /\ y e. ( Base ` A ) ) /\ h e. ( x ( Hom ` A ) y ) ) -> ( ( ( x ( 2nd ` g ) y ) ` h ) ( <. ( r ` x ) , ( s ` x ) >. (comp ` C ) ( s ` y ) ) ( a ` x ) ) = ( ( ( x ( 2nd ` g ) y ) ` h ) ( <. ( r ` x ) , ( s ` x ) >. (comp ` D ) ( s ` y ) ) ( a ` x ) ) )
94	81, 93	eqeq12d 2637	. . . . . . . . . . . 12 |- ( ( ( ( ( ( ( ( ph /\ ( f e. ( A Func C ) /\ g e. ( A Func C ) ) ) /\ r = ( 1st ` f ) ) /\ s = ( 1st ` g ) ) /\ a e. X_ z e. ( Base ` A ) ( ( r ` z ) ( Hom ` C ) ( s ` z ) ) ) /\ x e. ( Base ` A ) ) /\ y e. ( Base ` A ) ) /\ h e. ( x ( Hom ` A ) y ) ) -> ( ( ( a ` y ) ( <. ( r ` x ) , ( r ` y ) >. (comp ` C ) ( s ` y ) ) ( ( x ( 2nd ` f ) y ) ` h ) ) = ( ( ( x ( 2nd ` g ) y ) ` h ) ( <. ( r ` x ) , ( s ` x ) >. (comp ` C ) ( s ` y ) ) ( a ` x ) ) <-> ( ( a ` y ) ( <. ( r ` x ) , ( r ` y ) >. (comp ` D ) ( s ` y ) ) ( ( x ( 2nd ` f ) y ) ` h ) ) = ( ( ( x ( 2nd ` g ) y ) ` h ) ( <. ( r ` x ) , ( s ` x ) >. (comp ` D ) ( s ` y ) ) ( a ` x ) ) ) )
95	58, 94	raleqbidva 3154	. . . . . . . . . . 11 |- ( ( ( ( ( ( ( ph /\ ( f e. ( A Func C ) /\ g e. ( A Func C ) ) ) /\ r = ( 1st ` f ) ) /\ s = ( 1st ` g ) ) /\ a e. X_ z e. ( Base ` A ) ( ( r ` z ) ( Hom ` C ) ( s ` z ) ) ) /\ x e. ( Base ` A ) ) /\ y e. ( Base ` A ) ) -> ( A. h e. ( x ( Hom ` A ) y ) ( ( a ` y ) ( <. ( r ` x ) , ( r ` y ) >. (comp ` C ) ( s ` y ) ) ( ( x ( 2nd ` f ) y ) ` h ) ) = ( ( ( x ( 2nd ` g ) y ) ` h ) ( <. ( r ` x ) , ( s ` x ) >. (comp ` C ) ( s ` y ) ) ( a ` x ) ) <-> A. h e. ( x ( Hom ` B ) y ) ( ( a ` y ) ( <. ( r ` x ) , ( r ` y ) >. (comp ` D ) ( s ` y ) ) ( ( x ( 2nd ` f ) y ) ` h ) ) = ( ( ( x ( 2nd ` g ) y ) ` h ) ( <. ( r ` x ) , ( s ` x ) >. (comp ` D ) ( s ` y ) ) ( a ` x ) ) ) )
96	52, 95	raleqbidva 3154	. . . . . . . . . 10 |- ( ( ( ( ( ( ph /\ ( f e. ( A Func C ) /\ g e. ( A Func C ) ) ) /\ r = ( 1st ` f ) ) /\ s = ( 1st ` g ) ) /\ a e. X_ z e. ( Base ` A ) ( ( r ` z ) ( Hom ` C ) ( s ` z ) ) ) /\ x e. ( Base ` A ) ) -> ( A. y e. ( Base ` A ) A. h e. ( x ( Hom ` A ) y ) ( ( a ` y ) ( <. ( r ` x ) , ( r ` y ) >. (comp ` C ) ( s ` y ) ) ( ( x ( 2nd ` f ) y ) ` h ) ) = ( ( ( x ( 2nd ` g ) y ) ` h ) ( <. ( r ` x ) , ( s ` x ) >. (comp ` C ) ( s ` y ) ) ( a ` x ) ) <-> A. y e. ( Base ` B ) A. h e. ( x ( Hom ` B ) y ) ( ( a ` y ) ( <. ( r ` x ) , ( r ` y ) >. (comp ` D ) ( s ` y ) ) ( ( x ( 2nd ` f ) y ) ` h ) ) = ( ( ( x ( 2nd ` g ) y ) ` h ) ( <. ( r ` x ) , ( s ` x ) >. (comp ` D ) ( s ` y ) ) ( a ` x ) ) ) )
