Theorem curfuncf 16878

Description: Cancellation of curry with uncurry. (Contributed by Mario Carneiro, 13-Jan-2017.)

Hypotheses

Ref	Expression
uncfval.g	|- F = ( <" C D E "> uncurryF G )
uncfval.c	|- ( ph -> D e. Cat )
uncfval.d	|- ( ph -> E e. Cat )
uncfval.f	|- ( ph -> G e. ( C Func ( D FuncCat E ) ) )

Assertion

Ref	Expression
curfuncf	|- ( ph -> ( <. C , D >. curryF F ) = G )

Proof of Theorem curfuncf

Dummy variables g x y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		uncfval.g	. . . . . . . . . 10 |- F = ( <" C D E "> uncurryF G )
2		uncfval.c	. . . . . . . . . . 11 |- ( ph -> D e. Cat )
3	2	ad2antrr 762	. . . . . . . . . 10 |- ( ( ( ph /\ x e. ( Base ` C ) ) /\ y e. ( Base ` D ) ) -> D e. Cat )
4		uncfval.d	. . . . . . . . . . 11 |- ( ph -> E e. Cat )
5	4	ad2antrr 762	. . . . . . . . . 10 |- ( ( ( ph /\ x e. ( Base ` C ) ) /\ y e. ( Base ` D ) ) -> E e. Cat )
6		uncfval.f	. . . . . . . . . . 11 |- ( ph -> G e. ( C Func ( D FuncCat E ) ) )
7	6	ad2antrr 762	. . . . . . . . . 10 |- ( ( ( ph /\ x e. ( Base ` C ) ) /\ y e. ( Base ` D ) ) -> G e. ( C Func ( D FuncCat E ) ) )
8		eqid 2622	. . . . . . . . . 10 |- ( Base ` C ) = ( Base ` C )
9		eqid 2622	. . . . . . . . . 10 |- ( Base ` D ) = ( Base ` D )
10		simplr 792	. . . . . . . . . 10 |- ( ( ( ph /\ x e. ( Base ` C ) ) /\ y e. ( Base ` D ) ) -> x e. ( Base ` C ) )
11		simpr 477	. . . . . . . . . 10 |- ( ( ( ph /\ x e. ( Base ` C ) ) /\ y e. ( Base ` D ) ) -> y e. ( Base ` D ) )
12	1, 3, 5, 7, 8, 9, 10, 11	uncf1 16876	. . . . . . . . 9 |- ( ( ( ph /\ x e. ( Base ` C ) ) /\ y e. ( Base ` D ) ) -> ( x ( 1st ` F ) y ) = ( ( 1st ` ( ( 1st ` G ) ` x ) ) ` y ) )
13	12	mpteq2dva 4744	. . . . . . . 8 |- ( ( ph /\ x e. ( Base ` C ) ) -> ( y e. ( Base ` D ) |-> ( x ( 1st ` F ) y ) ) = ( y e. ( Base ` D ) |-> ( ( 1st ` ( ( 1st ` G ) ` x ) ) ` y ) ) )
14		eqid 2622	. . . . . . . . . 10 |- ( Base ` E ) = ( Base ` E )
15		relfunc 16522	. . . . . . . . . . 11 |- Rel ( D Func E )
16		eqid 2622	. . . . . . . . . . . . . 14 |- ( D FuncCat E ) = ( D FuncCat E )
17	16	fucbas 16620	. . . . . . . . . . . . 13 |- ( D Func E ) = ( Base ` ( D FuncCat E ) )
18		relfunc 16522	. . . . . . . . . . . . . 14 |- Rel ( C Func ( D FuncCat E ) )
19		1st2ndbr 7217	. . . . . . . . . . . . . 14 |- ( ( Rel ( C Func ( D FuncCat E ) ) /\ G e. ( C Func ( D FuncCat E ) ) ) -> ( 1st ` G ) ( C Func ( D FuncCat E ) ) ( 2nd ` G ) )
20	18, 6, 19	sylancr 695	. . . . . . . . . . . . 13 |- ( ph -> ( 1st ` G ) ( C Func ( D FuncCat E ) ) ( 2nd ` G ) )
21	8, 17, 20	funcf1 16526	. . . . . . . . . . . 12 |- ( ph -> ( 1st ` G ) : ( Base ` C ) --> ( D Func E ) )
22	21	ffvelrnda 6359	. . . . . . . . . . 11 |- ( ( ph /\ x e. ( Base ` C ) ) -> ( ( 1st ` G ) ` x ) e. ( D Func E ) )
23		1st2ndbr 7217	. . . . . . . . . . 11 |- ( ( Rel ( D Func E ) /\ ( ( 1st ` G ) ` x ) e. ( D Func E ) ) -> ( 1st ` ( ( 1st ` G ) ` x ) ) ( D Func E ) ( 2nd ` ( ( 1st ` G ) ` x ) ) )
24	15, 22, 23	sylancr 695	. . . . . . . . . 10 |- ( ( ph /\ x e. ( Base ` C ) ) -> ( 1st ` ( ( 1st ` G ) ` x ) ) ( D Func E ) ( 2nd ` ( ( 1st ` G ) ` x ) ) )
25	9, 14, 24	funcf1 16526	. . . . . . . . 9 |- ( ( ph /\ x e. ( Base ` C ) ) -> ( 1st ` ( ( 1st ` G ) ` x ) ) : ( Base ` D ) --> ( Base ` E ) )
26	25	feqmptd 6249	. . . . . . . 8 |- ( ( ph /\ x e. ( Base ` C ) ) -> ( 1st ` ( ( 1st ` G ) ` x ) ) = ( y e. ( Base ` D ) |-> ( ( 1st ` ( ( 1st ` G ) ` x ) ) ` y ) ) )
27	13, 26	eqtr4d 2659	. . . . . . 7 |- ( ( ph /\ x e. ( Base ` C ) ) -> ( y e. ( Base ` D ) |-> ( x ( 1st ` F ) y ) ) = ( 1st ` ( ( 1st ` G ) ` x ) ) )
28	2	ad3antrrr 766	. . . . . . . . . . . . . 14 |- ( ( ( ( ph /\ x e. ( Base ` C ) ) /\ ( y e. ( Base ` D ) /\ z e. ( Base ` D ) ) ) /\ g e. ( y ( Hom ` D ) z ) ) -> D e. Cat )
29	4	ad3antrrr 766	. . . . . . . . . . . . . 14 |- ( ( ( ( ph /\ x e. ( Base ` C ) ) /\ ( y e. ( Base ` D ) /\ z e. ( Base ` D ) ) ) /\ g e. ( y ( Hom ` D ) z ) ) -> E e. Cat )
