Theorem unccur 33392

Description: Uncurrying of currying. (Contributed by Brendan Leahy, 5-Jun-2021.)

Assertion

Ref	Expression
unccur	|- ( ( F : ( A X. B ) --> C /\ B e. ( V \ { (/) } ) /\ C e. W ) -> uncurry curry F = F )

Proof of Theorem unccur

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

Step	Hyp	Ref	Expression
1		ffn 6045	. . . . . . . . 9 |- ( F : ( A X. B ) --> C -> F Fn ( A X. B ) )
2	1	anim1i 592	. . . . . . . 8 |- ( ( F : ( A X. B ) --> C /\ B e. ( V \ { (/) } ) ) -> ( F Fn ( A X. B ) /\ B e. ( V \ { (/) } ) ) )
3	2	3adant3 1081	. . . . . . 7 |- ( ( F : ( A X. B ) --> C /\ B e. ( V \ { (/) } ) /\ C e. W ) -> ( F Fn ( A X. B ) /\ B e. ( V \ { (/) } ) ) )
4		3anass 1042	. . . . . . . . . . 11 |- ( ( F Fn ( A X. B ) /\ x e. A /\ y e. B ) <-> ( F Fn ( A X. B ) /\ ( x e. A /\ y e. B ) ) )
5		curfv 33389	. . . . . . . . . . 11 |- ( ( ( F Fn ( A X. B ) /\ x e. A /\ y e. B ) /\ B e. ( V \ { (/) } ) ) -> ( (curry F ` x ) ` y ) = ( x F y ) )
6	4, 5	sylanbr 490	. . . . . . . . . 10 |- ( ( ( F Fn ( A X. B ) /\ ( x e. A /\ y e. B ) ) /\ B e. ( V \ { (/) } ) ) -> ( (curry F ` x ) ` y ) = ( x F y ) )
7	6	an32s 846	. . . . . . . . 9 |- ( ( ( F Fn ( A X. B ) /\ B e. ( V \ { (/) } ) ) /\ ( x e. A /\ y e. B ) ) -> ( (curry F ` x ) ` y ) = ( x F y ) )
8	7	eqeq1d 2624	. . . . . . . 8 |- ( ( ( F Fn ( A X. B ) /\ B e. ( V \ { (/) } ) ) /\ ( x e. A /\ y e. B ) ) -> ( ( (curry F ` x ) ` y ) = z <-> ( x F y ) = z ) )
9		eqcom 2629	. . . . . . . 8 |- ( ( x F y ) = z <-> z = ( x F y ) )
10	8, 9	syl6bb 276	. . . . . . 7 |- ( ( ( F Fn ( A X. B ) /\ B e. ( V \ { (/) } ) ) /\ ( x e. A /\ y e. B ) ) -> ( ( (curry F ` x ) ` y ) = z <-> z = ( x F y ) ) )
11	3, 10	sylan 488	. . . . . 6 |- ( ( ( F : ( A X. B ) --> C /\ B e. ( V \ { (/) } ) /\ C e. W ) /\ ( x e. A /\ y e. B ) ) -> ( ( (curry F ` x ) ` y ) = z <-> z = ( x F y ) ) )
12		curf 33387	. . . . . . . . . 10 |- ( ( F : ( A X. B ) --> C /\ B e. ( V \ { (/) } ) /\ C e. W ) -> curry F : A --> ( C ^m B ) )
13	12	ffvelrnda 6359	. . . . . . . . 9 |- ( ( ( F : ( A X. B ) --> C /\ B e. ( V \ { (/) } ) /\ C e. W ) /\ x e. A ) -> (curry F ` x ) e. ( C ^m B ) )
14		elmapfn 7880	. . . . . . . . 9 |- ( (curry F ` x ) e. ( C ^m B ) -> (curry F ` x ) Fn B )
15	13, 14	syl 17	. . . . . . . 8 |- ( ( ( F : ( A X. B ) --> C /\ B e. ( V \ { (/) } ) /\ C e. W ) /\ x e. A ) -> (curry F ` x ) Fn B )
16		fnbrfvb 6236	. . . . . . . 8 |- ( ( (curry F ` x ) Fn B /\ y e. B ) -> ( ( (curry F ` x ) ` y ) = z <-> y (curry F ` x ) z ) )
17	15, 16	sylan 488	. . . . . . 7 |- ( ( ( ( F : ( A X. B ) --> C /\ B e. ( V \ { (/) } ) /\ C e. W ) /\ x e. A ) /\ y e. B ) -> ( ( (curry F ` x ) ` y ) = z <-> y (curry F ` x ) z ) )
