Theorem curfval 16863

Description: Value of the curry functor. (Contributed by Mario Carneiro, 12-Jan-2017.)

Hypotheses

Ref	Expression
curfval.g	|- G = ( <. C , D >. curryF F )
curfval.a	|- A = ( Base ` C )
curfval.c	|- ( ph -> C e. Cat )
curfval.d	|- ( ph -> D e. Cat )
curfval.f	|- ( ph -> F e. ( ( C X.c D ) Func E ) )
curfval.b	|- B = ( Base ` D )
curfval.j	|- J = ( Hom ` D )
curfval.1	|- .1. = ( Id ` C )
curfval.h	|- H = ( Hom ` C )
curfval.i	|- I = ( Id ` D )

Assertion

Ref	Expression
curfval	|- ( ph -> G = <. ( x e. A |-> <. ( y e. B |-> ( x ( 1st ` F ) y ) ) , ( y e. B , z e. B |-> ( g e. ( y J z ) |-> ( ( .1. ` x ) ( <. x , y >. ( 2nd ` F ) <. x , z >. ) g ) ) ) >. ) , ( x e. A , y e. A |-> ( g e. ( x H y ) |-> ( z e. B |-> ( g ( <. x , z >. ( 2nd ` F ) <. y , z >. ) ( I ` z ) ) ) ) ) >. )

Distinct variable groups:   x, g, y, z, .1.   x, A, y   B, g, x, y, z   C, g, x, y, z   D, g, x, y, z   g, H, y, z   ph, g, x, y, z   g, E, y, z   g, J, x   g, F, x, y, z

Allowed substitution hints:   A( z, g)   E( x)   G( x, y, z, g)   H( x)   I( x, y, z, g)   J( y, z)

