Theorem curf2ndf 16887

Description: As shown in diagval 16880, the currying of the first projection is the diagonal functor. On the other hand, the currying of the second projection is x e. C |-> ( y e. D |-> y ), which is a constant functor of the identity functor at D. (Contributed by Mario Carneiro, 15-Jan-2017.)

Hypotheses

Ref	Expression
curf2ndf.q	|- Q = ( D FuncCat D )
curf2ndf.c	|- ( ph -> C e. Cat )
curf2ndf.d	|- ( ph -> D e. Cat )

Assertion

Ref	Expression
curf2ndf	|- ( ph -> ( <. C , D >. curryF ( C 2ndF D ) ) = ( ( 1st ` ( QΔfunc C ) ) ` (idfunc ` D ) ) )

Proof of Theorem curf2ndf

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

Step	Hyp	Ref	Expression
1		df-ov 6653	. . . . . . . . . . 11 |- ( x ( 1st ` ( C 2ndF D ) ) y ) = ( ( 1st ` ( C 2ndF D ) ) ` <. x , y >. )
2		eqid 2622	. . . . . . . . . . . . 13 |- ( C X.c D ) = ( C X.c D )
3		eqid 2622	. . . . . . . . . . . . . 14 |- ( Base ` C ) = ( Base ` C )
4		eqid 2622	. . . . . . . . . . . . . 14 |- ( Base ` D ) = ( Base ` D )
5	2, 3, 4	xpcbas 16818	. . . . . . . . . . . . 13 |- ( ( Base ` C ) X. ( Base ` D ) ) = ( Base ` ( C X.c D ) )
6		eqid 2622	. . . . . . . . . . . . 13 |- ( Hom ` ( C X.c D ) ) = ( Hom ` ( C X.c D ) )
7		curf2ndf.c	. . . . . . . . . . . . . 14 |- ( ph -> C e. Cat )
8	7	ad2antrr 762	. . . . . . . . . . . . 13 |- ( ( ( ph /\ x e. ( Base ` C ) ) /\ y e. ( Base ` D ) ) -> C e. Cat )
9		curf2ndf.d	. . . . . . . . . . . . . 14 |- ( ph -> D e. Cat )
10	9	ad2antrr 762	. . . . . . . . . . . . 13 |- ( ( ( ph /\ x e. ( Base ` C ) ) /\ y e. ( Base ` D ) ) -> D e. Cat )
11		eqid 2622	. . . . . . . . . . . . 13 |- ( C 2ndF D ) = ( C 2ndF D )
12		opelxpi 5148	. . . . . . . . . . . . . 14 |- ( ( x e. ( Base ` C ) /\ y e. ( Base ` D ) ) -> <. x , y >. e. ( ( Base ` C ) X. ( Base ` D ) ) )
13	12	adantll 750	. . . . . . . . . . . . 13 |- ( ( ( ph /\ x e. ( Base ` C ) ) /\ y e. ( Base ` D ) ) -> <. x , y >. e. ( ( Base ` C ) X. ( Base ` D ) ) )
14	2, 5, 6, 8, 10, 11, 13	2ndf1 16835	. . . . . . . . . . . 12 |- ( ( ( ph /\ x e. ( Base ` C ) ) /\ y e. ( Base ` D ) ) -> ( ( 1st ` ( C 2ndF D ) ) ` <. x , y >. ) = ( 2nd ` <. x , y >. ) )
15		vex 3203	. . . . . . . . . . . . 13 |- x e. _V
16		vex 3203	. . . . . . . . . . . . 13 |- y e. _V
17	15, 16	op2nd 7177	. . . . . . . . . . . 12 |- ( 2nd ` <. x , y >. ) = y
18	14, 17	syl6eq 2672	. . . . . . . . . . 11 |- ( ( ( ph /\ x e. ( Base ` C ) ) /\ y e. ( Base ` D ) ) -> ( ( 1st ` ( C 2ndF D ) ) ` <. x , y >. ) = y )
19	1, 18	syl5eq 2668	. . . . . . . . . 10 |- ( ( ( ph /\ x e. ( Base ` C ) ) /\ y e. ( Base ` D ) ) -> ( x ( 1st ` ( C 2ndF D ) ) y ) = y )
20	19	mpteq2dva 4744	. . . . . . . . 9 |- ( ( ph /\ x e. ( Base ` C ) ) -> ( y e. ( Base ` D ) |-> ( x ( 1st ` ( C 2ndF D ) ) y ) ) = ( y e. ( Base ` D ) |-> y ) )
21		mptresid 5456	. . . . . . . . 9 |- ( y e. ( Base ` D ) |-> y ) = ( _I |` ( Base ` D ) )
22	20, 21	syl6eq 2672	. . . . . . . 8 |- ( ( ph /\ x e. ( Base ` C ) ) -> ( y e. ( Base ` D ) |-> ( x ( 1st ` ( C 2ndF D ) ) y ) ) = ( _I |` ( Base ` D ) ) )
23		df-ov 6653	. . . . . . . . . . . . . . 15 |- ( ( ( Id ` C ) ` x ) ( <. x , y >. ( 2nd ` ( C 2ndF D ) ) <. x , z >. ) f ) = ( ( <. x , y >. ( 2nd ` ( C 2ndF D ) ) <. x , z >. ) ` <. ( ( Id ` C ) ` x ) , f >. )
24	8	ad2antrr 762	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( ( ph /\ x e. ( Base ` C ) ) /\ y e. ( Base ` D ) ) /\ z e. ( Base ` D ) ) /\ f e. ( y ( Hom ` D ) z ) ) -> C e. Cat )
25	10	ad2antrr 762	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( ( ph /\ x e. ( Base ` C ) ) /\ y e. ( Base ` D ) ) /\ z e. ( Base ` D ) ) /\ f e. ( y ( Hom ` D ) z ) ) -> D e. Cat )
26	13	ad2antrr 762	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( ( ph /\ x e. ( Base ` C ) ) /\ y e. ( Base ` D ) ) /\ z e. ( Base ` D ) ) /\ f e. ( y ( Hom ` D ) z ) ) -> <. x , y >. e. ( ( Base ` C ) X. ( Base ` D ) ) )
27		simp-4r 807	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ( ( ph /\ x e. ( Base ` C ) ) /\ y e. ( Base ` D ) ) /\ z e. ( Base ` D ) ) /\ f e. ( y ( Hom ` D ) z ) ) -> x e. ( Base ` C ) )
