Theorem curfv 33389

Description: Value of currying. (Contributed by Brendan Leahy, 2-Jun-2021.)

Assertion

Ref	Expression
curfv	|- ( ( ( F Fn ( V X. W ) /\ A e. V /\ B e. W ) /\ W e. X ) -> ( (curry F ` A ) ` B ) = ( A F B ) )

Proof of Theorem curfv

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

Step	Hyp	Ref	Expression
1		dffn5 6241	. . . . . . . . . 10 |- ( F Fn ( V X. W ) <-> F = ( z e. ( V X. W ) |-> ( F ` z ) ) )
2		cureq 33385	. . . . . . . . . 10 |- ( F = ( z e. ( V X. W ) |-> ( F ` z ) ) -> curry F = curry ( z e. ( V X. W ) |-> ( F ` z ) ) )
3	1, 2	sylbi 207	. . . . . . . . 9 |- ( F Fn ( V X. W ) -> curry F = curry ( z e. ( V X. W ) |-> ( F ` z ) ) )
4	3	adantr 481	. . . . . . . 8 |- ( ( F Fn ( V X. W ) /\ B e. W ) -> curry F = curry ( z e. ( V X. W ) |-> ( F ` z ) ) )
5		fveq2 6191	. . . . . . . . . 10 |- ( z = <. x , y >. -> ( F ` z ) = ( F ` <. x , y >. ) )
6	5	mpt2mpt 6752	. . . . . . . . 9 |- ( z e. ( V X. W ) |-> ( F ` z ) ) = ( x e. V , y e. W |-> ( F ` <. x , y >. ) )
7		fvex 6201	. . . . . . . . . . 11 |- ( F ` <. x , y >. ) e. _V
8	7	rgen2w 2925	. . . . . . . . . 10 |- A. x e. V A. y e. W ( F ` <. x , y >. ) e. _V
9	8	a1i 11	. . . . . . . . 9 |- ( ( F Fn ( V X. W ) /\ B e. W ) -> A. x e. V A. y e. W ( F ` <. x , y >. ) e. _V )
10		ne0i 3921	. . . . . . . . . 10 |- ( B e. W -> W =/= (/) )
11	10	adantl 482	. . . . . . . . 9 |- ( ( F Fn ( V X. W ) /\ B e. W ) -> W =/= (/) )
12	6, 9, 11	mpt2curryd 7395	. . . . . . . 8 |- ( ( F Fn ( V X. W ) /\ B e. W ) -> curry ( z e. ( V X. W ) |-> ( F ` z ) ) = ( x e. V |-> ( y e. W |-> ( F ` <. x , y >. ) ) ) )
13	4, 12	eqtrd 2656	. . . . . . 7 |- ( ( F Fn ( V X. W ) /\ B e. W ) -> curry F = ( x e. V |-> ( y e. W |-> ( F ` <. x , y >. ) ) ) )
14	13	3adant2 1080	. . . . . 6 |- ( ( F Fn ( V X. W ) /\ A e. V /\ B e. W ) -> curry F = ( x e. V |-> ( y e. W |-> ( F ` <. x , y >. ) ) ) )
15	14	fveq1d 6193	. . . . 5 |- ( ( F Fn ( V X. W ) /\ A e. V /\ B e. W ) -> (curry F ` A ) = ( ( x e. V |-> ( y e. W |-> ( F ` <. x , y >. ) ) ) ` A ) )
16	15	adantr 481	. . . 4 |- ( ( ( F Fn ( V X. W ) /\ A e. V /\ B e. W ) /\ W e. X ) -> (curry F ` A ) = ( ( x e. V |-> ( y e. W |-> ( F ` <. x , y >. ) ) ) ` A ) )
17		mptexg 6484	. . . . . 6 |- ( W e. X -> ( y e. W |-> ( F ` <. A , y >. ) ) e. _V )
18		opeq1 4402	. . . . . . . . 9 |- ( x = A -> <. x , y >. = <. A , y >. )
19	18	fveq2d 6195	. . . . . . . 8 |- ( x = A -> ( F ` <. x , y >. ) = ( F ` <. A , y >. ) )
20	19	mpteq2dv 4745	. . . . . . 7 |- ( x = A -> ( y e. W |-> ( F ` <. x , y >. ) ) = ( y e. W |-> ( F ` <. A , y >. ) ) )