97	51, 96	raleqbidva 3154	. . . . . . . . 9 |- ( ( ( ( ( ph /\ ( f e. ( A Func C ) /\ g e. ( A Func C ) ) ) /\ r = ( 1st ` f ) ) /\ s = ( 1st ` g ) ) /\ a e. X_ z e. ( Base ` A ) ( ( r ` z ) ( Hom ` C ) ( s ` z ) ) ) -> ( A. x e. ( Base ` A ) A. y e. ( Base ` A ) A. h e. ( x ( Hom ` A ) y ) ( ( a ` y ) ( <. ( r ` x ) , ( r ` y ) >. (comp ` C ) ( s ` y ) ) ( ( x ( 2nd ` f ) y ) ` h ) ) = ( ( ( x ( 2nd ` g ) y ) ` h ) ( <. ( r ` x ) , ( s ` x ) >. (comp ` C ) ( s ` y ) ) ( a ` x ) ) <-> A. x e. ( Base ` B ) A. y e. ( Base ` B ) A. h e. ( x ( Hom ` B ) y ) ( ( a ` y ) ( <. ( r ` x ) , ( r ` y ) >. (comp ` D ) ( s ` y ) ) ( ( x ( 2nd ` f ) y ) ` h ) ) = ( ( ( x ( 2nd ` g ) y ) ` h ) ( <. ( r ` x ) , ( s ` x ) >. (comp ` D ) ( s ` y ) ) ( a ` x ) ) ) )
98	50, 97	sylan2b 492	. . . . . . . 8 |- ( ( ( ( ( ph /\ ( f e. ( A Func C ) /\ g e. ( A Func C ) ) ) /\ r = ( 1st ` f ) ) /\ s = ( 1st ` g ) ) /\ a e. X_ x e. ( Base ` A ) ( ( r ` x ) ( Hom ` C ) ( s ` x ) ) ) -> ( A. x e. ( Base ` A ) A. y e. ( Base ` A ) A. h e. ( x ( Hom ` A ) y ) ( ( a ` y ) ( <. ( r ` x ) , ( r ` y ) >. (comp ` C ) ( s ` y ) ) ( ( x ( 2nd ` f ) y ) ` h ) ) = ( ( ( x ( 2nd ` g ) y ) ` h ) ( <. ( r ` x ) , ( s ` x ) >. (comp ` C ) ( s ` y ) ) ( a ` x ) ) <-> A. x e. ( Base ` B ) A. y e. ( Base ` B ) A. h e. ( x ( Hom ` B ) y ) ( ( a ` y ) ( <. ( r ` x ) , ( r ` y ) >. (comp ` D ) ( s ` y ) ) ( ( x ( 2nd ` f ) y ) ` h ) ) = ( ( ( x ( 2nd ` g ) y ) ` h ) ( <. ( r ` x ) , ( s ` x ) >. (comp ` D ) ( s ` y ) ) ( a ` x ) ) ) )
99	45, 98	rabeqbidva 3196	. . . . . . 7 |- ( ( ( ( ph /\ ( f e. ( A Func C ) /\ g e. ( A Func C ) ) ) /\ r = ( 1st ` f ) ) /\ s = ( 1st ` g ) ) -> { a e. X_ x e. ( Base ` A ) ( ( r ` x ) ( Hom ` C ) ( s ` x ) ) | A. x e. ( Base ` A ) A. y e. ( Base ` A ) A. h e. ( x ( Hom ` A ) y ) ( ( a ` y ) ( <. ( r ` x ) , ( r ` y ) >. (comp ` C ) ( s ` y ) ) ( ( x ( 2nd ` f ) y ) ` h ) ) = ( ( ( x ( 2nd ` g ) y ) ` h ) ( <. ( r ` x ) , ( s ` x ) >. (comp ` C ) ( s ` y ) ) ( a ` x ) ) } = { a e. X_ x e. ( Base ` B ) ( ( r ` x ) ( Hom ` D ) ( s ` x ) ) | A. x e. ( Base ` B ) A. y e. ( Base ` B ) A. h e. ( x ( Hom ` B ) y ) ( ( a ` y ) ( <. ( r ` x ) , ( r ` y ) >. (comp ` D ) ( s ` y ) ) ( ( x ( 2nd ` f ) y ) ` h ) ) = ( ( ( x ( 2nd ` g ) y ) ` h ) ( <. ( r ` x ) , ( s ` x ) >. (comp ` D ) ( s ` y ) ) ( a ` x ) ) } )