28		simplr 792	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ( ( ph /\ x e. ( Base ` C ) ) /\ y e. ( Base ` D ) ) /\ z e. ( Base ` D ) ) /\ f e. ( y ( Hom ` D ) z ) ) -> z e. ( Base ` D ) )
29		opelxpi 5148	. . . . . . . . . . . . . . . . . 18 |- ( ( x e. ( Base ` C ) /\ z e. ( Base ` D ) ) -> <. x , z >. e. ( ( Base ` C ) X. ( Base ` D ) ) )
30	27, 28, 29	syl2anc 693	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( ( ph /\ x e. ( Base ` C ) ) /\ y e. ( Base ` D ) ) /\ z e. ( Base ` D ) ) /\ f e. ( y ( Hom ` D ) z ) ) -> <. x , z >. e. ( ( Base ` C ) X. ( Base ` D ) ) )
31	2, 5, 6, 24, 25, 11, 26, 30	2ndf2 16836	. . . . . . . . . . . . . . . 16 |- ( ( ( ( ( ph /\ x e. ( Base ` C ) ) /\ y e. ( Base ` D ) ) /\ z e. ( Base ` D ) ) /\ f e. ( y ( Hom ` D ) z ) ) -> ( <. x , y >. ( 2nd ` ( C 2ndF D ) ) <. x , z >. ) = ( 2nd |` ( <. x , y >. ( Hom ` ( C X.c D ) ) <. x , z >. ) ) )
32	31	fveq1d 6193	. . . . . . . . . . . . . . 15 |- ( ( ( ( ( ph /\ x e. ( Base ` C ) ) /\ y e. ( Base ` D ) ) /\ z e. ( Base ` D ) ) /\ f e. ( y ( Hom ` D ) z ) ) -> ( ( <. x , y >. ( 2nd ` ( C 2ndF D ) ) <. x , z >. ) ` <. ( ( Id ` C ) ` x ) , f >. ) = ( ( 2nd |` ( <. x , y >. ( Hom ` ( C X.c D ) ) <. x , z >. ) ) ` <. ( ( Id ` C ) ` x ) , f >. ) )
33	23, 32	syl5eq 2668	. . . . . . . . . . . . . 14 |- ( ( ( ( ( ph /\ x e. ( Base ` C ) ) /\ y e. ( Base ` D ) ) /\ z e. ( Base ` D ) ) /\ f e. ( y ( Hom ` D ) z ) ) -> ( ( ( Id ` C ) ` x ) ( <. x , y >. ( 2nd ` ( C 2ndF D ) ) <. x , z >. ) f ) = ( ( 2nd |` ( <. x , y >. ( Hom ` ( C X.c D ) ) <. x , z >. ) ) ` <. ( ( Id ` C ) ` x ) , f >. ) )
34		eqid 2622	. . . . . . . . . . . . . . . . . . . 20 |- ( Hom ` C ) = ( Hom ` C )
35		eqid 2622	. . . . . . . . . . . . . . . . . . . 20 |- ( Id ` C ) = ( Id ` C )
36	7	adantr 481	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ph /\ x e. ( Base ` C ) ) -> C e. Cat )
37		simpr 477	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ph /\ x e. ( Base ` C ) ) -> x e. ( Base ` C ) )
38	3, 34, 35, 36, 37	catidcl 16343	. . . . . . . . . . . . . . . . . . 19 |- ( ( ph /\ x e. ( Base ` C ) ) -> ( ( Id ` C ) ` x ) e. ( x ( Hom ` C ) x ) )
39	38	ad3antrrr 766	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ( ( ph /\ x e. ( Base ` C ) ) /\ y e. ( Base ` D ) ) /\ z e. ( Base ` D ) ) /\ f e. ( y ( Hom ` D ) z ) ) -> ( ( Id ` C ) ` x ) e. ( x ( Hom ` C ) x ) )
40		simpr 477	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ( ( ph /\ x e. ( Base ` C ) ) /\ y e. ( Base ` D ) ) /\ z e. ( Base ` D ) ) /\ f e. ( y ( Hom ` D ) z ) ) -> f e. ( y ( Hom ` D ) z ) )
41		opelxpi 5148	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ( Id ` C ) ` x ) e. ( x ( Hom ` C ) x ) /\ f e. ( y ( Hom ` D ) z ) ) -> <. ( ( Id ` C ) ` x ) , f >. e. ( ( x ( Hom ` C ) x ) X. ( y ( Hom ` D ) z ) ) )
42	39, 40, 41	syl2anc 693	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( ( ph /\ x e. ( Base ` C ) ) /\ y e. ( Base ` D ) ) /\ z e. ( Base ` D ) ) /\ f e. ( y ( Hom ` D ) z ) ) -> <. ( ( Id ` C ) ` x ) , f >. e. ( ( x ( Hom ` C ) x ) X. ( y ( Hom ` D ) z ) ) )
43		eqid 2622	. . . . . . . . . . . . . . . . . 18 |- ( Hom ` D ) = ( Hom ` D )
44		simpllr 799	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ( ( ph /\ x e. ( Base ` C ) ) /\ y e. ( Base ` D ) ) /\ z e. ( Base ` D ) ) /\ f e. ( y ( Hom ` D ) z ) ) -> y e. ( Base ` D ) )
45	2, 3, 4, 34, 43, 27, 44, 27, 28, 6	xpchom2 16826	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( ( ph /\ x e. ( Base ` C ) ) /\ y e. ( Base ` D ) ) /\ z e. ( Base ` D ) ) /\ f e. ( y ( Hom ` D ) z ) ) -> ( <. x , y >. ( Hom ` ( C X.c D ) ) <. x , z >. ) = ( ( x ( Hom ` C ) x ) X. ( y ( Hom ` D ) z ) ) )
46	42, 45	eleqtrrd 2704	. . . . . . . . . . . . . . . 16 |- ( ( ( ( ( ph /\ x e. ( Base ` C ) ) /\ y e. ( Base ` D ) ) /\ z e. ( Base ` D ) ) /\ f e. ( y ( Hom ` D ) z ) ) -> <. ( ( Id ` C ) ` x ) , f >. e. ( <. x , y >. ( Hom ` ( C X.c D ) ) <. x , z >. ) )
47		fvres 6207	. . . . . . . . . . . . . . . 16 |- ( <. ( ( Id ` C ) ` x ) , f >. e. ( <. x , y >. ( Hom ` ( C X.c D ) ) <. x , z >. ) -> ( ( 2nd |` ( <. x , y >. ( Hom ` ( C X.c D ) ) <. x , z >. ) ) ` <. ( ( Id ` C ) ` x ) , f >. ) = ( 2nd ` <. ( ( Id ` C ) ` x ) , f >. ) )