30	6	ad3antrrr 766	. . . . . . . . . . . . . 14 |- ( ( ( ( ph /\ x e. ( Base ` C ) ) /\ ( y e. ( Base ` D ) /\ z e. ( Base ` D ) ) ) /\ g e. ( y ( Hom ` D ) z ) ) -> G e. ( C Func ( D FuncCat E ) ) )
31		simpllr 799	. . . . . . . . . . . . . 14 |- ( ( ( ( ph /\ x e. ( Base ` C ) ) /\ ( y e. ( Base ` D ) /\ z e. ( Base ` D ) ) ) /\ g e. ( y ( Hom ` D ) z ) ) -> x e. ( Base ` C ) )
32		simplrl 800	. . . . . . . . . . . . . 14 |- ( ( ( ( ph /\ x e. ( Base ` C ) ) /\ ( y e. ( Base ` D ) /\ z e. ( Base ` D ) ) ) /\ g e. ( y ( Hom ` D ) z ) ) -> y e. ( Base ` D ) )
33		eqid 2622	. . . . . . . . . . . . . 14 |- ( Hom ` C ) = ( Hom ` C )
34		eqid 2622	. . . . . . . . . . . . . 14 |- ( Hom ` D ) = ( Hom ` D )
35		simprr 796	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ x e. ( Base ` C ) ) /\ ( y e. ( Base ` D ) /\ z e. ( Base ` D ) ) ) -> z e. ( Base ` D ) )
36	35	adantr 481	. . . . . . . . . . . . . 14 |- ( ( ( ( ph /\ x e. ( Base ` C ) ) /\ ( y e. ( Base ` D ) /\ z e. ( Base ` D ) ) ) /\ g e. ( y ( Hom ` D ) z ) ) -> z e. ( Base ` D ) )
37		eqid 2622	. . . . . . . . . . . . . . 15 |- ( Id ` C ) = ( Id ` C )
38		funcrcl 16523	. . . . . . . . . . . . . . . . . 18 |- ( G e. ( C Func ( D FuncCat E ) ) -> ( C e. Cat /\ ( D FuncCat E ) e. Cat ) )
39	6, 38	syl 17	. . . . . . . . . . . . . . . . 17 |- ( ph -> ( C e. Cat /\ ( D FuncCat E ) e. Cat ) )
40	39	simpld 475	. . . . . . . . . . . . . . . 16 |- ( ph -> C e. Cat )
41	40	ad3antrrr 766	. . . . . . . . . . . . . . 15 |- ( ( ( ( ph /\ x e. ( Base ` C ) ) /\ ( y e. ( Base ` D ) /\ z e. ( Base ` D ) ) ) /\ g e. ( y ( Hom ` D ) z ) ) -> C e. Cat )
42	8, 33, 37, 41, 31	catidcl 16343	. . . . . . . . . . . . . 14 |- ( ( ( ( ph /\ x e. ( Base ` C ) ) /\ ( y e. ( Base ` D ) /\ z e. ( Base ` D ) ) ) /\ g e. ( y ( Hom ` D ) z ) ) -> ( ( Id ` C ) ` x ) e. ( x ( Hom ` C ) x ) )
43		simpr 477	. . . . . . . . . . . . . 14 |- ( ( ( ( ph /\ x e. ( Base ` C ) ) /\ ( y e. ( Base ` D ) /\ z e. ( Base ` D ) ) ) /\ g e. ( y ( Hom ` D ) z ) ) -> g e. ( y ( Hom ` D ) z ) )
44	1, 28, 29, 30, 8, 9, 31, 32, 33, 34, 31, 36, 42, 43	uncf2 16877	. . . . . . . . . . . . 13 |- ( ( ( ( ph /\ x e. ( Base ` C ) ) /\ ( y e. ( Base ` D ) /\ z e. ( Base ` D ) ) ) /\ g e. ( y ( Hom ` D ) z ) ) -> ( ( ( Id ` C ) ` x ) ( <. x , y >. ( 2nd ` F ) <. x , z >. ) g ) = ( ( ( ( x ( 2nd ` G ) x ) ` ( ( Id ` C ) ` x ) ) ` z ) ( <. ( ( 1st ` ( ( 1st ` G ) ` x ) ) ` y ) , ( ( 1st ` ( ( 1st ` G ) ` x ) ) ` z ) >. (comp ` E ) ( ( 1st ` ( ( 1st ` G ) ` x ) ) ` z ) ) ( ( y ( 2nd ` ( ( 1st ` G ) ` x ) ) z ) ` g ) ) )
45		eqid 2622	. . . . . . . . . . . . . . . . . 18 |- ( Id ` ( D FuncCat E ) ) = ( Id ` ( D FuncCat E ) )
46	20	ad3antrrr 766	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ( ph /\ x e. ( Base ` C ) ) /\ ( y e. ( Base ` D ) /\ z e. ( Base ` D ) ) ) /\ g e. ( y ( Hom ` D ) z ) ) -> ( 1st ` G ) ( C Func ( D FuncCat E ) ) ( 2nd ` G ) )
47	8, 37, 45, 46, 31	funcid 16530	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( ph /\ x e. ( Base ` C ) ) /\ ( y e. ( Base ` D ) /\ z e. ( Base ` D ) ) ) /\ g e. ( y ( Hom ` D ) z ) ) -> ( ( x ( 2nd ` G ) x ) ` ( ( Id ` C ) ` x ) ) = ( ( Id ` ( D FuncCat E ) ) ` ( ( 1st ` G ) ` x ) ) )
48		eqid 2622	. . . . . . . . . . . . . . . . . 18 |- ( Id ` E ) = ( Id ` E )
49	22	ad2antrr 762	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ( ph /\ x e. ( Base ` C ) ) /\ ( y e. ( Base ` D ) /\ z e. ( Base ` D ) ) ) /\ g e. ( y ( Hom ` D ) z ) ) -> ( ( 1st ` G ) ` x ) e. ( D Func E ) )
50	16, 45, 48, 49	fucid 16631	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( ph /\ x e. ( Base ` C ) ) /\ ( y e. ( Base ` D ) /\ z e. ( Base ` D ) ) ) /\ g e. ( y ( Hom ` D ) z ) ) -> ( ( Id ` ( D FuncCat E ) ) ` ( ( 1st ` G ) ` x ) ) = ( ( Id ` E ) o. ( 1st ` ( ( 1st ` G ) ` x ) ) ) )
