Theorem curry1 6441

Description: Composition with `' ( 2nd |` ( { C } X. _V ) ) turns any binary operation F with a constant first operand into a function G of the second operand only. This transformation is called "currying." (Contributed by NM, 28-Mar-2008.) (Revised by Mario Carneiro, 26-Dec-2014.)

Hypothesis

Ref	Expression
curry1.1	|- G = ( F o. `' ( 2nd |` ( { C } X. _V ) ) )

Assertion

Ref	Expression
curry1	|- ( ( F Fn ( A X. B ) /\ C e. A ) -> G = ( x e. B |-> ( C F x ) ) )

Distinct variable groups:   x, A   x, B   x, C   x, F   x, G

Proof of Theorem curry1

Step	Hyp	Ref	Expression
1		fnfun 5545	. . . . 5 |- ( F Fn ( A X. B ) -> Fun F )
2		2ndconst 6439	. . . . . 6 |- ( C e. A -> ( 2nd |` ( { C } X. _V ) ) : ( { C } X. _V ) -1-1-onto-> _V )
3		dff1o3 5683	. . . . . . 7 |- ( ( 2nd |` ( { C } X. _V ) ) : ( { C } X. _V ) -1-1-onto-> _V <-> ( ( 2nd |` ( { C } X. _V ) ) : ( { C } X. _V ) -onto-> _V /\ Fun `' ( 2nd |` ( { C } X. _V ) ) ) )
4	3	simprbi 452	. . . . . 6 |- ( ( 2nd |` ( { C } X. _V ) ) : ( { C } X. _V ) -1-1-onto-> _V -> Fun `' ( 2nd |` ( { C } X. _V ) ) )
5	2, 4	syl 16	. . . . 5 |- ( C e. A -> Fun `' ( 2nd |` ( { C } X. _V ) ) )
6		funco 5494	. . . . 5 |- ( ( Fun F /\ Fun `' ( 2nd |` ( { C } X. _V ) ) ) -> Fun ( F o. `' ( 2nd |` ( { C } X. _V ) ) ) )
7	1, 5, 6	syl2an 465	. . . 4 |- ( ( F Fn ( A X. B ) /\ C e. A ) -> Fun ( F o. `' ( 2nd |` ( { C } X. _V ) ) ) )
8		dmco 5381	. . . . 5 |- dom ( F o. `' ( 2nd |` ( { C } X. _V ) ) ) = ( `' `' ( 2nd |` ( { C } X. _V ) ) " dom F )
9		fndm 5547	. . . . . . . 8 |- ( F Fn ( A X. B ) -> dom F = ( A X. B ) )
10	9	adantr 453	. . . . . . 7 |- ( ( F Fn ( A X. B ) /\ C e. A ) -> dom F = ( A X. B ) )
11	10	imaeq2d 5206	. . . . . 6 |- ( ( F Fn ( A X. B ) /\ C e. A ) -> ( `' `' ( 2nd |` ( { C } X. _V ) ) " dom F ) = ( `' `' ( 2nd |` ( { C } X. _V ) ) " ( A X. B ) ) )
12		imacnvcnv 5337	. . . . . . . . 9 |- ( `' `' ( 2nd |` ( { C } X. _V ) ) " ( A X. B ) ) = ( ( 2nd |` ( { C } X. _V ) ) " ( A X. B ) )
13		df-ima 4894	. . . . . . . . 9 |- ( ( 2nd |` ( { C } X. _V ) ) " ( A X. B ) ) = ran ( ( 2nd |` ( { C } X. _V ) ) |` ( A X. B ) )
14		resres 5162	. . . . . . . . . 10 |- ( ( 2nd |` ( { C } X. _V ) ) |` ( A X. B ) ) = ( 2nd |` ( ( { C } X. _V ) i^i ( A X. B ) ) )
15	14	rneqi 5099	. . . . . . . . 9 |- ran ( ( 2nd |` ( { C } X. _V ) ) |` ( A X. B ) ) = ran ( 2nd |` ( ( { C } X. _V ) i^i ( A X. B ) ) )
16	12, 13, 15	3eqtri 2462	. . . . . . . 8 |- ( `' `' ( 2nd |` ( { C } X. _V ) ) " ( A X. B ) ) = ran ( 2nd |` ( ( { C } X. _V ) i^i ( A X. B ) ) )