48	46, 47	syl 17	. . . . . . . . . . . . . . 15 |- ( ( ( ( ( ph /\ x e. ( Base ` C ) ) /\ y e. ( Base ` D ) ) /\ z e. ( Base ` D ) ) /\ f e. ( y ( Hom ` D ) z ) ) -> ( ( 2nd |` ( <. x , y >. ( Hom ` ( C X.c D ) ) <. x , z >. ) ) ` <. ( ( Id ` C ) ` x ) , f >. ) = ( 2nd ` <. ( ( Id ` C ) ` x ) , f >. ) )
49		fvex 6201	. . . . . . . . . . . . . . . 16 |- ( ( Id ` C ) ` x ) e. _V
50		vex 3203	. . . . . . . . . . . . . . . 16 |- f e. _V
51	49, 50	op2nd 7177	. . . . . . . . . . . . . . 15 |- ( 2nd ` <. ( ( Id ` C ) ` x ) , f >. ) = f
52	48, 51	syl6eq 2672	. . . . . . . . . . . . . 14 |- ( ( ( ( ( ph /\ x e. ( Base ` C ) ) /\ y e. ( Base ` D ) ) /\ z e. ( Base ` D ) ) /\ f e. ( y ( Hom ` D ) z ) ) -> ( ( 2nd |` ( <. x , y >. ( Hom ` ( C X.c D ) ) <. x , z >. ) ) ` <. ( ( Id ` C ) ` x ) , f >. ) = f )
53	33, 52	eqtrd 2656	. . . . . . . . . . . . 13 |- ( ( ( ( ( ph /\ x e. ( Base ` C ) ) /\ y e. ( Base ` D ) ) /\ z e. ( Base ` D ) ) /\ f e. ( y ( Hom ` D ) z ) ) -> ( ( ( Id ` C ) ` x ) ( <. x , y >. ( 2nd ` ( C 2ndF D ) ) <. x , z >. ) f ) = f )
54	53	mpteq2dva 4744	. . . . . . . . . . . 12 |- ( ( ( ( ph /\ x e. ( Base ` C ) ) /\ y e. ( Base ` D ) ) /\ z e. ( Base ` D ) ) -> ( f e. ( y ( Hom ` D ) z ) |-> ( ( ( Id ` C ) ` x ) ( <. x , y >. ( 2nd ` ( C 2ndF D ) ) <. x , z >. ) f ) ) = ( f e. ( y ( Hom ` D ) z ) |-> f ) )
55		mptresid 5456	. . . . . . . . . . . 12 |- ( f e. ( y ( Hom ` D ) z ) |-> f ) = ( _I |` ( y ( Hom ` D ) z ) )
56	54, 55	syl6eq 2672	. . . . . . . . . . 11 |- ( ( ( ( ph /\ x e. ( Base ` C ) ) /\ y e. ( Base ` D ) ) /\ z e. ( Base ` D ) ) -> ( f e. ( y ( Hom ` D ) z ) |-> ( ( ( Id ` C ) ` x ) ( <. x , y >. ( 2nd ` ( C 2ndF D ) ) <. x , z >. ) f ) ) = ( _I |` ( y ( Hom ` D ) z ) ) )
57	56	3impa 1259	. . . . . . . . . 10 |- ( ( ( ph /\ x e. ( Base ` C ) ) /\ y e. ( Base ` D ) /\ z e. ( Base ` D ) ) -> ( f e. ( y ( Hom ` D ) z ) |-> ( ( ( Id ` C ) ` x ) ( <. x , y >. ( 2nd ` ( C 2ndF D ) ) <. x , z >. ) f ) ) = ( _I |` ( y ( Hom ` D ) z ) ) )
58	57	mpt2eq3dva 6719	. . . . . . . . 9 |- ( ( ph /\ x e. ( Base ` C ) ) -> ( y e. ( Base ` D ) , z e. ( Base ` D ) |-> ( f e. ( y ( Hom ` D ) z ) |-> ( ( ( Id ` C ) ` x ) ( <. x , y >. ( 2nd ` ( C 2ndF D ) ) <. x , z >. ) f ) ) ) = ( y e. ( Base ` D ) , z e. ( Base ` D ) |-> ( _I |` ( y ( Hom ` D ) z ) ) ) )
59		fveq2 6191	. . . . . . . . . . . 12 |- ( u = <. y , z >. -> ( ( Hom ` D ) ` u ) = ( ( Hom ` D ) ` <. y , z >. ) )
60		df-ov 6653	. . . . . . . . . . . 12 |- ( y ( Hom ` D ) z ) = ( ( Hom ` D ) ` <. y , z >. )
61	59, 60	syl6eqr 2674	. . . . . . . . . . 11 |- ( u = <. y , z >. -> ( ( Hom ` D ) ` u ) = ( y ( Hom ` D ) z ) )
62	61	reseq2d 5396	. . . . . . . . . 10 |- ( u = <. y , z >. -> ( _I |` ( ( Hom ` D ) ` u ) ) = ( _I |` ( y ( Hom ` D ) z ) ) )
63	62	mpt2mpt 6752	. . . . . . . . 9 |- ( u e. ( ( Base ` D ) X. ( Base ` D ) ) |-> ( _I |` ( ( Hom ` D ) ` u ) ) ) = ( y e. ( Base ` D ) , z e. ( Base ` D ) |-> ( _I |` ( y ( Hom ` D ) z ) ) )
64	58, 63	syl6eqr 2674	. . . . . . . 8 |- ( ( ph /\ x e. ( Base ` C ) ) -> ( y e. ( Base ` D ) , z e. ( Base ` D ) |-> ( f e. ( y ( Hom ` D ) z ) |-> ( ( ( Id ` C ) ` x ) ( <. x , y >. ( 2nd ` ( C 2ndF D ) ) <. x , z >. ) f ) ) ) = ( u e. ( ( Base ` D ) X. ( Base ` D ) ) |-> ( _I |` ( ( Hom ` D ) ` u ) ) ) )
65	22, 64	opeq12d 4410	. . . . . . 7 |- ( ( ph /\ x e. ( Base ` C ) ) -> <. ( y e. ( Base ` D ) |-> ( x ( 1st ` ( C 2ndF D ) ) y ) ) , ( y e. ( Base ` D ) , z e. ( Base ` D ) |-> ( f e. ( y ( Hom ` D ) z ) |-> ( ( ( Id ` C ) ` x ) ( <. x , y >. ( 2nd ` ( C 2ndF D ) ) <. x , z >. ) f ) ) ) >. = <. ( _I |` ( Base ` D ) ) , ( u e. ( ( Base ` D ) X. ( Base ` D ) ) |-> ( _I |` ( ( Hom ` D ) ` u ) ) ) >. )
66		eqid 2622	. . . . . . . 8 |- ( <. C , D >. curryF ( C 2ndF D ) ) = ( <. C , D >. curryF ( C 2ndF D ) )
67	9	adantr 481	. . . . . . . 8 |- ( ( ph /\ x e. ( Base ` C ) ) -> D e. Cat )
68	2, 7, 9, 11	2ndfcl 16838	. . . . . . . . 9 |- ( ph -> ( C 2ndF D ) e. ( ( C X.c D ) Func D ) )
69	68	adantr 481	. . . . . . . 8 |- ( ( ph /\ x e. ( Base ` C ) ) -> ( C 2ndF D ) e. ( ( C X.c D ) Func D ) )
70		eqid 2622	. . . . . . . 8 |- ( ( 1st ` ( <. C , D >. curryF ( C 2ndF D ) ) ) ` x ) = ( ( 1st ` ( <. C , D >. curryF ( C 2ndF D ) ) ) ` x )
