Theorem curunc 33391

Description: Currying of uncurrying. (Contributed by Brendan Leahy, 2-Jun-2021.)

Assertion

Ref	Expression
curunc	|- ( ( F : A --> ( C ^m B ) /\ B =/= (/) ) -> curry uncurry F = F )

Proof of Theorem curunc

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

Step	Hyp	Ref	Expression
1		simpl 473	. . 3 |- ( ( F : A --> ( C ^m B ) /\ B =/= (/) ) -> F : A --> ( C ^m B ) )
2	1	feqmptd 6249	. 2 |- ( ( F : A --> ( C ^m B ) /\ B =/= (/) ) -> F = ( x e. A |-> ( F ` x ) ) )
3		uncf 33388	. . . . . . . 8 |- ( F : A --> ( C ^m B ) -> uncurry F : ( A X. B ) --> C )
4		fdm 6051	. . . . . . . 8 |- (uncurry F : ( A X. B ) --> C -> dom uncurry F = ( A X. B ) )
5	3, 4	syl 17	. . . . . . 7 |- ( F : A --> ( C ^m B ) -> dom uncurry F = ( A X. B ) )
6	5	dmeqd 5326	. . . . . 6 |- ( F : A --> ( C ^m B ) -> dom dom uncurry F = dom ( A X. B ) )
7		dmxp 5344	. . . . . 6 |- ( B =/= (/) -> dom ( A X. B ) = A )
8	6, 7	sylan9eq 2676	. . . . 5 |- ( ( F : A --> ( C ^m B ) /\ B =/= (/) ) -> dom dom uncurry F = A )
9	8	eqcomd 2628	. . . 4 |- ( ( F : A --> ( C ^m B ) /\ B =/= (/) ) -> A = dom dom uncurry F )
10		df-mpt 4730	. . . . . 6 |- ( y e. B |-> ( ( F ` x ) ` y ) ) = { <. y , z >. | ( y e. B /\ z = ( ( F ` x ) ` y ) ) }
11		ffvelrn 6357	. . . . . . . 8 |- ( ( F : A --> ( C ^m B ) /\ x e. A ) -> ( F ` x ) e. ( C ^m B ) )
12		elmapi 7879	. . . . . . . 8 |- ( ( F ` x ) e. ( C ^m B ) -> ( F ` x ) : B --> C )
13	11, 12	syl 17	. . . . . . 7 |- ( ( F : A --> ( C ^m B ) /\ x e. A ) -> ( F ` x ) : B --> C )
14	13	feqmptd 6249	. . . . . 6 |- ( ( F : A --> ( C ^m B ) /\ x e. A ) -> ( F ` x ) = ( y e. B |-> ( ( F ` x ) ` y ) ) )
15		ffun 6048	. . . . . . . . . 10 |- (uncurry F : ( A X. B ) --> C -> Fun uncurry F )
16		funbrfv2b 6240	. . . . . . . . . 10 |- ( Fun uncurry F -> ( <. x , y >.uncurry F z <-> ( <. x , y >. e. dom uncurry F /\ (uncurry F ` <. x , y >. ) = z ) ) )
17	3, 15, 16	3syl 18	. . . . . . . . 9 |- ( F : A --> ( C ^m B ) -> ( <. x , y >.uncurry F z <-> ( <. x , y >. e. dom uncurry F /\ (uncurry F ` <. x , y >. ) = z ) ) )
18	17	adantr 481	. . . . . . . 8 |- ( ( F : A --> ( C ^m B ) /\ x e. A ) -> ( <. x , y >.uncurry F z <-> ( <. x , y >. e. dom uncurry F /\ (uncurry F ` <. x , y >. ) = z ) ) )
19	5	eleq2d 2687	. . . . . . . . . 10 |- ( F : A --> ( C ^m B ) -> ( <. x , y >. e. dom uncurry F <-> <. x , y >. e. ( A X. B ) ) )
20		opelxp 5146	. . . . . . . . . . 11 |- ( <. x , y >. e. ( A X. B ) <-> ( x e. A /\ y e. B ) )
21	20	baib 944	. . . . . . . . . 10 |- ( x e. A -> ( <. x , y >. e. ( A X. B ) <-> y e. B ) )
22	19, 21	sylan9bb 736	. . . . . . . . 9 |- ( ( F : A --> ( C ^m B ) /\ x e. A ) -> ( <. x , y >. e. dom uncurry F <-> y e. B ) )