17		inxp 5010	. . . . . . . . . . . . 13 |- ( ( { C } X. _V ) i^i ( A X. B ) ) = ( ( { C } i^i A ) X. ( _V i^i B ) )
18		incom 3535	. . . . . . . . . . . . . . 15 |- ( _V i^i B ) = ( B i^i _V )
19		inv1 3656	. . . . . . . . . . . . . . 15 |- ( B i^i _V ) = B
20	18, 19	eqtri 2458	. . . . . . . . . . . . . 14 |- ( _V i^i B ) = B
21	20	xpeq2i 4902	. . . . . . . . . . . . 13 |- ( ( { C } i^i A ) X. ( _V i^i B ) ) = ( ( { C } i^i A ) X. B )
22	17, 21	eqtri 2458	. . . . . . . . . . . 12 |- ( ( { C } X. _V ) i^i ( A X. B ) ) = ( ( { C } i^i A ) X. B )
23		snssi 3944	. . . . . . . . . . . . . 14 |- ( C e. A -> { C } C_ A )
24		df-ss 3336	. . . . . . . . . . . . . 14 |- ( { C } C_ A <-> ( { C } i^i A ) = { C } )
25	23, 24	sylib 190	. . . . . . . . . . . . 13 |- ( C e. A -> ( { C } i^i A ) = { C } )
26	25	xpeq1d 4904	. . . . . . . . . . . 12 |- ( C e. A -> ( ( { C } i^i A ) X. B ) = ( { C } X. B ) )
27	22, 26	syl5eq 2482	. . . . . . . . . . 11 |- ( C e. A -> ( ( { C } X. _V ) i^i ( A X. B ) ) = ( { C } X. B ) )
28	27	reseq2d 5149	. . . . . . . . . 10 |- ( C e. A -> ( 2nd |` ( ( { C } X. _V ) i^i ( A X. B ) ) ) = ( 2nd |` ( { C } X. B ) ) )
29	28	rneqd 5100	. . . . . . . . 9 |- ( C e. A -> ran ( 2nd |` ( ( { C } X. _V ) i^i ( A X. B ) ) ) = ran ( 2nd |` ( { C } X. B ) ) )
30		2ndconst 6439	. . . . . . . . . 10 |- ( C e. A -> ( 2nd |` ( { C } X. B ) ) : ( { C } X. B ) -1-1-onto-> B )
31		f1ofo 5684	. . . . . . . . . 10 |- ( ( 2nd |` ( { C } X. B ) ) : ( { C } X. B ) -1-1-onto-> B -> ( 2nd |` ( { C } X. B ) ) : ( { C } X. B ) -onto-> B )
32		forn 5659	. . . . . . . . . 10 |- ( ( 2nd |` ( { C } X. B ) ) : ( { C } X. B ) -onto-> B -> ran ( 2nd |` ( { C } X. B ) ) = B )
33	30, 31, 32	3syl 19	. . . . . . . . 9 |- ( C e. A -> ran ( 2nd |` ( { C } X. B ) ) = B )
34	29, 33	eqtrd 2470	. . . . . . . 8 |- ( C e. A -> ran ( 2nd |` ( ( { C } X. _V ) i^i ( A X. B ) ) ) = B )
35	16, 34	syl5eq 2482	. . . . . . 7 |- ( C e. A -> ( `' `' ( 2nd |` ( { C } X. _V ) ) " ( A X. B ) ) = B )
36	35	adantl 454	. . . . . 6 |- ( ( F Fn ( A X. B ) /\ C e. A ) -> ( `' `' ( 2nd |` ( { C } X. _V ) ) " ( A X. B ) ) = B )
37	11, 36	eqtrd 2470	. . . . 5 |- ( ( F Fn ( A X. B ) /\ C e. A ) -> ( `' `' ( 2nd |` ( { C } X. _V ) ) " dom F ) = B )
38	8, 37	syl5eq 2482	. . . 4 |- ( ( F Fn ( A X. B ) /\ C e. A ) -> dom ( F o. `' ( 2nd |` ( { C } X. _V ) ) ) = B )
39		curry1.1	. . . . . 6 |- G = ( F o. `' ( 2nd |` ( { C } X. _V ) ) )
40	39	fneq1i 5542	. . . . 5 |- ( G Fn B <-> ( F o. `' ( 2nd |` ( { C } X. _V ) ) ) Fn B )