51	47, 50	eqtrd 2656	. . . . . . . . . . . . . . . 16 |- ( ( ( ( ph /\ x e. ( Base ` C ) ) /\ ( y e. ( Base ` D ) /\ z e. ( Base ` D ) ) ) /\ g e. ( y ( Hom ` D ) z ) ) -> ( ( x ( 2nd ` G ) x ) ` ( ( Id ` C ) ` x ) ) = ( ( Id ` E ) o. ( 1st ` ( ( 1st ` G ) ` x ) ) ) )
52	51	fveq1d 6193	. . . . . . . . . . . . . . 15 |- ( ( ( ( ph /\ x e. ( Base ` C ) ) /\ ( y e. ( Base ` D ) /\ z e. ( Base ` D ) ) ) /\ g e. ( y ( Hom ` D ) z ) ) -> ( ( ( x ( 2nd ` G ) x ) ` ( ( Id ` C ) ` x ) ) ` z ) = ( ( ( Id ` E ) o. ( 1st ` ( ( 1st ` G ) ` x ) ) ) ` z ) )
53	25	ad2antrr 762	. . . . . . . . . . . . . . . 16 |- ( ( ( ( ph /\ x e. ( Base ` C ) ) /\ ( y e. ( Base ` D ) /\ z e. ( Base ` D ) ) ) /\ g e. ( y ( Hom ` D ) z ) ) -> ( 1st ` ( ( 1st ` G ) ` x ) ) : ( Base ` D ) --> ( Base ` E ) )
54		fvco3 6275	. . . . . . . . . . . . . . . 16 |- ( ( ( 1st ` ( ( 1st ` G ) ` x ) ) : ( Base ` D ) --> ( Base ` E ) /\ z e. ( Base ` D ) ) -> ( ( ( Id ` E ) o. ( 1st ` ( ( 1st ` G ) ` x ) ) ) ` z ) = ( ( Id ` E ) ` ( ( 1st ` ( ( 1st ` G ) ` x ) ) ` z ) ) )
55	53, 36, 54	syl2anc 693	. . . . . . . . . . . . . . 15 |- ( ( ( ( ph /\ x e. ( Base ` C ) ) /\ ( y e. ( Base ` D ) /\ z e. ( Base ` D ) ) ) /\ g e. ( y ( Hom ` D ) z ) ) -> ( ( ( Id ` E ) o. ( 1st ` ( ( 1st ` G ) ` x ) ) ) ` z ) = ( ( Id ` E ) ` ( ( 1st ` ( ( 1st ` G ) ` x ) ) ` z ) ) )
56	52, 55	eqtrd 2656	. . . . . . . . . . . . . 14 |- ( ( ( ( ph /\ x e. ( Base ` C ) ) /\ ( y e. ( Base ` D ) /\ z e. ( Base ` D ) ) ) /\ g e. ( y ( Hom ` D ) z ) ) -> ( ( ( x ( 2nd ` G ) x ) ` ( ( Id ` C ) ` x ) ) ` z ) = ( ( Id ` E ) ` ( ( 1st ` ( ( 1st ` G ) ` x ) ) ` z ) ) )
57	56	oveq1d 6665	. . . . . . . . . . . . 13 |- ( ( ( ( ph /\ x e. ( Base ` C ) ) /\ ( y e. ( Base ` D ) /\ z e. ( Base ` D ) ) ) /\ g e. ( y ( Hom ` D ) z ) ) -> ( ( ( ( x ( 2nd ` G ) x ) ` ( ( Id ` C ) ` x ) ) ` z ) ( <. ( ( 1st ` ( ( 1st ` G ) ` x ) ) ` y ) , ( ( 1st ` ( ( 1st ` G ) ` x ) ) ` z ) >. (comp ` E ) ( ( 1st ` ( ( 1st ` G ) ` x ) ) ` z ) ) ( ( y ( 2nd ` ( ( 1st ` G ) ` x ) ) z ) ` g ) ) = ( ( ( Id ` E ) ` ( ( 1st ` ( ( 1st ` G ) ` x ) ) ` z ) ) ( <. ( ( 1st ` ( ( 1st ` G ) ` x ) ) ` y ) , ( ( 1st ` ( ( 1st ` G ) ` x ) ) ` z ) >. (comp ` E ) ( ( 1st ` ( ( 1st ` G ) ` x ) ) ` z ) ) ( ( y ( 2nd ` ( ( 1st ` G ) ` x ) ) z ) ` g ) ) )
58		eqid 2622	. . . . . . . . . . . . . 14 |- ( Hom ` E ) = ( Hom ` E )
59	53, 32	ffvelrnd 6360	. . . . . . . . . . . . . 14 |- ( ( ( ( ph /\ x e. ( Base ` C ) ) /\ ( y e. ( Base ` D ) /\ z e. ( Base ` D ) ) ) /\ g e. ( y ( Hom ` D ) z ) ) -> ( ( 1st ` ( ( 1st ` G ) ` x ) ) ` y ) e. ( Base ` E ) )
60		eqid 2622	. . . . . . . . . . . . . 14 |- (comp ` E ) = (comp ` E )
61	53, 36	ffvelrnd 6360	. . . . . . . . . . . . . 14 |- ( ( ( ( ph /\ x e. ( Base ` C ) ) /\ ( y e. ( Base ` D ) /\ z e. ( Base ` D ) ) ) /\ g e. ( y ( Hom ` D ) z ) ) -> ( ( 1st ` ( ( 1st ` G ) ` x ) ) ` z ) e. ( Base ` E ) )
62	24	adantr 481	. . . . . . . . . . . . . . . 16 |- ( ( ( ph /\ x e. ( Base ` C ) ) /\ ( y e. ( Base ` D ) /\ z e. ( Base ` D ) ) ) -> ( 1st ` ( ( 1st ` G ) ` x ) ) ( D Func E ) ( 2nd ` ( ( 1st ` G ) ` x ) ) )
63		simprl 794	. . . . . . . . . . . . . . . 16 |- ( ( ( ph /\ x e. ( Base ` C ) ) /\ ( y e. ( Base ` D ) /\ z e. ( Base ` D ) ) ) -> y e. ( Base ` D ) )
64	9, 34, 58, 62, 63, 35	funcf2 16528	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ x e. ( Base ` C ) ) /\ ( y e. ( Base ` D ) /\ z e. ( Base ` D ) ) ) -> ( y ( 2nd ` ( ( 1st ` G ) ` x ) ) z ) : ( y ( Hom ` D ) z ) --> ( ( ( 1st ` ( ( 1st ` G ) ` x ) ) ` y ) ( Hom ` E ) ( ( 1st ` ( ( 1st ` G ) ` x ) ) ` z ) ) )
65	64	ffvelrnda 6359	. . . . . . . . . . . . . 14 |- ( ( ( ( ph /\ x e. ( Base ` C ) ) /\ ( y e. ( Base ` D ) /\ z e. ( Base ` D ) ) ) /\ g e. ( y ( Hom ` D ) z ) ) -> ( ( y ( 2nd ` ( ( 1st ` G ) ` x ) ) z ) ` g ) e. ( ( ( 1st ` ( ( 1st ` G ) ` x ) ) ` y ) ( Hom ` E ) ( ( 1st ` ( ( 1st ` G ) ` x ) ) ` z ) ) )