Proof of Theorem curfval

Dummy variables c d e f are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		curfval.g	. 2 |- G = ( <. C , D >. curryF F )
2		df-curf 16854	. . . 4 |- curryF = ( e e. _V , f e. _V |-> [_ ( 1st ` e ) / c ]_ [_ ( 2nd ` e ) / d ]_ <. ( 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 ) ) ) ) ) >. )
3	2	a1i 11	. . 3 |- ( ph -> curryF = ( e e. _V , f e. _V |-> [_ ( 1st ` e ) / c ]_ [_ ( 2nd ` e ) / d ]_ <. ( 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 ) ) ) ) ) >. ) )
4		fvexd 6203	. . . 4 |- ( ( ph /\ ( e = <. C , D >. /\ f = F ) ) -> ( 1st ` e ) e. _V )
5		simprl 794	. . . . . 6 |- ( ( ph /\ ( e = <. C , D >. /\ f = F ) ) -> e = <. C , D >. )
6	5	fveq2d 6195	. . . . 5 |- ( ( ph /\ ( e = <. C , D >. /\ f = F ) ) -> ( 1st ` e ) = ( 1st ` <. C , D >. ) )
7		curfval.c	. . . . . . 7 |- ( ph -> C e. Cat )
8		curfval.d	. . . . . . 7 |- ( ph -> D e. Cat )
9		op1stg 7180	. . . . . . 7 |- ( ( C e. Cat /\ D e. Cat ) -> ( 1st ` <. C , D >. ) = C )
10	7, 8, 9	syl2anc 693	. . . . . 6 |- ( ph -> ( 1st ` <. C , D >. ) = C )
11	10	adantr 481	. . . . 5 |- ( ( ph /\ ( e = <. C , D >. /\ f = F ) ) -> ( 1st ` <. C , D >. ) = C )
12	6, 11	eqtrd 2656	. . . 4 |- ( ( ph /\ ( e = <. C , D >. /\ f = F ) ) -> ( 1st ` e ) = C )
13		fvexd 6203	. . . . 5 |- ( ( ( ph /\ ( e = <. C , D >. /\ f = F ) ) /\ c = C ) -> ( 2nd ` e ) e. _V )
14	5	adantr 481	. . . . . . 7 |- ( ( ( ph /\ ( e = <. C , D >. /\ f = F ) ) /\ c = C ) -> e = <. C , D >. )
15	14	fveq2d 6195	. . . . . 6 |- ( ( ( ph /\ ( e = <. C , D >. /\ f = F ) ) /\ c = C ) -> ( 2nd ` e ) = ( 2nd ` <. C , D >. ) )
16		op2ndg 7181	. . . . . . . 8 |- ( ( C e. Cat /\ D e. Cat ) -> ( 2nd ` <. C , D >. ) = D )
17	7, 8, 16	syl2anc 693	. . . . . . 7 |- ( ph -> ( 2nd ` <. C , D >. ) = D )
18	17	ad2antrr 762	. . . . . 6 |- ( ( ( ph /\ ( e = <. C , D >. /\ f = F ) ) /\ c = C ) -> ( 2nd ` <. C , D >. ) = D )
19	15, 18	eqtrd 2656	. . . . 5 |- ( ( ( ph /\ ( e = <. C , D >. /\ f = F ) ) /\ c = C ) -> ( 2nd ` e ) = D )
20		simplr 792	. . . . . . . . 9 |- ( ( ( ( ph /\ ( e = <. C , D >. /\ f = F ) ) /\ c = C ) /\ d = D ) -> c = C )
21	20	fveq2d 6195	. . . . . . . 8 |- ( ( ( ( ph /\ ( e = <. C , D >. /\ f = F ) ) /\ c = C ) /\ d = D ) -> ( Base ` c ) = ( Base ` C ) )
22		curfval.a	. . . . . . . 8 |- A = ( Base ` C )
23	21, 22	syl6eqr 2674	. . . . . . 7 |- ( ( ( ( ph /\ ( e = <. C , D >. /\ f = F ) ) /\ c = C ) /\ d = D ) -> ( Base ` c ) = A )
24		simpr 477	. . . . . . . . . . 11 |- ( ( ( ( ph /\ ( e = <. C , D >. /\ f = F ) ) /\ c = C ) /\ d = D ) -> d = D )
25	24	fveq2d 6195	. . . . . . . . . 10 |- ( ( ( ( ph /\ ( e = <. C , D >. /\ f = F ) ) /\ c = C ) /\ d = D ) -> ( Base ` d ) = ( Base ` D ) )
26		curfval.b	. . . . . . . . . 10 |- B = ( Base ` D )
27	25, 26	syl6eqr 2674	. . . . . . . . 9 |- ( ( ( ( ph /\ ( e = <. C , D >. /\ f = F ) ) /\ c = C ) /\ d = D ) -> ( Base ` d ) = B )
28		simprr 796	. . . . . . . . . . . 12 |- ( ( ph /\ ( e = <. C , D >. /\ f = F ) ) -> f = F )
29	28	ad2antrr 762	. . . . . . . . . . 11 |- ( ( ( ( ph /\ ( e = <. C , D >. /\ f = F ) ) /\ c = C ) /\ d = D ) -> f = F )