18	17	anasss 679	. . . . . 6 |- ( ( ( F : ( A X. B ) --> C /\ B e. ( V \ { (/) } ) /\ C e. W ) /\ ( x e. A /\ y e. B ) ) -> ( ( (curry F ` x ) ` y ) = z <-> y (curry F ` x ) z ) )
19		ibar 525	. . . . . . 7 |- ( ( x e. A /\ y e. B ) -> ( z = ( x F y ) <-> ( ( x e. A /\ y e. B ) /\ z = ( x F y ) ) ) )
20	19	adantl 482	. . . . . 6 |- ( ( ( F : ( A X. B ) --> C /\ B e. ( V \ { (/) } ) /\ C e. W ) /\ ( x e. A /\ y e. B ) ) -> ( z = ( x F y ) <-> ( ( x e. A /\ y e. B ) /\ z = ( x F y ) ) ) )
21	11, 18, 20	3bitr3d 298	. . . . 5 |- ( ( ( F : ( A X. B ) --> C /\ B e. ( V \ { (/) } ) /\ C e. W ) /\ ( x e. A /\ y e. B ) ) -> ( y (curry F ` x ) z <-> ( ( x e. A /\ y e. B ) /\ z = ( x F y ) ) ) )
22		df-br 4654	. . . . . . . . . . 11 |- ( y (curry F ` x ) z <-> <. y , z >. e. (curry F ` x ) )
23		elfvdm 6220	. . . . . . . . . . 11 |- ( <. y , z >. e. (curry F ` x ) -> x e. dom curry F )
24	22, 23	sylbi 207	. . . . . . . . . 10 |- ( y (curry F ` x ) z -> x e. dom curry F )
25		fdm 6051	. . . . . . . . . . . 12 |- (curry F : A --> ( C ^m B ) -> dom curry F = A )
26	25	eleq2d 2687	. . . . . . . . . . 11 |- (curry F : A --> ( C ^m B ) -> ( x e. dom curry F <-> x e. A ) )
27	26	biimpa 501	. . . . . . . . . 10 |- ( (curry F : A --> ( C ^m B ) /\ x e. dom curry F ) -> x e. A )
28	24, 27	sylan2 491	. . . . . . . . 9 |- ( (curry F : A --> ( C ^m B ) /\ y (curry F ` x ) z ) -> x e. A )
29		ffvelrn 6357	. . . . . . . . . . . . 13 |- ( (curry F : A --> ( C ^m B ) /\ x e. A ) -> (curry F ` x ) e. ( C ^m B ) )
30		elmapi 7879	. . . . . . . . . . . . 13 |- ( (curry F ` x ) e. ( C ^m B ) -> (curry F ` x ) : B --> C )
31		fdm 6051	. . . . . . . . . . . . 13 |- ( (curry F ` x ) : B --> C -> dom (curry F ` x ) = B )
32	29, 30, 31	3syl 18	. . . . . . . . . . . 12 |- ( (curry F : A --> ( C ^m B ) /\ x e. A ) -> dom (curry F ` x ) = B )
33		vex 3203	. . . . . . . . . . . . 13 |- y e. _V
34		vex 3203	. . . . . . . . . . . . 13 |- z e. _V
35	33, 34	breldm 5329	. . . . . . . . . . . 12 |- ( y (curry F ` x ) z -> y e. dom (curry F ` x ) )
36		eleq2 2690	. . . . . . . . . . . . 13 |- ( dom (curry F ` x ) = B -> ( y e. dom (curry F ` x ) <-> y e. B ) )
37	36	biimpa 501	. . . . . . . . . . . 12 |- ( ( dom (curry F ` x ) = B /\ y e. dom (curry F ` x ) ) -> y e. B )
38	32, 35, 37	syl2an 494	. . . . . . . . . . 11 |- ( ( (curry F : A --> ( C ^m B ) /\ x e. A ) /\ y (curry F ` x ) z ) -> y e. B )
39	38	an32s 846	. . . . . . . . . 10 |- ( ( (curry F : A --> ( C ^m B ) /\ y (curry F ` x ) z ) /\ x e. A ) -> y e. B )
40	28, 39	mpdan 702	. . . . . . . . 9 |- ( (curry F : A --> ( C ^m B ) /\ y (curry F ` x ) z ) -> y e. B )
41	28, 40	jca 554	. . . . . . . 8 |- ( (curry F : A --> ( C ^m B ) /\ y (curry F ` x ) z ) -> ( x e. A /\ y e. B ) )