100		csbeq1a 3542	. . . . . . . 8 |- ( s = ( 1st ` g ) -> { a e. X_ x e. ( Base ` B ) ( ( r ` x ) ( Hom ` D ) ( s ` x ) ) | A. x e. ( Base ` B ) A. y e. ( Base ` B ) A. h e. ( x ( Hom ` B ) y ) ( ( a ` y ) ( <. ( r ` x ) , ( r ` y ) >. (comp ` D ) ( s ` y ) ) ( ( x ( 2nd ` f ) y ) ` h ) ) = ( ( ( x ( 2nd ` g ) y ) ` h ) ( <. ( r ` x ) , ( s ` x ) >. (comp ` D ) ( s ` y ) ) ( a ` x ) ) } = [_ ( 1st ` g ) / s ]_ { a e. X_ x e. ( Base ` B ) ( ( r ` x ) ( Hom ` D ) ( s ` x ) ) | A. x e. ( Base ` B ) A. y e. ( Base ` B ) A. h e. ( x ( Hom ` B ) y ) ( ( a ` y ) ( <. ( r ` x ) , ( r ` y ) >. (comp ` D ) ( s ` y ) ) ( ( x ( 2nd ` f ) y ) ` h ) ) = ( ( ( x ( 2nd ` g ) y ) ` h ) ( <. ( r ` x ) , ( s ` x ) >. (comp ` D ) ( s ` y ) ) ( a ` x ) ) } )
101	100	adantl 482	. . . . . . 7 |- ( ( ( ( ph /\ ( f e. ( A Func C ) /\ g e. ( A Func C ) ) ) /\ r = ( 1st ` f ) ) /\ s = ( 1st ` g ) ) -> { a e. X_ x e. ( Base ` B ) ( ( r ` x ) ( Hom ` D ) ( s ` x ) ) | A. x e. ( Base ` B ) A. y e. ( Base ` B ) A. h e. ( x ( Hom ` B ) y ) ( ( a ` y ) ( <. ( r ` x ) , ( r ` y ) >. (comp ` D ) ( s ` y ) ) ( ( x ( 2nd ` f ) y ) ` h ) ) = ( ( ( x ( 2nd ` g ) y ) ` h ) ( <. ( r ` x ) , ( s ` x ) >. (comp ` D ) ( s ` y ) ) ( a ` x ) ) } = [_ ( 1st ` g ) / s ]_ { a e. X_ x e. ( Base ` B ) ( ( r ` x ) ( Hom ` D ) ( s ` x ) ) | A. x e. ( Base ` B ) A. y e. ( Base ` B ) A. h e. ( x ( Hom ` B ) y ) ( ( a ` y ) ( <. ( r ` x ) , ( r ` y ) >. (comp ` D ) ( s ` y ) ) ( ( x ( 2nd ` f ) y ) ` h ) ) = ( ( ( x ( 2nd ` g ) y ) ` h ) ( <. ( r ` x ) , ( s ` x ) >. (comp ` D ) ( s ` y ) ) ( a ` x ) ) } )
102	99, 101	eqtrd 2656	. . . . . 6 |- ( ( ( ( ph /\ ( f e. ( A Func C ) /\ g e. ( A Func C ) ) ) /\ r = ( 1st ` f ) ) /\ s = ( 1st ` g ) ) -> { a e. X_ x e. ( Base ` A ) ( ( r ` x ) ( Hom ` C ) ( s ` x ) ) | A. x e. ( Base ` A ) A. y e. ( Base ` A ) A. h e. ( x ( Hom ` A ) y ) ( ( a ` y ) ( <. ( r ` x ) , ( r ` y ) >. (comp ` C ) ( s ` y ) ) ( ( x ( 2nd ` f ) y ) ` h ) ) = ( ( ( x ( 2nd ` g ) y ) ` h ) ( <. ( r ` x ) , ( s ` x ) >. (comp ` C ) ( s ` y ) ) ( a ` x ) ) } = [_ ( 1st ` g ) / s ]_ { a e. X_ x e. ( Base ` B ) ( ( r ` x ) ( Hom ` D ) ( s ` x ) ) | A. x e. ( Base ` B ) A. y e. ( Base ` B ) A. h e. ( x ( Hom ` B ) y ) ( ( a ` y ) ( <. ( r ` x ) , ( r ` y ) >. (comp ` D ) ( s ` y ) ) ( ( x ( 2nd ` f ) y ) ` h ) ) = ( ( ( x ( 2nd ` g ) y ) ` h ) ( <. ( r ` x ) , ( s ` x ) >. (comp ` D ) ( s ` y ) ) ( a ` x ) ) } )