30	29	fveq2d 6195	. . . . . . . . . 10 |- ( ( ( ( ph /\ ( e = <. C , D >. /\ f = F ) ) /\ c = C ) /\ d = D ) -> ( 1st ` f ) = ( 1st ` F ) )
31	30	oveqd 6667	. . . . . . . . 9 |- ( ( ( ( ph /\ ( e = <. C , D >. /\ f = F ) ) /\ c = C ) /\ d = D ) -> ( x ( 1st ` f ) y ) = ( x ( 1st ` F ) y ) )
32	27, 31	mpteq12dv 4733	. . . . . . . 8 |- ( ( ( ( ph /\ ( e = <. C , D >. /\ f = F ) ) /\ c = C ) /\ d = D ) -> ( y e. ( Base ` d ) |-> ( x ( 1st ` f ) y ) ) = ( y e. B |-> ( x ( 1st ` F ) y ) ) )
33	24	fveq2d 6195	. . . . . . . . . . . 12 |- ( ( ( ( ph /\ ( e = <. C , D >. /\ f = F ) ) /\ c = C ) /\ d = D ) -> ( Hom ` d ) = ( Hom ` D ) )
34		curfval.j	. . . . . . . . . . . 12 |- J = ( Hom ` D )
35	33, 34	syl6eqr 2674	. . . . . . . . . . 11 |- ( ( ( ( ph /\ ( e = <. C , D >. /\ f = F ) ) /\ c = C ) /\ d = D ) -> ( Hom ` d ) = J )
36	35	oveqd 6667	. . . . . . . . . 10 |- ( ( ( ( ph /\ ( e = <. C , D >. /\ f = F ) ) /\ c = C ) /\ d = D ) -> ( y ( Hom ` d ) z ) = ( y J z ) )
37	29	fveq2d 6195	. . . . . . . . . . . 12 |- ( ( ( ( ph /\ ( e = <. C , D >. /\ f = F ) ) /\ c = C ) /\ d = D ) -> ( 2nd ` f ) = ( 2nd ` F ) )
38	37	oveqd 6667	. . . . . . . . . . 11 |- ( ( ( ( ph /\ ( e = <. C , D >. /\ f = F ) ) /\ c = C ) /\ d = D ) -> ( <. x , y >. ( 2nd ` f ) <. x , z >. ) = ( <. x , y >. ( 2nd ` F ) <. x , z >. ) )
39	20	fveq2d 6195	. . . . . . . . . . . . 13 |- ( ( ( ( ph /\ ( e = <. C , D >. /\ f = F ) ) /\ c = C ) /\ d = D ) -> ( Id ` c ) = ( Id ` C ) )
40		curfval.1	. . . . . . . . . . . . 13 |- .1. = ( Id ` C )
41	39, 40	syl6eqr 2674	. . . . . . . . . . . 12 |- ( ( ( ( ph /\ ( e = <. C , D >. /\ f = F ) ) /\ c = C ) /\ d = D ) -> ( Id ` c ) = .1. )
42	41	fveq1d 6193	. . . . . . . . . . 11 |- ( ( ( ( ph /\ ( e = <. C , D >. /\ f = F ) ) /\ c = C ) /\ d = D ) -> ( ( Id ` c ) ` x ) = ( .1. ` x ) )
43		eqidd 2623	. . . . . . . . . . 11 |- ( ( ( ( ph /\ ( e = <. C , D >. /\ f = F ) ) /\ c = C ) /\ d = D ) -> g = g )
44	38, 42, 43	oveq123d 6671	. . . . . . . . . 10 |- ( ( ( ( ph /\ ( e = <. C , D >. /\ f = F ) ) /\ c = C ) /\ d = D ) -> ( ( ( Id ` c ) ` x ) ( <. x , y >. ( 2nd ` f ) <. x , z >. ) g ) = ( ( .1. ` x ) ( <. x , y >. ( 2nd ` F ) <. x , z >. ) g ) )
45	36, 44	mpteq12dv 4733	. . . . . . . . 9 |- ( ( ( ( ph /\ ( e = <. C , D >. /\ f = F ) ) /\ c = C ) /\ d = D ) -> ( g e. ( y ( Hom ` d ) z ) |-> ( ( ( Id ` c ) ` x ) ( <. x , y >. ( 2nd ` f ) <. x , z >. ) g ) ) = ( g e. ( y J z ) |-> ( ( .1. ` x ) ( <. x , y >. ( 2nd ` F ) <. x , z >. ) g ) ) )
46	27, 27, 45	mpt2eq123dv 6717	. . . . . . . 8 |- ( ( ( ( ph /\ ( e = <. C , D >. /\ f = F ) ) /\ c = C ) /\ d = D ) -> ( 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. B , z e. B |-> ( g e. ( y J z ) |-> ( ( .1. ` x ) ( <. x , y >. ( 2nd ` F ) <. x , z >. ) g ) ) ) )
47	32, 46	opeq12d 4410	. . . . . . 7 |- ( ( ( ( ph /\ ( e = <. C , D >. /\ f = F ) ) /\ c = C ) /\ d = D ) -> <. ( 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 ) ) ) >. = <. ( y e. B |-> ( x ( 1st ` F ) y ) ) , ( y e. B , z e. B |-> ( g e. ( y J z ) |-> ( ( .1. ` x ) ( <. x , y >. ( 2nd ` F ) <. x , z >. ) g ) ) ) >. )