71	66, 3, 36, 67, 69, 4, 37, 70, 43, 35	curf1 16865	. . . . . . 7 |- ( ( ph /\ x e. ( Base ` C ) ) -> ( ( 1st ` ( <. C , D >. curryF ( C 2ndF D ) ) ) ` x ) = <. ( y e. ( Base ` D ) |-> ( x ( 1st ` ( C 2ndF D ) ) y ) ) , ( y e. ( Base ` D ) , z e. ( Base ` D ) |-> ( f e. ( y ( Hom ` D ) z ) |-> ( ( ( Id ` C ) ` x ) ( <. x , y >. ( 2nd ` ( C 2ndF D ) ) <. x , z >. ) f ) ) ) >. )
72		eqid 2622	. . . . . . . 8 |- (idfunc ` D ) = (idfunc ` D )
73	72, 4, 67, 43	idfuval 16536	. . . . . . 7 |- ( ( ph /\ x e. ( Base ` C ) ) -> (idfunc ` D ) = <. ( _I |` ( Base ` D ) ) , ( u e. ( ( Base ` D ) X. ( Base ` D ) ) |-> ( _I |` ( ( Hom ` D ) ` u ) ) ) >. )
74	65, 71, 73	3eqtr4d 2666	. . . . . 6 |- ( ( ph /\ x e. ( Base ` C ) ) -> ( ( 1st ` ( <. C , D >. curryF ( C 2ndF D ) ) ) ` x ) = (idfunc ` D ) )
75		eqid 2622	. . . . . . 7 |- ( QΔfunc C ) = ( QΔfunc C )
76		curf2ndf.q	. . . . . . . . 9 |- Q = ( D FuncCat D )
77	76, 9, 9	fuccat 16630	. . . . . . . 8 |- ( ph -> Q e. Cat )
78	77	adantr 481	. . . . . . 7 |- ( ( ph /\ x e. ( Base ` C ) ) -> Q e. Cat )
79	76	fucbas 16620	. . . . . . 7 |- ( D Func D ) = ( Base ` Q )
80	72	idfucl 16541	. . . . . . . . 9 |- ( D e. Cat -> (idfunc ` D ) e. ( D Func D ) )
81	9, 80	syl 17	. . . . . . . 8 |- ( ph -> (idfunc ` D ) e. ( D Func D ) )
82	81	adantr 481	. . . . . . 7 |- ( ( ph /\ x e. ( Base ` C ) ) -> (idfunc ` D ) e. ( D Func D ) )
83		eqid 2622	. . . . . . 7 |- ( ( 1st ` ( QΔfunc C ) ) ` (idfunc ` D ) ) = ( ( 1st ` ( QΔfunc C ) ) ` (idfunc ` D ) )
84	75, 78, 36, 79, 82, 83, 3, 37	diag11 16883	. . . . . 6 |- ( ( ph /\ x e. ( Base ` C ) ) -> ( ( 1st ` ( ( 1st ` ( QΔfunc C ) ) ` (idfunc ` D ) ) ) ` x ) = (idfunc ` D ) )
85	74, 84	eqtr4d 2659	. . . . 5 |- ( ( ph /\ x e. ( Base ` C ) ) -> ( ( 1st ` ( <. C , D >. curryF ( C 2ndF D ) ) ) ` x ) = ( ( 1st ` ( ( 1st ` ( QΔfunc C ) ) ` (idfunc ` D ) ) ) ` x ) )
86	85	mpteq2dva 4744	. . . 4 |- ( ph -> ( x e. ( Base ` C ) |-> ( ( 1st ` ( <. C , D >. curryF ( C 2ndF D ) ) ) ` x ) ) = ( x e. ( Base ` C ) |-> ( ( 1st ` ( ( 1st ` ( QΔfunc C ) ) ` (idfunc ` D ) ) ) ` x ) ) )
87		relfunc 16522	. . . . . . 7 |- Rel ( C Func Q )
88	66, 76, 7, 9, 68	curfcl 16872	. . . . . . 7 |- ( ph -> ( <. C , D >. curryF ( C 2ndF D ) ) e. ( C Func Q ) )
89		1st2ndbr 7217	. . . . . . 7 |- ( ( Rel ( C Func Q ) /\ ( <. C , D >. curryF ( C 2ndF D ) ) e. ( C Func Q ) ) -> ( 1st ` ( <. C , D >. curryF ( C 2ndF D ) ) ) ( C Func Q ) ( 2nd ` ( <. C , D >. curryF ( C 2ndF D ) ) ) )
90	87, 88, 89	sylancr 695	. . . . . 6 |- ( ph -> ( 1st ` ( <. C , D >. curryF ( C 2ndF D ) ) ) ( C Func Q ) ( 2nd ` ( <. C , D >. curryF ( C 2ndF D ) ) ) )
91	3, 79, 90	funcf1 16526	. . . . 5 |- ( ph -> ( 1st ` ( <. C , D >. curryF ( C 2ndF D ) ) ) : ( Base ` C ) --> ( D Func D ) )
92	91	feqmptd 6249	. . . 4 |- ( ph -> ( 1st ` ( <. C , D >. curryF ( C 2ndF D ) ) ) = ( x e. ( Base ` C ) |-> ( ( 1st ` ( <. C , D >. curryF ( C 2ndF D ) ) ) ` x ) ) )
93	75, 77, 7, 79, 81, 83	diag1cl 16882	. . . . . . 7 |- ( ph -> ( ( 1st ` ( QΔfunc C ) ) ` (idfunc ` D ) ) e. ( C Func Q ) )
94		1st2ndbr 7217	. . . . . . 7 |- ( ( Rel ( C Func Q ) /\ ( ( 1st ` ( QΔfunc C ) ) ` (idfunc ` D ) ) e. ( C Func Q ) ) -> ( 1st ` ( ( 1st ` ( QΔfunc C ) ) ` (idfunc ` D ) ) ) ( C Func Q ) ( 2nd ` ( ( 1st ` ( QΔfunc C ) ) ` (idfunc ` D ) ) ) )
95	87, 93, 94	sylancr 695	. . . . . 6 |- ( ph -> ( 1st ` ( ( 1st ` ( QΔfunc C ) ) ` (idfunc ` D ) ) ) ( C Func Q ) ( 2nd ` ( ( 1st ` ( QΔfunc C ) ) ` (idfunc ` D ) ) ) )
96	3, 79, 95	funcf1 16526	. . . . 5 |- ( ph -> ( 1st ` ( ( 1st ` ( QΔfunc C ) ) ` (idfunc ` D ) ) ) : ( Base ` C ) --> ( D Func D ) )
97	96	feqmptd 6249	. . . 4 |- ( ph -> ( 1st ` ( ( 1st ` ( QΔfunc C ) ) ` (idfunc ` D ) ) ) = ( x e. ( Base ` C ) |-> ( ( 1st ` ( ( 1st ` ( QΔfunc C ) ) ` (idfunc ` D ) ) ) ` x ) ) )
98	86, 92, 97	3eqtr4d 2666	. . 3 |- ( ph -> ( 1st ` ( <. C , D >. curryF ( C 2ndF D ) ) ) = ( 1st ` ( ( 1st ` ( QΔfunc C ) ) ` (idfunc ` D ) ) ) )
99	9	ad2antrr 762	. . . . . . . . . . . 12 |- ( ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) /\ f e. ( x ( Hom ` C ) y ) ) -> D e. Cat )