41		df-fn 5460	. . . . 5 |- ( ( F o. `' ( 2nd |` ( { C } X. _V ) ) ) Fn B <-> ( Fun ( F o. `' ( 2nd |` ( { C } X. _V ) ) ) /\ dom ( F o. `' ( 2nd |` ( { C } X. _V ) ) ) = B ) )
42	40, 41	bitri 242	. . . 4 |- ( G Fn B <-> ( Fun ( F o. `' ( 2nd |` ( { C } X. _V ) ) ) /\ dom ( F o. `' ( 2nd |` ( { C } X. _V ) ) ) = B ) )
43	7, 38, 42	sylanbrc 647	. . 3 |- ( ( F Fn ( A X. B ) /\ C e. A ) -> G Fn B )
44		dffn5 5775	. . 3 |- ( G Fn B <-> G = ( x e. B |-> ( G ` x ) ) )
45	43, 44	sylib 190	. 2 |- ( ( F Fn ( A X. B ) /\ C e. A ) -> G = ( x e. B |-> ( G ` x ) ) )
46	39	fveq1i 5732	. . . . 5 |- ( G ` x ) = ( ( F o. `' ( 2nd |` ( { C } X. _V ) ) ) ` x )
47		dff1o4 5685	. . . . . . . . 9 |- ( ( 2nd |` ( { C } X. _V ) ) : ( { C } X. _V ) -1-1-onto-> _V <-> ( ( 2nd |` ( { C } X. _V ) ) Fn ( { C } X. _V ) /\ `' ( 2nd |` ( { C } X. _V ) ) Fn _V ) )
48	2, 47	sylib 190	. . . . . . . 8 |- ( C e. A -> ( ( 2nd |` ( { C } X. _V ) ) Fn ( { C } X. _V ) /\ `' ( 2nd |` ( { C } X. _V ) ) Fn _V ) )
49	48	simprd 451	. . . . . . 7 |- ( C e. A -> `' ( 2nd |` ( { C } X. _V ) ) Fn _V )
50		vex 2961	. . . . . . . 8 |- x e. _V
51		fvco2 5801	. . . . . . . 8 |- ( ( `' ( 2nd |` ( { C } X. _V ) ) Fn _V /\ x e. _V ) -> ( ( F o. `' ( 2nd |` ( { C } X. _V ) ) ) ` x ) = ( F ` ( `' ( 2nd |` ( { C } X. _V ) ) ` x ) ) )
52	50, 51	mpan2 654	. . . . . . 7 |- ( `' ( 2nd |` ( { C } X. _V ) ) Fn _V -> ( ( F o. `' ( 2nd |` ( { C } X. _V ) ) ) ` x ) = ( F ` ( `' ( 2nd |` ( { C } X. _V ) ) ` x ) ) )
53	49, 52	syl 16	. . . . . 6 |- ( C e. A -> ( ( F o. `' ( 2nd |` ( { C } X. _V ) ) ) ` x ) = ( F ` ( `' ( 2nd |` ( { C } X. _V ) ) ` x ) ) )
54	53	ad2antlr 709	. . . . 5 |- ( ( ( F Fn ( A X. B ) /\ C e. A ) /\ x e. B ) -> ( ( F o. `' ( 2nd |` ( { C } X. _V ) ) ) ` x ) = ( F ` ( `' ( 2nd |` ( { C } X. _V ) ) ` x ) ) )
55	46, 54	syl5eq 2482	. . . 4 |- ( ( ( F Fn ( A X. B ) /\ C e. A ) /\ x e. B ) -> ( G ` x ) = ( F ` ( `' ( 2nd |` ( { C } X. _V ) ) ` x ) ) )
56	2	adantr 453	. . . . . . . . 9 |- ( ( C e. A /\ x e. B ) -> ( 2nd |` ( { C } X. _V ) ) : ( { C } X. _V ) -1-1-onto-> _V )
57		snidg 3841	. . . . . . . . . . . 12 |- ( C e. A -> C e. { C } )
58	57, 50	jctir 526	. . . . . . . . . . 11 |- ( C e. A -> ( C e. { C } /\ x e. _V ) )
59		opelxp 4911	. . . . . . . . . . 11 |- ( <. C , x >. e. ( { C } X. _V ) <-> ( C e. { C } /\ x e. _V ) )
60	58, 59	sylibr 205	. . . . . . . . . 10 |- ( C e. A -> <. C , x >. e. ( { C } X. _V ) )
61	60	adantr 453	. . . . . . . . 9 |- ( ( C e. A /\ x e. B ) -> <. C , x >. e. ( { C } X. _V ) )