103	15, 17, 18, 102	csbiedf 3554	. . . . 5 |- ( ( ( ph /\ ( f e. ( A Func C ) /\ g e. ( A Func C ) ) ) /\ r = ( 1st ` f ) ) -> [_ ( 1st ` g ) / s ]_ { a e. X_ x e. ( Base ` A ) ( ( r ` x ) ( Hom ` C ) ( s ` x ) ) | A. x e. ( Base ` A ) A. y e. ( Base ` A ) A. h e. ( x ( Hom ` A ) y ) ( ( a ` y ) ( <. ( r ` x ) , ( r ` y ) >. (comp ` C ) ( s ` y ) ) ( ( x ( 2nd ` f ) y ) ` h ) ) = ( ( ( x ( 2nd ` g ) y ) ` h ) ( <. ( r ` x ) , ( s ` x ) >. (comp ` C ) ( s ` y ) ) ( a ` x ) ) } = [_ ( 1st ` g ) / s ]_ { a e. X_ x e. ( Base ` B ) ( ( r ` x ) ( Hom ` D ) ( s ` x ) ) | A. x e. ( Base ` B ) A. y e. ( Base ` B ) A. h e. ( x ( Hom ` B ) y ) ( ( a ` y ) ( <. ( r ` x ) , ( r ` y ) >. (comp ` D ) ( s ` y ) ) ( ( x ( 2nd ` f ) y ) ` h ) ) = ( ( ( x ( 2nd ` g ) y ) ` h ) ( <. ( r ` x ) , ( s ` x ) >. (comp ` D ) ( s ` y ) ) ( a ` x ) ) } )
104		csbeq1a 3542	. . . . . 6 |- ( r = ( 1st ` f ) -> [_ ( 1st ` g ) / s ]_ { a e. X_ x e. ( Base ` B ) ( ( r ` x ) ( Hom ` D ) ( s ` x ) ) | A. x e. ( Base ` B ) A. y e. ( Base ` B ) A. h e. ( x ( Hom ` B ) y ) ( ( a ` y ) ( <. ( r ` x ) , ( r ` y ) >. (comp ` D ) ( s ` y ) ) ( ( x ( 2nd ` f ) y ) ` h ) ) = ( ( ( x ( 2nd ` g ) y ) ` h ) ( <. ( r ` x ) , ( s ` x ) >. (comp ` D ) ( s ` y ) ) ( a ` x ) ) } = [_ ( 1st ` f ) / r ]_ [_ ( 1st ` g ) / s ]_ { a e. X_ x e. ( Base ` B ) ( ( r ` x ) ( Hom ` D ) ( s ` x ) ) | A. x e. ( Base ` B ) A. y e. ( Base ` B ) A. h e. ( x ( Hom ` B ) y ) ( ( a ` y ) ( <. ( r ` x ) , ( r ` y ) >. (comp ` D ) ( s ` y ) ) ( ( x ( 2nd ` f ) y ) ` h ) ) = ( ( ( x ( 2nd ` g ) y ) ` h ) ( <. ( r ` x ) , ( s ` x ) >. (comp ` D ) ( s ` y ) ) ( a ` x ) ) } )
105	104	adantl 482	. . . . 5 |- ( ( ( ph /\ ( f e. ( A Func C ) /\ g e. ( A Func C ) ) ) /\ r = ( 1st ` f ) ) -> [_ ( 1st ` g ) / s ]_ { a e. X_ x e. ( Base ` B ) ( ( r ` x ) ( Hom ` D ) ( s ` x ) ) | A. x e. ( Base ` B ) A. y e. ( Base ` B ) A. h e. ( x ( Hom ` B ) y ) ( ( a ` y ) ( <. ( r ` x ) , ( r ` y ) >. (comp ` D ) ( s ` y ) ) ( ( x ( 2nd ` f ) y ) ` h ) ) = ( ( ( x ( 2nd ` g ) y ) ` h ) ( <. ( r ` x ) , ( s ` x ) >. (comp ` D ) ( s ` y ) ) ( a ` x ) ) } = [_ ( 1st ` f ) / r ]_ [_ ( 1st ` g ) / s ]_ { a e. X_ x e. ( Base ` B ) ( ( r ` x ) ( Hom ` D ) ( s ` x ) ) | A. x e. ( Base ` B ) A. y e. ( Base ` B ) A. h e. ( x ( Hom ` B ) y ) ( ( a ` y ) ( <. ( r ` x ) , ( r ` y ) >. (comp ` D ) ( s ` y ) ) ( ( x ( 2nd ` f ) y ) ` h ) ) = ( ( ( x ( 2nd ` g ) y ) ` h ) ( <. ( r ` x ) , ( s ` x ) >. (comp ` D ) ( s ` y ) ) ( a ` x ) ) } )