100	72, 4, 99	idfu1st 16539	. . . . . . . . . . 11 |- ( ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) /\ f e. ( x ( Hom ` C ) y ) ) -> ( 1st ` (idfunc ` D ) ) = ( _I |` ( Base ` D ) ) )
101	100	coeq2d 5284	. . . . . . . . . 10 |- ( ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) /\ f e. ( x ( Hom ` C ) y ) ) -> ( ( Id ` D ) o. ( 1st ` (idfunc ` D ) ) ) = ( ( Id ` D ) o. ( _I |` ( Base ` D ) ) ) )
102		eqid 2622	. . . . . . . . . . 11 |- ( Id ` Q ) = ( Id ` Q )
103		eqid 2622	. . . . . . . . . . 11 |- ( Id ` D ) = ( Id ` D )
104	81	ad2antrr 762	. . . . . . . . . . 11 |- ( ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) /\ f e. ( x ( Hom ` C ) y ) ) -> (idfunc ` D ) e. ( D Func D ) )
105	76, 102, 103, 104	fucid 16631	. . . . . . . . . 10 |- ( ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) /\ f e. ( x ( Hom ` C ) y ) ) -> ( ( Id ` Q ) ` (idfunc ` D ) ) = ( ( Id ` D ) o. ( 1st ` (idfunc ` D ) ) ) )
106	4, 103	cidfn 16340	. . . . . . . . . . . . . 14 |- ( D e. Cat -> ( Id ` D ) Fn ( Base ` D ) )
107	99, 106	syl 17	. . . . . . . . . . . . 13 |- ( ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) /\ f e. ( x ( Hom ` C ) y ) ) -> ( Id ` D ) Fn ( Base ` D ) )
108		dffn2 6047	. . . . . . . . . . . . 13 |- ( ( Id ` D ) Fn ( Base ` D ) <-> ( Id ` D ) : ( Base ` D ) --> _V )
109	107, 108	sylib 208	. . . . . . . . . . . 12 |- ( ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) /\ f e. ( x ( Hom ` C ) y ) ) -> ( Id ` D ) : ( Base ` D ) --> _V )
110	109	feqmptd 6249	. . . . . . . . . . 11 |- ( ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) /\ f e. ( x ( Hom ` C ) y ) ) -> ( Id ` D ) = ( z e. ( Base ` D ) |-> ( ( Id ` D ) ` z ) ) )
111		fcoi1 6078	. . . . . . . . . . . 12 |- ( ( Id ` D ) : ( Base ` D ) --> _V -> ( ( Id ` D ) o. ( _I |` ( Base ` D ) ) ) = ( Id ` D ) )
112	109, 111	syl 17	. . . . . . . . . . 11 |- ( ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) /\ f e. ( x ( Hom ` C ) y ) ) -> ( ( Id ` D ) o. ( _I |` ( Base ` D ) ) ) = ( Id ` D ) )
113	7	ad2antrr 762	. . . . . . . . . . . . . . . 16 |- ( ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) /\ f e. ( x ( Hom ` C ) y ) ) -> C e. Cat )
114	113	adantr 481	. . . . . . . . . . . . . . 15 |- ( ( ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) /\ f e. ( x ( Hom ` C ) y ) ) /\ z e. ( Base ` D ) ) -> C e. Cat )
115	99	adantr 481	. . . . . . . . . . . . . . 15 |- ( ( ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) /\ f e. ( x ( Hom ` C ) y ) ) /\ z e. ( Base ` D ) ) -> D e. Cat )
116		simplrl 800	. . . . . . . . . . . . . . . 16 |- ( ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) /\ f e. ( x ( Hom ` C ) y ) ) -> x e. ( Base ` C ) )
117	116, 29	sylan 488	. . . . . . . . . . . . . . 15 |- ( ( ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) /\ f e. ( x ( Hom ` C ) y ) ) /\ z e. ( Base ` D ) ) -> <. x , z >. e. ( ( Base ` C ) X. ( Base ` D ) ) )
118		simplrr 801	. . . . . . . . . . . . . . . 16 |- ( ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) /\ f e. ( x ( Hom ` C ) y ) ) -> y e. ( Base ` C ) )
119		opelxpi 5148	. . . . . . . . . . . . . . . 16 |- ( ( y e. ( Base ` C ) /\ z e. ( Base ` D ) ) -> <. y , z >. e. ( ( Base ` C ) X. ( Base ` D ) ) )
120	118, 119	sylan 488	. . . . . . . . . . . . . . 15 |- ( ( ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) /\ f e. ( x ( Hom ` C ) y ) ) /\ z e. ( Base ` D ) ) -> <. y , z >. e. ( ( Base ` C ) X. ( Base ` D ) ) )
121	2, 5, 6, 114, 115, 11, 117, 120	2ndf2 16836	. . . . . . . . . . . . . 14 |- ( ( ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) /\ f e. ( x ( Hom ` C ) y ) ) /\ z e. ( Base ` D ) ) -> ( <. x , z >. ( 2nd ` ( C 2ndF D ) ) <. y , z >. ) = ( 2nd |` ( <. x , z >. ( Hom ` ( C X.c D ) ) <. y , z >. ) ) )
122	121	oveqd 6667	. . . . . . . . . . . . 13 |- ( ( ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) /\ f e. ( x ( Hom ` C ) y ) ) /\ z e. ( Base ` D ) ) -> ( f ( <. x , z >. ( 2nd ` ( C 2ndF D ) ) <. y , z >. ) ( ( Id ` D ) ` z ) ) = ( f ( 2nd |` ( <. x , z >. ( Hom ` ( C X.c D ) ) <. y , z >. ) ) ( ( Id ` D ) ` z ) ) )