66	14, 58, 48, 29, 59, 60, 61, 65	catlid 16344	. . . . . . . . . . . . 13 |- ( ( ( ( ph /\ x e. ( Base ` C ) ) /\ ( y e. ( Base ` D ) /\ z e. ( Base ` D ) ) ) /\ g e. ( y ( Hom ` D ) z ) ) -> ( ( ( Id ` E ) ` ( ( 1st ` ( ( 1st ` G ) ` x ) ) ` z ) ) ( <. ( ( 1st ` ( ( 1st ` G ) ` x ) ) ` y ) , ( ( 1st ` ( ( 1st ` G ) ` x ) ) ` z ) >. (comp ` E ) ( ( 1st ` ( ( 1st ` G ) ` x ) ) ` z ) ) ( ( y ( 2nd ` ( ( 1st ` G ) ` x ) ) z ) ` g ) ) = ( ( y ( 2nd ` ( ( 1st ` G ) ` x ) ) z ) ` g ) )
67	44, 57, 66	3eqtrd 2660	. . . . . . . . . . . 12 |- ( ( ( ( ph /\ x e. ( Base ` C ) ) /\ ( y e. ( Base ` D ) /\ z e. ( Base ` D ) ) ) /\ g e. ( y ( Hom ` D ) z ) ) -> ( ( ( Id ` C ) ` x ) ( <. x , y >. ( 2nd ` F ) <. x , z >. ) g ) = ( ( y ( 2nd ` ( ( 1st ` G ) ` x ) ) z ) ` g ) )
68	67	mpteq2dva 4744	. . . . . . . . . . 11 |- ( ( ( ph /\ x e. ( Base ` C ) ) /\ ( y e. ( Base ` D ) /\ z e. ( Base ` D ) ) ) -> ( g e. ( y ( Hom ` D ) z ) |-> ( ( ( Id ` C ) ` x ) ( <. x , y >. ( 2nd ` F ) <. x , z >. ) g ) ) = ( g e. ( y ( Hom ` D ) z ) |-> ( ( y ( 2nd ` ( ( 1st ` G ) ` x ) ) z ) ` g ) ) )
69	64	feqmptd 6249	. . . . . . . . . . 11 |- ( ( ( ph /\ x e. ( Base ` C ) ) /\ ( y e. ( Base ` D ) /\ z e. ( Base ` D ) ) ) -> ( y ( 2nd ` ( ( 1st ` G ) ` x ) ) z ) = ( g e. ( y ( Hom ` D ) z ) |-> ( ( y ( 2nd ` ( ( 1st ` G ) ` x ) ) z ) ` g ) ) )
70	68, 69	eqtr4d 2659	. . . . . . . . . 10 |- ( ( ( ph /\ x e. ( Base ` C ) ) /\ ( y e. ( Base ` D ) /\ z e. ( Base ` D ) ) ) -> ( g e. ( y ( Hom ` D ) z ) |-> ( ( ( Id ` C ) ` x ) ( <. x , y >. ( 2nd ` F ) <. x , z >. ) g ) ) = ( y ( 2nd ` ( ( 1st ` G ) ` x ) ) z ) )
71	70	3impb 1260	. . . . . . . . 9 |- ( ( ( ph /\ x e. ( Base ` C ) ) /\ y e. ( Base ` D ) /\ z e. ( Base ` D ) ) -> ( g e. ( y ( Hom ` D ) z ) |-> ( ( ( Id ` C ) ` x ) ( <. x , y >. ( 2nd ` F ) <. x , z >. ) g ) ) = ( y ( 2nd ` ( ( 1st ` G ) ` x ) ) z ) )
72	71	mpt2eq3dva 6719	. . . . . . . 8 |- ( ( ph /\ x e. ( Base ` C ) ) -> ( y e. ( Base ` D ) , z e. ( Base ` D ) |-> ( g e. ( y ( Hom ` D ) z ) |-> ( ( ( Id ` C ) ` x ) ( <. x , y >. ( 2nd ` F ) <. x , z >. ) g ) ) ) = ( y e. ( Base ` D ) , z e. ( Base ` D ) |-> ( y ( 2nd ` ( ( 1st ` G ) ` x ) ) z ) ) )
73	9, 24	funcfn2 16529	. . . . . . . . 9 |- ( ( ph /\ x e. ( Base ` C ) ) -> ( 2nd ` ( ( 1st ` G ) ` x ) ) Fn ( ( Base ` D ) X. ( Base ` D ) ) )
74		fnov 6768	. . . . . . . . 9 |- ( ( 2nd ` ( ( 1st ` G ) ` x ) ) Fn ( ( Base ` D ) X. ( Base ` D ) ) <-> ( 2nd ` ( ( 1st ` G ) ` x ) ) = ( y e. ( Base ` D ) , z e. ( Base ` D ) |-> ( y ( 2nd ` ( ( 1st ` G ) ` x ) ) z ) ) )
75	73, 74	sylib 208	. . . . . . . 8 |- ( ( ph /\ x e. ( Base ` C ) ) -> ( 2nd ` ( ( 1st ` G ) ` x ) ) = ( y e. ( Base ` D ) , z e. ( Base ` D ) |-> ( y ( 2nd ` ( ( 1st ` G ) ` x ) ) z ) ) )
76	72, 75	eqtr4d 2659	. . . . . . 7 |- ( ( ph /\ x e. ( Base ` C ) ) -> ( y e. ( Base ` D ) , z e. ( Base ` D ) |-> ( g e. ( y ( Hom ` D ) z ) |-> ( ( ( Id ` C ) ` x ) ( <. x , y >. ( 2nd ` F ) <. x , z >. ) g ) ) ) = ( 2nd ` ( ( 1st ` G ) ` x ) ) )
77	27, 76	opeq12d 4410	. . . . . 6 |- ( ( ph /\ x e. ( Base ` C ) ) -> <. ( y e. ( Base ` D ) |-> ( x ( 1st ` F ) y ) ) , ( y e. ( Base ` D ) , z e. ( Base ` D ) |-> ( g e. ( y ( Hom ` D ) z ) |-> ( ( ( Id ` C ) ` x ) ( <. x , y >. ( 2nd ` F ) <. x , z >. ) g ) ) ) >. = <. ( 1st ` ( ( 1st ` G ) ` x ) ) , ( 2nd ` ( ( 1st ` G ) ` x ) ) >. )
78		1st2nd 7214	. . . . . . 7 |- ( ( Rel ( D Func E ) /\ ( ( 1st ` G ) ` x ) e. ( D Func E ) ) -> ( ( 1st ` G ) ` x ) = <. ( 1st ` ( ( 1st ` G ) ` x ) ) , ( 2nd ` ( ( 1st ` G ) ` x ) ) >. )
79	15, 22, 78	sylancr 695	. . . . . 6 |- ( ( ph /\ x e. ( Base ` C ) ) -> ( ( 1st ` G ) ` x ) = <. ( 1st ` ( ( 1st ` G ) ` x ) ) , ( 2nd ` ( ( 1st ` G ) ` x ) ) >. )