48	23, 47	mpteq12dv 4733	. . . . . 6 |- ( ( ( ( ph /\ ( e = <. C , D >. /\ f = F ) ) /\ c = C ) /\ d = D ) -> ( 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. A |-> <. ( y e. B |-> ( x ( 1st ` F ) y ) ) , ( y e. B , z e. B |-> ( g e. ( y J z ) |-> ( ( .1. ` x ) ( <. x , y >. ( 2nd ` F ) <. x , z >. ) g ) ) ) >. ) )
49	20	fveq2d 6195	. . . . . . . . . 10 |- ( ( ( ( ph /\ ( e = <. C , D >. /\ f = F ) ) /\ c = C ) /\ d = D ) -> ( Hom ` c ) = ( Hom ` C ) )
50		curfval.h	. . . . . . . . . 10 |- H = ( Hom ` C )
51	49, 50	syl6eqr 2674	. . . . . . . . 9 |- ( ( ( ( ph /\ ( e = <. C , D >. /\ f = F ) ) /\ c = C ) /\ d = D ) -> ( Hom ` c ) = H )
52	51	oveqd 6667	. . . . . . . 8 |- ( ( ( ( ph /\ ( e = <. C , D >. /\ f = F ) ) /\ c = C ) /\ d = D ) -> ( x ( Hom ` c ) y ) = ( x H y ) )
53	37	oveqd 6667	. . . . . . . . . 10 |- ( ( ( ( ph /\ ( e = <. C , D >. /\ f = F ) ) /\ c = C ) /\ d = D ) -> ( <. x , z >. ( 2nd ` f ) <. y , z >. ) = ( <. x , z >. ( 2nd ` F ) <. y , z >. ) )
54	24	fveq2d 6195	. . . . . . . . . . . 12 |- ( ( ( ( ph /\ ( e = <. C , D >. /\ f = F ) ) /\ c = C ) /\ d = D ) -> ( Id ` d ) = ( Id ` D ) )
55		curfval.i	. . . . . . . . . . . 12 |- I = ( Id ` D )
56	54, 55	syl6eqr 2674	. . . . . . . . . . 11 |- ( ( ( ( ph /\ ( e = <. C , D >. /\ f = F ) ) /\ c = C ) /\ d = D ) -> ( Id ` d ) = I )
57	56	fveq1d 6193	. . . . . . . . . 10 |- ( ( ( ( ph /\ ( e = <. C , D >. /\ f = F ) ) /\ c = C ) /\ d = D ) -> ( ( Id ` d ) ` z ) = ( I ` z ) )
58	53, 43, 57	oveq123d 6671	. . . . . . . . 9 |- ( ( ( ( ph /\ ( e = <. C , D >. /\ f = F ) ) /\ c = C ) /\ d = D ) -> ( g ( <. x , z >. ( 2nd ` f ) <. y , z >. ) ( ( Id ` d ) ` z ) ) = ( g ( <. x , z >. ( 2nd ` F ) <. y , z >. ) ( I ` z ) ) )
59	27, 58	mpteq12dv 4733	. . . . . . . 8 |- ( ( ( ( ph /\ ( e = <. C , D >. /\ f = F ) ) /\ c = C ) /\ d = D ) -> ( z e. ( Base ` d ) |-> ( g ( <. x , z >. ( 2nd ` f ) <. y , z >. ) ( ( Id ` d ) ` z ) ) ) = ( z e. B |-> ( g ( <. x , z >. ( 2nd ` F ) <. y , z >. ) ( I ` z ) ) ) )
60	52, 59	mpteq12dv 4733	. . . . . . 7 |- ( ( ( ( ph /\ ( e = <. C , D >. /\ f = F ) ) /\ c = C ) /\ d = D ) -> ( g e. ( x ( Hom ` c ) y ) |-> ( z e. ( Base ` d ) |-> ( g ( <. x , z >. ( 2nd ` f ) <. y , z >. ) ( ( Id ` d ) ` z ) ) ) ) = ( g e. ( x H y ) |-> ( z e. B |-> ( g ( <. x , z >. ( 2nd ` F ) <. y , z >. ) ( I ` z ) ) ) ) )
61	23, 23, 60	mpt2eq123dv 6717	. . . . . 6 |- ( ( ( ( ph /\ ( e = <. C , D >. /\ f = F ) ) /\ c = C ) /\ d = D ) -> ( 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. A , y e. A |-> ( g e. ( x H y ) |-> ( z e. B |-> ( g ( <. x , z >. ( 2nd ` F ) <. y , z >. ) ( I ` z ) ) ) ) ) )
62	48, 61	opeq12d 4410	. . . . 5 |- ( ( ( ( ph /\ ( e = <. C , D >. /\ f = F ) ) /\ c = C ) /\ d = D ) -> <. ( 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 ) ) ) ) ) >. = <. ( x e. A |-> <. ( y e. B |-> ( x ( 1st ` F ) y ) ) , ( y e. B , z e. B |-> ( g e. ( y J z ) |-> ( ( .1. ` x ) ( <. x , y >. ( 2nd ` F ) <. x , z >. ) g ) ) ) >. ) , ( x e. A , y e. A |-> ( g e. ( x H y ) |-> ( z e. B |-> ( g ( <. x , z >. ( 2nd ` F ) <. y , z >. ) ( I ` z ) ) ) ) ) >. )