106	103, 105	eqtrd 2656	. . . 4 |- ( ( ( ph /\ ( f e. ( A Func C ) /\ g e. ( A Func C ) ) ) /\ r = ( 1st ` f ) ) -> [_ ( 1st ` g ) / s ]_ { a e. X_ x e. ( Base ` A ) ( ( r ` x ) ( Hom ` C ) ( s ` x ) ) | A. x e. ( Base ` A ) A. y e. ( Base ` A ) A. h e. ( x ( Hom ` A ) y ) ( ( a ` y ) ( <. ( r ` x ) , ( r ` y ) >. (comp ` C ) ( s ` y ) ) ( ( x ( 2nd ` f ) y ) ` h ) ) = ( ( ( x ( 2nd ` g ) y ) ` h ) ( <. ( r ` x ) , ( s ` x ) >. (comp ` C ) ( s ` y ) ) ( a ` x ) ) } = [_ ( 1st ` f ) / r ]_ [_ ( 1st ` g ) / s ]_ { a e. X_ x e. ( Base ` B ) ( ( r ` x ) ( Hom ` D ) ( s ` x ) ) | A. x e. ( Base ` B ) A. y e. ( Base ` B ) A. h e. ( x ( Hom ` B ) y ) ( ( a ` y ) ( <. ( r ` x ) , ( r ` y ) >. (comp ` D ) ( s ` y ) ) ( ( x ( 2nd ` f ) y ) ` h ) ) = ( ( ( x ( 2nd ` g ) y ) ` h ) ( <. ( r ` x ) , ( s ` x ) >. (comp ` D ) ( s ` y ) ) ( a ` x ) ) } )
107	11, 13, 14, 106	csbiedf 3554	. . 3 |- ( ( ph /\ ( f e. ( A Func C ) /\ g e. ( A Func C ) ) ) -> [_ ( 1st ` f ) / r ]_ [_ ( 1st ` g ) / s ]_ { a e. X_ x e. ( Base ` A ) ( ( r ` x ) ( Hom ` C ) ( s ` x ) ) | A. x e. ( Base ` A ) A. y e. ( Base ` A ) A. h e. ( x ( Hom ` A ) y ) ( ( a ` y ) ( <. ( r ` x ) , ( r ` y ) >. (comp ` C ) ( s ` y ) ) ( ( x ( 2nd ` f ) y ) ` h ) ) = ( ( ( x ( 2nd ` g ) y ) ` h ) ( <. ( r ` x ) , ( s ` x ) >. (comp ` C ) ( s ` y ) ) ( a ` x ) ) } = [_ ( 1st ` f ) / r ]_ [_ ( 1st ` g ) / s ]_ { a e. X_ x e. ( Base ` B ) ( ( r ` x ) ( Hom ` D ) ( s ` x ) ) | A. x e. ( Base ` B ) A. y e. ( Base ` B ) A. h e. ( x ( Hom ` B ) y ) ( ( a ` y ) ( <. ( r ` x ) , ( r ` y ) >. (comp ` D ) ( s ` y ) ) ( ( x ( 2nd ` f ) y ) ` h ) ) = ( ( ( x ( 2nd ` g ) y ) ` h ) ( <. ( r ` x ) , ( s ` x ) >. (comp ` D ) ( s ` y ) ) ( a ` x ) ) } )