80	77, 79	eqtr4d 2659	. . . . 5 |- ( ( ph /\ x e. ( Base ` C ) ) -> <. ( y e. ( Base ` D ) |-> ( x ( 1st ` F ) y ) ) , ( y e. ( Base ` D ) , z e. ( Base ` D ) |-> ( g e. ( y ( Hom ` D ) z ) |-> ( ( ( Id ` C ) ` x ) ( <. x , y >. ( 2nd ` F ) <. x , z >. ) g ) ) ) >. = ( ( 1st ` G ) ` x ) )
81	80	mpteq2dva 4744	. . . 4 |- ( ph -> ( x e. ( Base ` C ) |-> <. ( y e. ( Base ` D ) |-> ( x ( 1st ` F ) y ) ) , ( y e. ( Base ` D ) , z e. ( Base ` D ) |-> ( g e. ( y ( Hom ` D ) z ) |-> ( ( ( Id ` C ) ` x ) ( <. x , y >. ( 2nd ` F ) <. x , z >. ) g ) ) ) >. ) = ( x e. ( Base ` C ) |-> ( ( 1st ` G ) ` x ) ) )
82	21	feqmptd 6249	. . . 4 |- ( ph -> ( 1st ` G ) = ( x e. ( Base ` C ) |-> ( ( 1st ` G ) ` x ) ) )
83	81, 82	eqtr4d 2659	. . 3 |- ( ph -> ( x e. ( Base ` C ) |-> <. ( y e. ( Base ` D ) |-> ( x ( 1st ` F ) y ) ) , ( y e. ( Base ` D ) , z e. ( Base ` D ) |-> ( g e. ( y ( Hom ` D ) z ) |-> ( ( ( Id ` C ) ` x ) ( <. x , y >. ( 2nd ` F ) <. x , z >. ) g ) ) ) >. ) = ( 1st ` G ) )
84	2	ad3antrrr 766	. . . . . . . . . . . 12 |- ( ( ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) /\ g e. ( x ( Hom ` C ) y ) ) /\ z e. ( Base ` D ) ) -> D e. Cat )
85	4	ad3antrrr 766	. . . . . . . . . . . 12 |- ( ( ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) /\ g e. ( x ( Hom ` C ) y ) ) /\ z e. ( Base ` D ) ) -> E e. Cat )
86	6	ad3antrrr 766	. . . . . . . . . . . 12 |- ( ( ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) /\ g e. ( x ( Hom ` C ) y ) ) /\ z e. ( Base ` D ) ) -> G e. ( C Func ( D FuncCat E ) ) )
87		simprl 794	. . . . . . . . . . . . 13 |- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) -> x e. ( Base ` C ) )
88	87	ad2antrr 762	. . . . . . . . . . . 12 |- ( ( ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) /\ g e. ( x ( Hom ` C ) y ) ) /\ z e. ( Base ` D ) ) -> x e. ( Base ` C ) )
89		simpr 477	. . . . . . . . . . . 12 |- ( ( ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) /\ g e. ( x ( Hom ` C ) y ) ) /\ z e. ( Base ` D ) ) -> z e. ( Base ` D ) )
90		simprr 796	. . . . . . . . . . . . 13 |- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) -> y e. ( Base ` C ) )
91	90	ad2antrr 762	. . . . . . . . . . . 12 |- ( ( ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) /\ g e. ( x ( Hom ` C ) y ) ) /\ z e. ( Base ` D ) ) -> y e. ( Base ` C ) )
92		simplr 792	. . . . . . . . . . . 12 |- ( ( ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) /\ g e. ( x ( Hom ` C ) y ) ) /\ z e. ( Base ` D ) ) -> g e. ( x ( Hom ` C ) y ) )
93		eqid 2622	. . . . . . . . . . . . 13 |- ( Id ` D ) = ( Id ` D )
94	9, 34, 93, 84, 89	catidcl 16343	. . . . . . . . . . . 12 |- ( ( ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) /\ g e. ( x ( Hom ` C ) y ) ) /\ z e. ( Base ` D ) ) -> ( ( Id ` D ) ` z ) e. ( z ( Hom ` D ) z ) )
95	1, 84, 85, 86, 8, 9, 88, 89, 33, 34, 91, 89, 92, 94	uncf2 16877	. . . . . . . . . . 11 |- ( ( ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) /\ g e. ( x ( Hom ` C ) y ) ) /\ z e. ( Base ` D ) ) -> ( g ( <. x , z >. ( 2nd ` F ) <. y , z >. ) ( ( Id ` D ) ` z ) ) = ( ( ( ( x ( 2nd ` G ) y ) ` g ) ` z ) ( <. ( ( 1st ` ( ( 1st ` G ) ` x ) ) ` z ) , ( ( 1st ` ( ( 1st ` G ) ` x ) ) ` z ) >. (comp ` E ) ( ( 1st ` ( ( 1st ` G ) ` y ) ) ` z ) ) ( ( z ( 2nd ` ( ( 1st ` G ) ` x ) ) z ) ` ( ( Id ` D ) ` z ) ) ) )
96	22	adantrr 753	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) -> ( ( 1st ` G ) ` x ) e. ( D Func E ) )
97	96	adantr 481	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) /\ g e. ( x ( Hom ` C ) y ) ) -> ( ( 1st ` G ) ` x ) e. ( D Func E ) )
98	15, 97, 23	sylancr 695	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) /\ g e. ( x ( Hom ` C ) y ) ) -> ( 1st ` ( ( 1st ` G ) ` x ) ) ( D Func E ) ( 2nd ` ( ( 1st ` G ) ` x ) ) )
99	98	adantr 481	. . . . . . . . . . . . 13 |- ( ( ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) /\ g e. ( x ( Hom ` C ) y ) ) /\ z e. ( Base ` D ) ) -> ( 1st ` ( ( 1st ` G ) ` x ) ) ( D Func E ) ( 2nd ` ( ( 1st ` G ) ` x ) ) )