63	13, 19, 62	csbied2 3561	. . . 4 |- ( ( ( ph /\ ( e = <. C , D >. /\ f = F ) ) /\ c = C ) -> [_ ( 2nd ` e ) / d ]_ <. ( 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 ) ) ) ) ) >. = <. ( x e. A |-> <. ( y e. B |-> ( x ( 1st ` F ) y ) ) , ( y e. B , z e. B |-> ( g e. ( y J z ) |-> ( ( .1. ` x ) ( <. x , y >. ( 2nd ` F ) <. x , z >. ) g ) ) ) >. ) , ( x e. A , y e. A |-> ( g e. ( x H y ) |-> ( z e. B |-> ( g ( <. x , z >. ( 2nd ` F ) <. y , z >. ) ( I ` z ) ) ) ) ) >. )
64	4, 12, 63	csbied2 3561	. . 3 |- ( ( ph /\ ( e = <. C , D >. /\ f = F ) ) -> [_ ( 1st ` e ) / c ]_ [_ ( 2nd ` e ) / d ]_ <. ( 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 ) ) ) ) ) >. = <. ( x e. A |-> <. ( y e. B |-> ( x ( 1st ` F ) y ) ) , ( y e. B , z e. B |-> ( g e. ( y J z ) |-> ( ( .1. ` x ) ( <. x , y >. ( 2nd ` F ) <. x , z >. ) g ) ) ) >. ) , ( x e. A , y e. A |-> ( g e. ( x H y ) |-> ( z e. B |-> ( g ( <. x , z >. ( 2nd ` F ) <. y , z >. ) ( I ` z ) ) ) ) ) >. )
65		opex 4932	. . . 4 |- <. C , D >. e. _V
66	65	a1i 11	. . 3 |- ( ph -> <. C , D >. e. _V )
67		curfval.f	. . . 4 |- ( ph -> F e. ( ( C X.c D ) Func E ) )
68		elex 3212	. . . 4 |- ( F e. ( ( C X.c D ) Func E ) -> F e. _V )
69	67, 68	syl 17	. . 3 |- ( ph -> F e. _V )
70		opex 4932	. . . 4 |- <. ( x e. A |-> <. ( y e. B |-> ( x ( 1st ` F ) y ) ) , ( y e. B , z e. B |-> ( g e. ( y J z ) |-> ( ( .1. ` x ) ( <. x , y >. ( 2nd ` F ) <. x , z >. ) g ) ) ) >. ) , ( x e. A , y e. A |-> ( g e. ( x H y ) |-> ( z e. B |-> ( g ( <. x , z >. ( 2nd ` F ) <. y , z >. ) ( I ` z ) ) ) ) ) >. e. _V
71	70	a1i 11	. . 3 |- ( ph -> <. ( x e. A |-> <. ( y e. B |-> ( x ( 1st ` F ) y ) ) , ( y e. B , z e. B |-> ( g e. ( y J z ) |-> ( ( .1. ` x ) ( <. x , y >. ( 2nd ` F ) <. x , z >. ) g ) ) ) >. ) , ( x e. A , y e. A |-> ( g e. ( x H y ) |-> ( z e. B |-> ( g ( <. x , z >. ( 2nd ` F ) <. y , z >. ) ( I ` z ) ) ) ) ) >. e. _V )
72	3, 64, 66, 69, 71	ovmpt2d 6788	. 2 |- ( ph -> ( <. C , D >. curryF F ) = <. ( x e. A |-> <. ( y e. B |-> ( x ( 1st ` F ) y ) ) , ( y e. B , z e. B |-> ( g e. ( y J z ) |-> ( ( .1. ` x ) ( <. x , y >. ( 2nd ` F ) <. x , z >. ) g ) ) ) >. ) , ( x e. A , y e. A |-> ( g e. ( x H y ) |-> ( z e. B |-> ( g ( <. x , z >. ( 2nd ` F ) <. y , z >. ) ( I ` z ) ) ) ) ) >. )
73	1, 72	syl5eq 2668	1 |- ( ph -> G = <. ( x e. A |-> <. ( y e. B |-> ( x ( 1st ` F ) y ) ) , ( y e. B , z e. B |-> ( g e. ( y J z ) |-> ( ( .1. ` x ) ( <. x , y >. ( 2nd ` F ) <. x , z >. ) g ) ) ) >. ) , ( x e. A , y e. A |-> ( g e. ( x H y ) |-> ( z e. B |-> ( g ( <. x , z >. ( 2nd ` F ) <. y , z >. ) ( I ` z ) ) ) ) ) >. )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   [_csb 3533   <.cop 4183   |-> cmpt 4729   ` 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   X._c cxpc 16808   curry_F ccurf 16850

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-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-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-ral 2917  df-rex 2918  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-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  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-iota 5851  df-fun 5890  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-curf 16854

This theorem is referenced by:  curf1fval  16864  curf2  16869  curfcl  16872  curfpropd  16873  curfuncf  16878