23		df-ov 6653	. . . . . . . . . . . . 13 |- ( xuncurry F y ) = (uncurry F ` <. x , y >. )
24		vex 3203	. . . . . . . . . . . . . 14 |- x e. _V
25		vex 3203	. . . . . . . . . . . . . 14 |- y e. _V
26		uncov 33390	. . . . . . . . . . . . . 14 |- ( ( x e. _V /\ y e. _V ) -> ( xuncurry F y ) = ( ( F ` x ) ` y ) )
27	24, 25, 26	mp2an 708	. . . . . . . . . . . . 13 |- ( xuncurry F y ) = ( ( F ` x ) ` y )
28	23, 27	eqtr3i 2646	. . . . . . . . . . . 12 |- (uncurry F ` <. x , y >. ) = ( ( F ` x ) ` y )
29	28	eqeq1i 2627	. . . . . . . . . . 11 |- ( (uncurry F ` <. x , y >. ) = z <-> ( ( F ` x ) ` y ) = z )
30		eqcom 2629	. . . . . . . . . . 11 |- ( ( ( F ` x ) ` y ) = z <-> z = ( ( F ` x ) ` y ) )
31	29, 30	bitri 264	. . . . . . . . . 10 |- ( (uncurry F ` <. x , y >. ) = z <-> z = ( ( F ` x ) ` y ) )
32	31	a1i 11	. . . . . . . . 9 |- ( ( F : A --> ( C ^m B ) /\ x e. A ) -> ( (uncurry F ` <. x , y >. ) = z <-> z = ( ( F ` x ) ` y ) ) )
33	22, 32	anbi12d 747	. . . . . . . 8 |- ( ( F : A --> ( C ^m B ) /\ x e. A ) -> ( ( <. x , y >. e. dom uncurry F /\ (uncurry F ` <. x , y >. ) = z ) <-> ( y e. B /\ z = ( ( F ` x ) ` y ) ) ) )
34	18, 33	bitrd 268	. . . . . . 7 |- ( ( F : A --> ( C ^m B ) /\ x e. A ) -> ( <. x , y >.uncurry F z <-> ( y e. B /\ z = ( ( F ` x ) ` y ) ) ) )
35	34	opabbidv 4716	. . . . . 6 |- ( ( F : A --> ( C ^m B ) /\ x e. A ) -> { <. y , z >. | <. x , y >.uncurry F z } = { <. y , z >. | ( y e. B /\ z = ( ( F ` x ) ` y ) ) } )
36	10, 14, 35	3eqtr4a 2682	. . . . 5 |- ( ( F : A --> ( C ^m B ) /\ x e. A ) -> ( F ` x ) = { <. y , z >. | <. x , y >.uncurry F z } )
37	36	adantlr 751	. . . 4 |- ( ( ( F : A --> ( C ^m B ) /\ B =/= (/) ) /\ x e. A ) -> ( F ` x ) = { <. y , z >. | <. x , y >.uncurry F z } )
38	9, 37	mpteq12dva 4732	. . 3 |- ( ( F : A --> ( C ^m B ) /\ B =/= (/) ) -> ( x e. A |-> ( F ` x ) ) = ( x e. dom dom uncurry F |-> { <. y , z >. | <. x , y >.uncurry F z } ) )
39		df-cur 7393	. . 3 |- curry uncurry F = ( x e. dom dom uncurry F |-> { <. y , z >. | <. x , y >.uncurry F z } )
40	38, 39	syl6eqr 2674	. 2 |- ( ( F : A --> ( C ^m B ) /\ B =/= (/) ) -> ( x e. A |-> ( F ` x ) ) = curry uncurry F )
41	2, 40	eqtr2d 2657	1 |- ( ( F : A --> ( C ^m B ) /\ B =/= (/) ) -> curry uncurry F = F )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   _Vcvv 3200   (/)c0 3915   <.cop 4183   class class class wbr 4653   {copab 4712   |-> cmpt 4729   X. cxp 5112   dom cdm 5114   Fun wfun 5882   -->wf 5884   ` cfv 5888  (class class class)co 6650  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-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-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-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)