123		df-ov 6653	. . . . . . . . . . . . . . 15 |- ( f ( 2nd |` ( <. x , z >. ( Hom ` ( C X.c D ) ) <. y , z >. ) ) ( ( Id ` D ) ` z ) ) = ( ( 2nd |` ( <. x , z >. ( Hom ` ( C X.c D ) ) <. y , z >. ) ) ` <. f , ( ( Id ` D ) ` z ) >. )
124		simplr 792	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) /\ f e. ( x ( Hom ` C ) y ) ) /\ z e. ( Base ` D ) ) -> f e. ( x ( Hom ` C ) y ) )
125		simpr 477	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) /\ f e. ( x ( Hom ` C ) y ) ) /\ z e. ( Base ` D ) ) -> z e. ( Base ` D ) )
126	4, 43, 103, 115, 125	catidcl 16343	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) /\ f e. ( x ( Hom ` C ) y ) ) /\ z e. ( Base ` D ) ) -> ( ( Id ` D ) ` z ) e. ( z ( Hom ` D ) z ) )
127		opelxpi 5148	. . . . . . . . . . . . . . . . . 18 |- ( ( f e. ( x ( Hom ` C ) y ) /\ ( ( Id ` D ) ` z ) e. ( z ( Hom ` D ) z ) ) -> <. f , ( ( Id ` D ) ` z ) >. e. ( ( x ( Hom ` C ) y ) X. ( z ( Hom ` D ) z ) ) )
128	124, 126, 127	syl2anc 693	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) /\ f e. ( x ( Hom ` C ) y ) ) /\ z e. ( Base ` D ) ) -> <. f , ( ( Id ` D ) ` z ) >. e. ( ( x ( Hom ` C ) y ) X. ( z ( Hom ` D ) z ) ) )
129	116	adantr 481	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) /\ f e. ( x ( Hom ` C ) y ) ) /\ z e. ( Base ` D ) ) -> x e. ( Base ` C ) )
130	118	adantr 481	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) /\ f e. ( x ( Hom ` C ) y ) ) /\ z e. ( Base ` D ) ) -> y e. ( Base ` C ) )
131	2, 3, 4, 34, 43, 129, 125, 130, 125, 6	xpchom2 16826	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) /\ f e. ( x ( Hom ` C ) y ) ) /\ z e. ( Base ` D ) ) -> ( <. x , z >. ( Hom ` ( C X.c D ) ) <. y , z >. ) = ( ( x ( Hom ` C ) y ) X. ( z ( Hom ` D ) z ) ) )
132	128, 131	eleqtrrd 2704	. . . . . . . . . . . . . . . 16 |- ( ( ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) /\ f e. ( x ( Hom ` C ) y ) ) /\ z e. ( Base ` D ) ) -> <. f , ( ( Id ` D ) ` z ) >. e. ( <. x , z >. ( Hom ` ( C X.c D ) ) <. y , z >. ) )
133		fvres 6207	. . . . . . . . . . . . . . . 16 |- ( <. f , ( ( Id ` D ) ` z ) >. e. ( <. x , z >. ( Hom ` ( C X.c D ) ) <. y , z >. ) -> ( ( 2nd |` ( <. x , z >. ( Hom ` ( C X.c D ) ) <. y , z >. ) ) ` <. f , ( ( Id ` D ) ` z ) >. ) = ( 2nd ` <. f , ( ( Id ` D ) ` z ) >. ) )
134	132, 133	syl 17	. . . . . . . . . . . . . . 15 |- ( ( ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) /\ f e. ( x ( Hom ` C ) y ) ) /\ z e. ( Base ` D ) ) -> ( ( 2nd |` ( <. x , z >. ( Hom ` ( C X.c D ) ) <. y , z >. ) ) ` <. f , ( ( Id ` D ) ` z ) >. ) = ( 2nd ` <. f , ( ( Id ` D ) ` z ) >. ) )
135	123, 134	syl5eq 2668	. . . . . . . . . . . . . 14 |- ( ( ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) /\ f e. ( x ( Hom ` C ) y ) ) /\ z e. ( Base ` D ) ) -> ( f ( 2nd |` ( <. x , z >. ( Hom ` ( C X.c D ) ) <. y , z >. ) ) ( ( Id ` D ) ` z ) ) = ( 2nd ` <. f , ( ( Id ` D ) ` z ) >. ) )
136		fvex 6201	. . . . . . . . . . . . . . 15 |- ( ( Id ` D ) ` z ) e. _V
137	50, 136	op2nd 7177	. . . . . . . . . . . . . 14 |- ( 2nd ` <. f , ( ( Id ` D ) ` z ) >. ) = ( ( Id ` D ) ` z )
138	135, 137	syl6eq 2672	. . . . . . . . . . . . 13 |- ( ( ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) /\ f e. ( x ( Hom ` C ) y ) ) /\ z e. ( Base ` D ) ) -> ( f ( 2nd |` ( <. x , z >. ( Hom ` ( C X.c D ) ) <. y , z >. ) ) ( ( Id ` D ) ` z ) ) = ( ( Id ` D ) ` z ) )
139	122, 138	eqtrd 2656	. . . . . . . . . . . 12 |- ( ( ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) /\ f e. ( x ( Hom ` C ) y ) ) /\ z e. ( Base ` D ) ) -> ( f ( <. x , z >. ( 2nd ` ( C 2ndF D ) ) <. y , z >. ) ( ( Id ` D ) ` z ) ) = ( ( Id ` D ) ` z ) )
140	139	mpteq2dva 4744	. . . . . . . . . . 11 |- ( ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) /\ f e. ( x ( Hom ` C ) y ) ) -> ( z e. ( Base ` D ) |-> ( f ( <. x , z >. ( 2nd ` ( C 2ndF D ) ) <. y , z >. ) ( ( Id ` D ) ` z ) ) ) = ( z e. ( Base ` D ) |-> ( ( Id ` D ) ` z ) ) )
141	110, 112, 140	3eqtr4rd 2667	. . . . . . . . . 10 |- ( ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) /\ f e. ( x ( Hom ` C ) y ) ) -> ( z e. ( Base ` D ) |-> ( f ( <. x , z >. ( 2nd ` ( C 2ndF D ) ) <. y , z >. ) ( ( Id ` D ) ` z ) ) ) = ( ( Id ` D ) o. ( _I |` ( Base ` D ) ) ) )