62	56, 61	jca 520	. . . . . . . 8 |- ( ( C e. A /\ x e. B ) -> ( ( 2nd |` ( { C } X. _V ) ) : ( { C } X. _V ) -1-1-onto-> _V /\ <. C , x >. e. ( { C } X. _V ) ) )
63		fvres 5748	. . . . . . . . . . 11 |- ( <. C , x >. e. ( { C } X. _V ) -> ( ( 2nd |` ( { C } X. _V ) ) ` <. C , x >. ) = ( 2nd ` <. C , x >. ) )
64	60, 63	syl 16	. . . . . . . . . 10 |- ( C e. A -> ( ( 2nd |` ( { C } X. _V ) ) ` <. C , x >. ) = ( 2nd ` <. C , x >. ) )
65		op2ndg 6363	. . . . . . . . . . 11 |- ( ( C e. A /\ x e. _V ) -> ( 2nd ` <. C , x >. ) = x )
66	50, 65	mpan2 654	. . . . . . . . . 10 |- ( C e. A -> ( 2nd ` <. C , x >. ) = x )
67	64, 66	eqtrd 2470	. . . . . . . . 9 |- ( C e. A -> ( ( 2nd |` ( { C } X. _V ) ) ` <. C , x >. ) = x )
68	67	adantr 453	. . . . . . . 8 |- ( ( C e. A /\ x e. B ) -> ( ( 2nd |` ( { C } X. _V ) ) ` <. C , x >. ) = x )
69		f1ocnvfv 6019	. . . . . . . 8 |- ( ( ( 2nd |` ( { C } X. _V ) ) : ( { C } X. _V ) -1-1-onto-> _V /\ <. C , x >. e. ( { C } X. _V ) ) -> ( ( ( 2nd |` ( { C } X. _V ) ) ` <. C , x >. ) = x -> ( `' ( 2nd |` ( { C } X. _V ) ) ` x ) = <. C , x >. ) )
70	62, 68, 69	sylc 59	. . . . . . 7 |- ( ( C e. A /\ x e. B ) -> ( `' ( 2nd |` ( { C } X. _V ) ) ` x ) = <. C , x >. )
71	70	fveq2d 5735	. . . . . 6 |- ( ( C e. A /\ x e. B ) -> ( F ` ( `' ( 2nd |` ( { C } X. _V ) ) ` x ) ) = ( F ` <. C , x >. ) )
72	71	adantll 696	. . . . 5 |- ( ( ( F Fn ( A X. B ) /\ C e. A ) /\ x e. B ) -> ( F ` ( `' ( 2nd |` ( { C } X. _V ) ) ` x ) ) = ( F ` <. C , x >. ) )
73		df-ov 6087	. . . . 5 |- ( C F x ) = ( F ` <. C , x >. )
74	72, 73	syl6eqr 2488	. . . 4 |- ( ( ( F Fn ( A X. B ) /\ C e. A ) /\ x e. B ) -> ( F ` ( `' ( 2nd |` ( { C } X. _V ) ) ` x ) ) = ( C F x ) )
75	55, 74	eqtrd 2470	. . 3 |- ( ( ( F Fn ( A X. B ) /\ C e. A ) /\ x e. B ) -> ( G ` x ) = ( C F x ) )
76	75	mpteq2dva 4298	. 2 |- ( ( F Fn ( A X. B ) /\ C e. A ) -> ( x e. B |-> ( G ` x ) ) = ( x e. B |-> ( C F x ) ) )
77	45, 76	eqtrd 2470	1 |- ( ( F Fn ( A X. B ) /\ C e. A ) -> G = ( x e. B |-> ( C F x ) ) )

Syntax hints:   -> wi 4   /\ wa 360   = wceq 1653   e. wcel 1726   _Vcvv 2958   i^i cin 3321   C_ wss 3322   {csn 3816   <.cop 3819   e. cmpt 4269   X. cxp 4879   `'ccnv 4880   dom cdm 4881   ran crn 4882   |` cres 4883   "cima 4884   o. ccom 4885   Fun wfun 5451   Fn wfn 5452   -onto->wfo 5455   -1-1-onto->wf1o 5456   ` cfv 5457  (class class class)co 6084   2ndc2nd 6351

This theorem is referenced by:  curry1val  6442  curry1f  6443

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704

This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-ov 6087  df-1st 6352  df-2nd 6353