108	9, 10, 107	mpt2eq123dva 6716	. 2 |- ( ph -> ( f e. ( A Func C ) , g e. ( A Func C ) |-> [_ ( 1st ` f ) / r ]_ [_ ( 1st ` g ) / s ]_ { a e. X_ x e. ( Base ` A ) ( ( r ` x ) ( Hom ` C ) ( s ` x ) ) | A. x e. ( Base ` A ) A. y e. ( Base ` A ) A. h e. ( x ( Hom ` A ) y ) ( ( a ` y ) ( <. ( r ` x ) , ( r ` y ) >. (comp ` C ) ( s ` y ) ) ( ( x ( 2nd ` f ) y ) ` h ) ) = ( ( ( x ( 2nd ` g ) y ) ` h ) ( <. ( r ` x ) , ( s ` x ) >. (comp ` C ) ( s ` y ) ) ( a ` x ) ) } ) = ( f e. ( B Func D ) , g e. ( B Func D ) |-> [_ ( 1st ` f ) / r ]_ [_ ( 1st ` g ) / s ]_ { a e. X_ x e. ( Base ` B ) ( ( r ` x ) ( Hom ` D ) ( s ` x ) ) | A. x e. ( Base ` B ) A. y e. ( Base ` B ) A. h e. ( x ( Hom ` B ) y ) ( ( a ` y ) ( <. ( r ` x ) , ( r ` y ) >. (comp ` D ) ( s ` y ) ) ( ( x ( 2nd ` f ) y ) ` h ) ) = ( ( ( x ( 2nd ` g ) y ) ` h ) ( <. ( r ` x ) , ( s ` x ) >. (comp ` D ) ( s ` y ) ) ( a ` x ) ) } ) )
109		eqid 2622	. . 3 |- ( A Nat C ) = ( A Nat C )
110	109, 23, 53, 20, 59	natfval 16606	. 2 |- ( A Nat C ) = ( f e. ( A Func C ) , g e. ( A Func C ) |-> [_ ( 1st ` f ) / r ]_ [_ ( 1st ` g ) / s ]_ { a e. X_ x e. ( Base ` A ) ( ( r ` x ) ( Hom ` C ) ( s ` x ) ) | A. x e. ( Base ` A ) A. y e. ( Base ` A ) A. h e. ( x ( Hom ` A ) y ) ( ( a ` y ) ( <. ( r ` x ) , ( r ` y ) >. (comp ` C ) ( s ` y ) ) ( ( x ( 2nd ` f ) y ) ` h ) ) = ( ( ( x ( 2nd ` g ) y ) ` h ) ( <. ( r ` x ) , ( s ` x ) >. (comp ` C ) ( s ` y ) ) ( a ` x ) ) } )
111		eqid 2622	. . 3 |- ( B Nat D ) = ( B Nat D )
112		eqid 2622	. . 3 |- ( Base ` B ) = ( Base ` B )
113	111, 112, 54, 21, 60	natfval 16606	. 2 |- ( B Nat D ) = ( f e. ( B Func D ) , g e. ( B Func D ) |-> [_ ( 1st ` f ) / r ]_ [_ ( 1st ` g ) / s ]_ { a e. X_ x e. ( Base ` B ) ( ( r ` x ) ( Hom ` D ) ( s ` x ) ) | A. x e. ( Base ` B ) A. y e. ( Base ` B ) A. h e. ( x ( Hom ` B ) y ) ( ( a ` y ) ( <. ( r ` x ) , ( r ` y ) >. (comp ` D ) ( s ` y ) ) ( ( x ( 2nd ` f ) y ) ` h ) ) = ( ( ( x ( 2nd ` g ) y ) ` h ) ( <. ( r ` x ) , ( s ` x ) >. (comp ` D ) ( s ` y ) ) ( a ` x ) ) } )
114	108, 110, 113	3eqtr4g 2681	1 |- ( ph -> ( A Nat C ) = ( B Nat D ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   F/_wnfc 2751   A.wral 2912   {crab 2916   _Vcvv 3200   [_csb 3533   <.cop 4183   class class class wbr 4653   Rel wrel 5119   -->wf 5884   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   1stc1st 7166   2ndc2nd 7167   X_cixp 7908   Basecbs 15857   Hom chom 15952  compcco 15953   Catccat 16325   Hom _f chomf 16327  comp_fccomf 16328   Func cfunc 16514   Nat cnat 16601

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-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

This theorem is referenced by:  fucpropd  16637