100	9, 93, 48, 99, 89	funcid 16530	. . . . . . . . . . . 12 |- ( ( ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) /\ g e. ( x ( Hom ` C ) y ) ) /\ z e. ( Base ` D ) ) -> ( ( z ( 2nd ` ( ( 1st ` G ) ` x ) ) z ) ` ( ( Id ` D ) ` z ) ) = ( ( Id ` E ) ` ( ( 1st ` ( ( 1st ` G ) ` x ) ) ` z ) ) )
101	100	oveq2d 6666	. . . . . . . . . . 11 |- ( ( ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) /\ g e. ( x ( Hom ` C ) y ) ) /\ z e. ( Base ` D ) ) -> ( ( ( ( x ( 2nd ` G ) y ) ` g ) ` z ) ( <. ( ( 1st ` ( ( 1st ` G ) ` x ) ) ` z ) , ( ( 1st ` ( ( 1st ` G ) ` x ) ) ` z ) >. (comp ` E ) ( ( 1st ` ( ( 1st ` G ) ` y ) ) ` z ) ) ( ( z ( 2nd ` ( ( 1st ` G ) ` x ) ) z ) ` ( ( Id ` D ) ` z ) ) ) = ( ( ( ( x ( 2nd ` G ) y ) ` g ) ` z ) ( <. ( ( 1st ` ( ( 1st ` G ) ` x ) ) ` z ) , ( ( 1st ` ( ( 1st ` G ) ` x ) ) ` z ) >. (comp ` E ) ( ( 1st ` ( ( 1st ` G ) ` y ) ) ` z ) ) ( ( Id ` E ) ` ( ( 1st ` ( ( 1st ` G ) ` x ) ) ` z ) ) ) )
102	9, 14, 98	funcf1 16526	. . . . . . . . . . . . 13 |- ( ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) /\ g e. ( x ( Hom ` C ) y ) ) -> ( 1st ` ( ( 1st ` G ) ` x ) ) : ( Base ` D ) --> ( Base ` E ) )
103	102	ffvelrnda 6359	. . . . . . . . . . . 12 |- ( ( ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) /\ g e. ( x ( Hom ` C ) y ) ) /\ z e. ( Base ` D ) ) -> ( ( 1st ` ( ( 1st ` G ) ` x ) ) ` z ) e. ( Base ` E ) )
104	21	ffvelrnda 6359	. . . . . . . . . . . . . . . . 17 |- ( ( ph /\ y e. ( Base ` C ) ) -> ( ( 1st ` G ) ` y ) e. ( D Func E ) )
105	104	adantrl 752	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) -> ( ( 1st ` G ) ` y ) e. ( D Func E ) )
106	105	adantr 481	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) /\ g e. ( x ( Hom ` C ) y ) ) -> ( ( 1st ` G ) ` y ) e. ( D Func E ) )
107		1st2ndbr 7217	. . . . . . . . . . . . . . 15 |- ( ( Rel ( D Func E ) /\ ( ( 1st ` G ) ` y ) e. ( D Func E ) ) -> ( 1st ` ( ( 1st ` G ) ` y ) ) ( D Func E ) ( 2nd ` ( ( 1st ` G ) ` y ) ) )
108	15, 106, 107	sylancr 695	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) /\ g e. ( x ( Hom ` C ) y ) ) -> ( 1st ` ( ( 1st ` G ) ` y ) ) ( D Func E ) ( 2nd ` ( ( 1st ` G ) ` y ) ) )
109	9, 14, 108	funcf1 16526	. . . . . . . . . . . . 13 |- ( ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) /\ g e. ( x ( Hom ` C ) y ) ) -> ( 1st ` ( ( 1st ` G ) ` y ) ) : ( Base ` D ) --> ( Base ` E ) )
110	109	ffvelrnda 6359	. . . . . . . . . . . 12 |- ( ( ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) /\ g e. ( x ( Hom ` C ) y ) ) /\ z e. ( Base ` D ) ) -> ( ( 1st ` ( ( 1st ` G ) ` y ) ) ` z ) e. ( Base ` E ) )
111		eqid 2622	. . . . . . . . . . . . 13 |- ( D Nat E ) = ( D Nat E )
112	16, 111	fuchom 16621	. . . . . . . . . . . . . . . 16 |- ( D Nat E ) = ( Hom ` ( D FuncCat E ) )
113	20	ad3antrrr 766	. . . . . . . . . . . . . . . 16 |- ( ( ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) /\ g e. ( x ( Hom ` C ) y ) ) /\ z e. ( Base ` D ) ) -> ( 1st ` G ) ( C Func ( D FuncCat E ) ) ( 2nd ` G ) )
114	8, 33, 112, 113, 88, 91	funcf2 16528	. . . . . . . . . . . . . . 15 |- ( ( ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) /\ g e. ( x ( Hom ` C ) y ) ) /\ z e. ( Base ` D ) ) -> ( x ( 2nd ` G ) y ) : ( x ( Hom ` C ) y ) --> ( ( ( 1st ` G ) ` x ) ( D Nat E ) ( ( 1st ` G ) ` y ) ) )
115	114, 92	ffvelrnd 6360	. . . . . . . . . . . . . 14 |- ( ( ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) /\ g e. ( x ( Hom ` C ) y ) ) /\ z e. ( Base ` D ) ) -> ( ( x ( 2nd ` G ) y ) ` g ) e. ( ( ( 1st ` G ) ` x ) ( D Nat E ) ( ( 1st ` G ) ` y ) ) )
116	111, 115	nat1st2nd 16611	. . . . . . . . . . . . 13 |- ( ( ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) /\ g e. ( x ( Hom ` C ) y ) ) /\ z e. ( Base ` D ) ) -> ( ( x ( 2nd ` G ) y ) ` g ) e. ( <. ( 1st ` ( ( 1st ` G ) ` x ) ) , ( 2nd ` ( ( 1st ` G ) ` x ) ) >. ( D Nat E ) <. ( 1st ` ( ( 1st ` G ) ` y ) ) , ( 2nd ` ( ( 1st ` G ) ` y ) ) >. ) )