142	101, 105, 141	3eqtr4rd 2667	. . . . . . . . 9 |- ( ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) /\ f e. ( x ( Hom ` C ) y ) ) -> ( z e. ( Base ` D ) |-> ( f ( <. x , z >. ( 2nd ` ( C 2ndF D ) ) <. y , z >. ) ( ( Id ` D ) ` z ) ) ) = ( ( Id ` Q ) ` (idfunc ` D ) ) )
143	68	ad2antrr 762	. . . . . . . . . 10 |- ( ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) /\ f e. ( x ( Hom ` C ) y ) ) -> ( C 2ndF D ) e. ( ( C X.c D ) Func D ) )
144		simpr 477	. . . . . . . . . 10 |- ( ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) /\ f e. ( x ( Hom ` C ) y ) ) -> f e. ( x ( Hom ` C ) y ) )
145		eqid 2622	. . . . . . . . . 10 |- ( ( x ( 2nd ` ( <. C , D >. curryF ( C 2ndF D ) ) ) y ) ` f ) = ( ( x ( 2nd ` ( <. C , D >. curryF ( C 2ndF D ) ) ) y ) ` f )
146	66, 3, 113, 99, 143, 4, 34, 103, 116, 118, 144, 145	curf2 16869	. . . . . . . . 9 |- ( ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) /\ f e. ( x ( Hom ` C ) y ) ) -> ( ( x ( 2nd ` ( <. C , D >. curryF ( C 2ndF D ) ) ) y ) ` f ) = ( z e. ( Base ` D ) |-> ( f ( <. x , z >. ( 2nd ` ( C 2ndF D ) ) <. y , z >. ) ( ( Id ` D ) ` z ) ) ) )
147	77	ad2antrr 762	. . . . . . . . . 10 |- ( ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) /\ f e. ( x ( Hom ` C ) y ) ) -> Q e. Cat )
148	75, 147, 113, 79, 104, 83, 3, 116, 34, 102, 118, 144	diag12 16884	. . . . . . . . 9 |- ( ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) /\ f e. ( x ( Hom ` C ) y ) ) -> ( ( x ( 2nd ` ( ( 1st ` ( QΔfunc C ) ) ` (idfunc ` D ) ) ) y ) ` f ) = ( ( Id ` Q ) ` (idfunc ` D ) ) )
149	142, 146, 148	3eqtr4d 2666	. . . . . . . 8 |- ( ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) /\ f e. ( x ( Hom ` C ) y ) ) -> ( ( x ( 2nd ` ( <. C , D >. curryF ( C 2ndF D ) ) ) y ) ` f ) = ( ( x ( 2nd ` ( ( 1st ` ( QΔfunc C ) ) ` (idfunc ` D ) ) ) y ) ` f ) )
150	149	mpteq2dva 4744	. . . . . . 7 |- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) -> ( f e. ( x ( Hom ` C ) y ) |-> ( ( x ( 2nd ` ( <. C , D >. curryF ( C 2ndF D ) ) ) y ) ` f ) ) = ( f e. ( x ( Hom ` C ) y ) |-> ( ( x ( 2nd ` ( ( 1st ` ( QΔfunc C ) ) ` (idfunc ` D ) ) ) y ) ` f ) ) )
151		eqid 2622	. . . . . . . . . 10 |- ( D Nat D ) = ( D Nat D )
152	76, 151	fuchom 16621	. . . . . . . . 9 |- ( D Nat D ) = ( Hom ` Q )
153	90	adantr 481	. . . . . . . . 9 |- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) -> ( 1st ` ( <. C , D >. curryF ( C 2ndF D ) ) ) ( C Func Q ) ( 2nd ` ( <. C , D >. curryF ( C 2ndF D ) ) ) )
154		simprl 794	. . . . . . . . 9 |- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) -> x e. ( Base ` C ) )
155		simprr 796	. . . . . . . . 9 |- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) -> y e. ( Base ` C ) )
156	3, 34, 152, 153, 154, 155	funcf2 16528	. . . . . . . 8 |- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) -> ( x ( 2nd ` ( <. C , D >. curryF ( C 2ndF D ) ) ) y ) : ( x ( Hom ` C ) y ) --> ( ( ( 1st ` ( <. C , D >. curryF ( C 2ndF D ) ) ) ` x ) ( D Nat D ) ( ( 1st ` ( <. C , D >. curryF ( C 2ndF D ) ) ) ` y ) ) )
157	156	feqmptd 6249	. . . . . . 7 |- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) -> ( x ( 2nd ` ( <. C , D >. curryF ( C 2ndF D ) ) ) y ) = ( f e. ( x ( Hom ` C ) y ) |-> ( ( x ( 2nd ` ( <. C , D >. curryF ( C 2ndF D ) ) ) y ) ` f ) ) )
158	95	adantr 481	. . . . . . . . 9 |- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) -> ( 1st ` ( ( 1st ` ( QΔfunc C ) ) ` (idfunc ` D ) ) ) ( C Func Q ) ( 2nd ` ( ( 1st ` ( QΔfunc C ) ) ` (idfunc ` D ) ) ) )
159	3, 34, 152, 158, 154, 155	funcf2 16528	. . . . . . . 8 |- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) -> ( x ( 2nd ` ( ( 1st ` ( QΔfunc C ) ) ` (idfunc ` D ) ) ) y ) : ( x ( Hom ` C ) y ) --> ( ( ( 1st ` ( ( 1st ` ( QΔfunc C ) ) ` (idfunc ` D ) ) ) ` x ) ( D Nat D ) ( ( 1st ` ( ( 1st ` ( QΔfunc C ) ) ` (idfunc ` D ) ) ) ` y ) ) )
160	159	feqmptd 6249	. . . . . . 7 |- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) -> ( x ( 2nd ` ( ( 1st ` ( QΔfunc C ) ) ` (idfunc ` D ) ) ) y ) = ( f e. ( x ( Hom ` C ) y ) |-> ( ( x ( 2nd ` ( ( 1st ` ( QΔfunc C ) ) ` (idfunc ` D ) ) ) y ) ` f ) ) )