42	12, 41	sylan 488	. . . . . . 7 |- ( ( ( F : ( A X. B ) --> C /\ B e. ( V \ { (/) } ) /\ C e. W ) /\ y (curry F ` x ) z ) -> ( x e. A /\ y e. B ) )
43	42	stoic1a 1697	. . . . . 6 |- ( ( ( F : ( A X. B ) --> C /\ B e. ( V \ { (/) } ) /\ C e. W ) /\ -. ( x e. A /\ y e. B ) ) -> -. y (curry F ` x ) z )
44		simpl 473	. . . . . . . 8 |- ( ( ( x e. A /\ y e. B ) /\ z = ( x F y ) ) -> ( x e. A /\ y e. B ) )
45	44	con3i 150	. . . . . . 7 |- ( -. ( x e. A /\ y e. B ) -> -. ( ( x e. A /\ y e. B ) /\ z = ( x F y ) ) )
46	45	adantl 482	. . . . . 6 |- ( ( ( F : ( A X. B ) --> C /\ B e. ( V \ { (/) } ) /\ C e. W ) /\ -. ( x e. A /\ y e. B ) ) -> -. ( ( x e. A /\ y e. B ) /\ z = ( x F y ) ) )
47	43, 46	2falsed 366	. . . . 5 |- ( ( ( F : ( A X. B ) --> C /\ B e. ( V \ { (/) } ) /\ C e. W ) /\ -. ( x e. A /\ y e. B ) ) -> ( y (curry F ` x ) z <-> ( ( x e. A /\ y e. B ) /\ z = ( x F y ) ) ) )
48	21, 47	pm2.61dan 832	. . . 4 |- ( ( F : ( A X. B ) --> C /\ B e. ( V \ { (/) } ) /\ C e. W ) -> ( y (curry F ` x ) z <-> ( ( x e. A /\ y e. B ) /\ z = ( x F y ) ) ) )
49	48	oprabbidv 6709	. . 3 |- ( ( F : ( A X. B ) --> C /\ B e. ( V \ { (/) } ) /\ C e. W ) -> { <. <. x , y >. , z >. | y (curry F ` x ) z } = { <. <. x , y >. , z >. | ( ( x e. A /\ y e. B ) /\ z = ( x F y ) ) } )
50		df-unc 7394	. . 3 |- uncurry curry F = { <. <. x , y >. , z >. | y (curry F ` x ) z }
51		df-mpt2 6655	. . 3 |- ( x e. A , y e. B |-> ( x F y ) ) = { <. <. x , y >. , z >. | ( ( x e. A /\ y e. B ) /\ z = ( x F y ) ) }
52	49, 50, 51	3eqtr4g 2681	. 2 |- ( ( F : ( A X. B ) --> C /\ B e. ( V \ { (/) } ) /\ C e. W ) -> uncurry curry F = ( x e. A , y e. B |-> ( x F y ) ) )
53		fnov 6768	. . . 4 |- ( F Fn ( A X. B ) <-> F = ( x e. A , y e. B |-> ( x F y ) ) )
54	1, 53	sylib 208	. . 3 |- ( F : ( A X. B ) --> C -> F = ( x e. A , y e. B |-> ( x F y ) ) )
55	54	3ad2ant1 1082	. 2 |- ( ( F : ( A X. B ) --> C /\ B e. ( V \ { (/) } ) /\ C e. W ) -> F = ( x e. A , y e. B |-> ( x F y ) ) )
56	52, 55	eqtr4d 2659	1 |- ( ( F : ( A X. B ) --> C /\ B e. ( V \ { (/) } ) /\ C e. W ) -> uncurry curry F = F )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   \ cdif 3571   (/)c0 3915   {csn 4177   <.cop 4183   class class class wbr 4653   X. cxp 5112   dom cdm 5114   Fn wfn 5883   -->wf 5884   ` cfv 5888  (class class class)co 6650   {coprab 6651   |-> cmpt2 6652  curry ccur 7391  uncurry cunc 7392   ^m cmap 7857

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-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-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-cur 7393  df-unc 7394  df-map 7859

This theorem is referenced by: (None)