21		eqid 2622	. . . . . . 7 |- ( x e. V |-> ( y e. W |-> ( F ` <. x , y >. ) ) ) = ( x e. V |-> ( y e. W |-> ( F ` <. x , y >. ) ) )
22	20, 21	fvmptg 6280	. . . . . 6 |- ( ( A e. V /\ ( y e. W |-> ( F ` <. A , y >. ) ) e. _V ) -> ( ( x e. V |-> ( y e. W |-> ( F ` <. x , y >. ) ) ) ` A ) = ( y e. W |-> ( F ` <. A , y >. ) ) )
23	17, 22	sylan2 491	. . . . 5 |- ( ( A e. V /\ W e. X ) -> ( ( x e. V |-> ( y e. W |-> ( F ` <. x , y >. ) ) ) ` A ) = ( y e. W |-> ( F ` <. A , y >. ) ) )
24	23	3ad2antl2 1224	. . . 4 |- ( ( ( F Fn ( V X. W ) /\ A e. V /\ B e. W ) /\ W e. X ) -> ( ( x e. V |-> ( y e. W |-> ( F ` <. x , y >. ) ) ) ` A ) = ( y e. W |-> ( F ` <. A , y >. ) ) )
25	16, 24	eqtrd 2656	. . 3 |- ( ( ( F Fn ( V X. W ) /\ A e. V /\ B e. W ) /\ W e. X ) -> (curry F ` A ) = ( y e. W |-> ( F ` <. A , y >. ) ) )
26	25	fveq1d 6193	. 2 |- ( ( ( F Fn ( V X. W ) /\ A e. V /\ B e. W ) /\ W e. X ) -> ( (curry F ` A ) ` B ) = ( ( y e. W |-> ( F ` <. A , y >. ) ) ` B ) )
27		opeq2 4403	. . . . . . 7 |- ( y = B -> <. A , y >. = <. A , B >. )
28	27	fveq2d 6195	. . . . . 6 |- ( y = B -> ( F ` <. A , y >. ) = ( F ` <. A , B >. ) )
29		eqid 2622	. . . . . 6 |- ( y e. W |-> ( F ` <. A , y >. ) ) = ( y e. W |-> ( F ` <. A , y >. ) )
30		fvex 6201	. . . . . 6 |- ( F ` <. A , B >. ) e. _V
31	28, 29, 30	fvmpt 6282	. . . . 5 |- ( B e. W -> ( ( y e. W |-> ( F ` <. A , y >. ) ) ` B ) = ( F ` <. A , B >. ) )
32		df-ov 6653	. . . . 5 |- ( A F B ) = ( F ` <. A , B >. )
33	31, 32	syl6eqr 2674	. . . 4 |- ( B e. W -> ( ( y e. W |-> ( F ` <. A , y >. ) ) ` B ) = ( A F B ) )
34	33	3ad2ant3 1084	. . 3 |- ( ( F Fn ( V X. W ) /\ A e. V /\ B e. W ) -> ( ( y e. W |-> ( F ` <. A , y >. ) ) ` B ) = ( A F B ) )
35	34	adantr 481	. 2 |- ( ( ( F Fn ( V X. W ) /\ A e. V /\ B e. W ) /\ W e. X ) -> ( ( y e. W |-> ( F ` <. A , y >. ) ) ` B ) = ( A F B ) )
36	26, 35	eqtrd 2656	1 |- ( ( ( F Fn ( V X. W ) /\ A e. V /\ B e. W ) /\ W e. X ) -> ( (curry F ` A ) ` B ) = ( A F B ) )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   _Vcvv 3200   (/)c0 3915   <.cop 4183   |-> cmpt 4729   X. cxp 5112   Fn wfn 5883   ` cfv 5888  (class class class)co 6650  curry ccur 7391

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-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-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-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-cur 7393

This theorem is referenced by:  unccur  33392  matunitlindflem1  33405  matunitlindflem2  33406