161	150, 157, 160	3eqtr4d 2666	. . . . . 6 |- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) -> ( x ( 2nd ` ( <. C , D >. curryF ( C 2ndF D ) ) ) y ) = ( x ( 2nd ` ( ( 1st ` ( QΔfunc C ) ) ` (idfunc ` D ) ) ) y ) )
162	161	3impb 1260	. . . . 5 |- ( ( ph /\ x e. ( Base ` C ) /\ y e. ( Base ` C ) ) -> ( x ( 2nd ` ( <. C , D >. curryF ( C 2ndF D ) ) ) y ) = ( x ( 2nd ` ( ( 1st ` ( QΔfunc C ) ) ` (idfunc ` D ) ) ) y ) )
163	162	mpt2eq3dva 6719	. . . 4 |- ( ph -> ( x e. ( Base ` C ) , y e. ( Base ` C ) |-> ( x ( 2nd ` ( <. C , D >. curryF ( C 2ndF D ) ) ) y ) ) = ( x e. ( Base ` C ) , y e. ( Base ` C ) |-> ( x ( 2nd ` ( ( 1st ` ( QΔfunc C ) ) ` (idfunc ` D ) ) ) y ) ) )
164	3, 90	funcfn2 16529	. . . . 5 |- ( ph -> ( 2nd ` ( <. C , D >. curryF ( C 2ndF D ) ) ) Fn ( ( Base ` C ) X. ( Base ` C ) ) )
165		fnov 6768	. . . . 5 |- ( ( 2nd ` ( <. C , D >. curryF ( C 2ndF D ) ) ) Fn ( ( Base ` C ) X. ( Base ` C ) ) <-> ( 2nd ` ( <. C , D >. curryF ( C 2ndF D ) ) ) = ( x e. ( Base ` C ) , y e. ( Base ` C ) |-> ( x ( 2nd ` ( <. C , D >. curryF ( C 2ndF D ) ) ) y ) ) )
166	164, 165	sylib 208	. . . 4 |- ( ph -> ( 2nd ` ( <. C , D >. curryF ( C 2ndF D ) ) ) = ( x e. ( Base ` C ) , y e. ( Base ` C ) |-> ( x ( 2nd ` ( <. C , D >. curryF ( C 2ndF D ) ) ) y ) ) )
167	3, 95	funcfn2 16529	. . . . 5 |- ( ph -> ( 2nd ` ( ( 1st ` ( QΔfunc C ) ) ` (idfunc ` D ) ) ) Fn ( ( Base ` C ) X. ( Base ` C ) ) )
168		fnov 6768	. . . . 5 |- ( ( 2nd ` ( ( 1st ` ( QΔfunc C ) ) ` (idfunc ` D ) ) ) Fn ( ( Base ` C ) X. ( Base ` C ) ) <-> ( 2nd ` ( ( 1st ` ( QΔfunc C ) ) ` (idfunc ` D ) ) ) = ( x e. ( Base ` C ) , y e. ( Base ` C ) |-> ( x ( 2nd ` ( ( 1st ` ( QΔfunc C ) ) ` (idfunc ` D ) ) ) y ) ) )
169	167, 168	sylib 208	. . . 4 |- ( ph -> ( 2nd ` ( ( 1st ` ( QΔfunc C ) ) ` (idfunc ` D ) ) ) = ( x e. ( Base ` C ) , y e. ( Base ` C ) |-> ( x ( 2nd ` ( ( 1st ` ( QΔfunc C ) ) ` (idfunc ` D ) ) ) y ) ) )
170	163, 166, 169	3eqtr4d 2666	. . 3 |- ( ph -> ( 2nd ` ( <. C , D >. curryF ( C 2ndF D ) ) ) = ( 2nd ` ( ( 1st ` ( QΔfunc C ) ) ` (idfunc ` D ) ) ) )
171	98, 170	opeq12d 4410	. 2 |- ( ph -> <. ( 1st ` ( <. C , D >. curryF ( C 2ndF D ) ) ) , ( 2nd ` ( <. C , D >. curryF ( C 2ndF D ) ) ) >. = <. ( 1st ` ( ( 1st ` ( QΔfunc C ) ) ` (idfunc ` D ) ) ) , ( 2nd ` ( ( 1st ` ( QΔfunc C ) ) ` (idfunc ` D ) ) ) >. )
172		1st2nd 7214	. . 3 |- ( ( Rel ( C Func Q ) /\ ( <. C , D >. curryF ( C 2ndF D ) ) e. ( C Func Q ) ) -> ( <. C , D >. curryF ( C 2ndF D ) ) = <. ( 1st ` ( <. C , D >. curryF ( C 2ndF D ) ) ) , ( 2nd ` ( <. C , D >. curryF ( C 2ndF D ) ) ) >. )
173	87, 88, 172	sylancr 695	. 2 |- ( ph -> ( <. C , D >. curryF ( C 2ndF D ) ) = <. ( 1st ` ( <. C , D >. curryF ( C 2ndF D ) ) ) , ( 2nd ` ( <. C , D >. curryF ( C 2ndF D ) ) ) >. )
174		1st2nd 7214	. . 3 |- ( ( Rel ( C Func Q ) /\ ( ( 1st ` ( QΔfunc C ) ) ` (idfunc ` D ) ) e. ( C Func Q ) ) -> ( ( 1st ` ( QΔfunc C ) ) ` (idfunc ` D ) ) = <. ( 1st ` ( ( 1st ` ( QΔfunc C ) ) ` (idfunc ` D ) ) ) , ( 2nd ` ( ( 1st ` ( QΔfunc C ) ) ` (idfunc ` D ) ) ) >. )
175	87, 93, 174	sylancr 695	. 2 |- ( ph -> ( ( 1st ` ( QΔfunc C ) ) ` (idfunc ` D ) ) = <. ( 1st ` ( ( 1st ` ( QΔfunc C ) ) ` (idfunc ` D ) ) ) , ( 2nd ` ( ( 1st ` ( QΔfunc C ) ) ` (idfunc ` D ) ) ) >. )
176	171, 173, 175	3eqtr4d 2666	1 |- ( ph -> ( <. C , D >. curryF ( C 2ndF D ) ) = ( ( 1st ` ( QΔfunc C ) ) ` (idfunc ` D ) ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   <.cop 4183   class class class wbr 4653   |-> cmpt 4729   _I cid 5023   X. cxp 5112   |` cres 5116   o. ccom 5118   Rel wrel 5119   Fn wfn 5883   -->wf 5884   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   1stc1st 7166   2ndc2nd 7167   Basecbs 15857   Hom chom 15952   Catccat 16325   Idccid 16326   Func cfunc 16514  id_funccidfu 16515   Nat cnat 16601   FuncCat cfuc 16602   X._c cxpc 16808   2nd_F c2ndf 16810   curry_F ccurf 16850  Δ_funccdiag 16852

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-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-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-idfu 16519  df-nat 16603  df-fuc 16604  df-xpc 16812  df-1stf 16813  df-2ndf 16814  df-curf 16854  df-diag 16856

This theorem is referenced by: (None)