117	111, 116, 9, 58, 89	natcl 16613	. . . . . . . . . . . 12 |- ( ( ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) /\ g e. ( x ( Hom ` C ) y ) ) /\ z e. ( Base ` D ) ) -> ( ( ( x ( 2nd ` G ) y ) ` g ) ` z ) e. ( ( ( 1st ` ( ( 1st ` G ) ` x ) ) ` z ) ( Hom ` E ) ( ( 1st ` ( ( 1st ` G ) ` y ) ) ` z ) ) )
118	14, 58, 48, 85, 103, 60, 110, 117	catrid 16345	. . . . . . . . . . 11 |- ( ( ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) /\ g e. ( x ( Hom ` C ) y ) ) /\ z e. ( Base ` D ) ) -> ( ( ( ( x ( 2nd ` G ) y ) ` g ) ` z ) ( <. ( ( 1st ` ( ( 1st ` G ) ` x ) ) ` z ) , ( ( 1st ` ( ( 1st ` G ) ` x ) ) ` z ) >. (comp ` E ) ( ( 1st ` ( ( 1st ` G ) ` y ) ) ` z ) ) ( ( Id ` E ) ` ( ( 1st ` ( ( 1st ` G ) ` x ) ) ` z ) ) ) = ( ( ( x ( 2nd ` G ) y ) ` g ) ` z ) )
119	95, 101, 118	3eqtrd 2660	. . . . . . . . . 10 |- ( ( ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) /\ g e. ( x ( Hom ` C ) y ) ) /\ z e. ( Base ` D ) ) -> ( g ( <. x , z >. ( 2nd ` F ) <. y , z >. ) ( ( Id ` D ) ` z ) ) = ( ( ( x ( 2nd ` G ) y ) ` g ) ` z ) )
120	119	mpteq2dva 4744	. . . . . . . . 9 |- ( ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) /\ g e. ( x ( Hom ` C ) y ) ) -> ( z e. ( Base ` D ) |-> ( g ( <. x , z >. ( 2nd ` F ) <. y , z >. ) ( ( Id ` D ) ` z ) ) ) = ( z e. ( Base ` D ) |-> ( ( ( x ( 2nd ` G ) y ) ` g ) ` z ) ) )
121	20	adantr 481	. . . . . . . . . . . . . 14 |- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) -> ( 1st ` G ) ( C Func ( D FuncCat E ) ) ( 2nd ` G ) )
122	8, 33, 112, 121, 87, 90	funcf2 16528	. . . . . . . . . . . . 13 |- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) -> ( x ( 2nd ` G ) y ) : ( x ( Hom ` C ) y ) --> ( ( ( 1st ` G ) ` x ) ( D Nat E ) ( ( 1st ` G ) ` y ) ) )
123	122	ffvelrnda 6359	. . . . . . . . . . . 12 |- ( ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) /\ g e. ( x ( Hom ` C ) y ) ) -> ( ( x ( 2nd ` G ) y ) ` g ) e. ( ( ( 1st ` G ) ` x ) ( D Nat E ) ( ( 1st ` G ) ` y ) ) )
124	111, 123	nat1st2nd 16611	. . . . . . . . . . 11 |- ( ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) /\ g e. ( x ( Hom ` C ) y ) ) -> ( ( x ( 2nd ` G ) y ) ` g ) e. ( <. ( 1st ` ( ( 1st ` G ) ` x ) ) , ( 2nd ` ( ( 1st ` G ) ` x ) ) >. ( D Nat E ) <. ( 1st ` ( ( 1st ` G ) ` y ) ) , ( 2nd ` ( ( 1st ` G ) ` y ) ) >. ) )
125	111, 124, 9	natfn 16614	. . . . . . . . . 10 |- ( ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) /\ g e. ( x ( Hom ` C ) y ) ) -> ( ( x ( 2nd ` G ) y ) ` g ) Fn ( Base ` D ) )
126		dffn5 6241	. . . . . . . . . 10 |- ( ( ( x ( 2nd ` G ) y ) ` g ) Fn ( Base ` D ) <-> ( ( x ( 2nd ` G ) y ) ` g ) = ( z e. ( Base ` D ) |-> ( ( ( x ( 2nd ` G ) y ) ` g ) ` z ) ) )
127	125, 126	sylib 208	. . . . . . . . 9 |- ( ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) /\ g e. ( x ( Hom ` C ) y ) ) -> ( ( x ( 2nd ` G ) y ) ` g ) = ( z e. ( Base ` D ) |-> ( ( ( x ( 2nd ` G ) y ) ` g ) ` z ) ) )
128	120, 127	eqtr4d 2659	. . . . . . . 8 |- ( ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) /\ g e. ( x ( Hom ` C ) y ) ) -> ( z e. ( Base ` D ) |-> ( g ( <. x , z >. ( 2nd ` F ) <. y , z >. ) ( ( Id ` D ) ` z ) ) ) = ( ( x ( 2nd ` G ) y ) ` g ) )
129	128	mpteq2dva 4744	. . . . . . 7 |- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) -> ( g e. ( x ( Hom ` C ) y ) |-> ( z e. ( Base ` D ) |-> ( g ( <. x , z >. ( 2nd ` F ) <. y , z >. ) ( ( Id ` D ) ` z ) ) ) ) = ( g e. ( x ( Hom ` C ) y ) |-> ( ( x ( 2nd ` G ) y ) ` g ) ) )
130	122	feqmptd 6249	. . . . . . 7 |- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) -> ( x ( 2nd ` G ) y ) = ( g e. ( x ( Hom ` C ) y ) |-> ( ( x ( 2nd ` G ) y ) ` g ) ) )
131	129, 130	eqtr4d 2659	. . . . . 6 |- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) -> ( g e. ( x ( Hom ` C ) y ) |-> ( z e. ( Base ` D ) |-> ( g ( <. x , z >. ( 2nd ` F ) <. y , z >. ) ( ( Id ` D ) ` z ) ) ) ) = ( x ( 2nd ` G ) y ) )
132	131	3impb 1260	. . . . 5 |- ( ( ph /\ x e. ( Base ` C ) /\ y e. ( Base ` C ) ) -> ( g e. ( x ( Hom ` C ) y ) |-> ( z e. ( Base ` D ) |-> ( g ( <. x , z >. ( 2nd ` F ) <. y , z >. ) ( ( Id ` D ) ` z ) ) ) ) = ( x ( 2nd ` G ) y ) )
133	132	mpt2eq3dva 6719	. . . 4 |- ( ph -> ( x e. ( Base ` C ) , y e. ( Base ` C ) |-> ( g e. ( x ( Hom ` C ) y ) |-> ( z e. ( Base ` D ) |-> ( g ( <. x , z >. ( 2nd ` F ) <. y , z >. ) ( ( Id ` D ) ` z ) ) ) ) ) = ( x e. ( Base ` C ) , y e. ( Base ` C ) |-> ( x ( 2nd ` G ) y ) ) )
134	8, 20	funcfn2 16529	. . . . 5 |- ( ph -> ( 2nd ` G ) Fn ( ( Base ` C ) X. ( Base ` C ) ) )
135		fnov 6768	. . . . 5 |- ( ( 2nd ` G ) Fn ( ( Base ` C ) X. ( Base ` C ) ) <-> ( 2nd ` G ) = ( x e. ( Base ` C ) , y e. ( Base ` C ) |-> ( x ( 2nd ` G ) y ) ) )
136	134, 135	sylib 208	. . . 4 |- ( ph -> ( 2nd ` G ) = ( x e. ( Base ` C ) , y e. ( Base ` C ) |-> ( x ( 2nd ` G ) y ) ) )
137	133, 136	eqtr4d 2659	. . 3 |- ( ph -> ( x e. ( Base ` C ) , y e. ( Base ` C ) |-> ( g e. ( x ( Hom ` C ) y ) |-> ( z e. ( Base ` D ) |-> ( g ( <. x , z >. ( 2nd ` F ) <. y , z >. ) ( ( Id ` D ) ` z ) ) ) ) ) = ( 2nd ` G ) )
138	83, 137	opeq12d 4410	. 2 |- ( ph -> <. ( x e. ( Base ` C ) |-> <. ( y e. ( Base ` D ) |-> ( x ( 1st ` F ) y ) ) , ( y e. ( Base ` D ) , z e. ( Base ` D ) |-> ( g e. ( y ( Hom ` D ) z ) |-> ( ( ( Id ` C ) ` x ) ( <. x , y >. ( 2nd ` F ) <. x , z >. ) g ) ) ) >. ) , ( x e. ( Base ` C ) , y e. ( Base ` C ) |-> ( g e. ( x ( Hom ` C ) y ) |-> ( z e. ( Base ` D ) |-> ( g ( <. x , z >. ( 2nd ` F ) <. y , z >. ) ( ( Id ` D ) ` z ) ) ) ) ) >. = <. ( 1st ` G ) , ( 2nd ` G ) >. )
139		eqid 2622	. . 3 |- ( <. C , D >. curryF F ) = ( <. C , D >. curryF F )
140	1, 2, 4, 6	uncfcl 16875	. . 3 |- ( ph -> F e. ( ( C X.c D ) Func E ) )
141	139, 8, 40, 2, 140, 9, 34, 37, 33, 93	curfval 16863	. 2 |- ( ph -> ( <. C , D >. curryF F ) = <. ( x e. ( Base ` C ) |-> <. ( y e. ( Base ` D ) |-> ( x ( 1st ` F ) y ) ) , ( y e. ( Base ` D ) , z e. ( Base ` D ) |-> ( g e. ( y ( Hom ` D ) z ) |-> ( ( ( Id ` C ) ` x ) ( <. x , y >. ( 2nd ` F ) <. x , z >. ) g ) ) ) >. ) , ( x e. ( Base ` C ) , y e. ( Base ` C ) |-> ( g e. ( x ( Hom ` C ) y ) |-> ( z e. ( Base ` D ) |-> ( g ( <. x , z >. ( 2nd ` F ) <. y , z >. ) ( ( Id ` D ) ` z ) ) ) ) ) >. )
142		1st2nd 7214	. . 3 |- ( ( Rel ( C Func ( D FuncCat E ) ) /\ G e. ( C Func ( D FuncCat E ) ) ) -> G = <. ( 1st ` G ) , ( 2nd ` G ) >. )
143	18, 6, 142	sylancr 695	. 2 |- ( ph -> G = <. ( 1st ` G ) , ( 2nd ` G ) >. )
144	138, 141, 143	3eqtr4d 2666	1 |- ( ph -> ( <. C , D >. curryF F ) = G )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   <.cop 4183   class class class wbr 4653   |-> cmpt 4729   X. cxp 5112   o. ccom 5118   Rel wrel 5119   Fn wfn 5883   -->wf 5884   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   1stc1st 7166   2ndc2nd 7167   <"cs3 13587   Basecbs 15857   Hom chom 15952  compcco 15953   Catccat 16325   Idccid 16326   Func cfunc 16514   Nat cnat 16601   FuncCat cfuc 16602   curry_F ccurf 16850   uncurry_F cuncf 16851

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-map 7859  df-ixp 7909  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-card 8765  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-z 11378  df-dec 11494  df-uz 11688  df-fz 12327  df-fzo 12466  df-hash 13118  df-word 13299  df-concat 13301  df-s1 13302  df-s2 13593  df-s3 13594  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-hom 15966  df-cco 15967  df-cat 16329  df-cid 16330  df-func 16518  df-cofu 16520  df-nat 16603  df-fuc 16604  df-xpc 16812  df-1stf 16813  df-2ndf 16814  df-prf 16815  df-evlf 16853  df-curf 16854  df-uncf 16855

This theorem is referenced by: (None)