Theorem curry2 7272

Description: Composition with `' ( 1st |` ( _V X. { C } ) ) turns any binary operation F with a constant second operand into a function G of the first operand only. This transformation is called "currying." (If this becomes frequently used, we can introduce a new notation for the hypothesis.) (Contributed by NM, 16-Dec-2008.)

Hypothesis

Ref	Expression
curry2.1	|- G = ( F o. `' ( 1st |` ( _V X. { C } ) ) )

Assertion

Ref	Expression
curry2	|- ( ( F Fn ( A X. B ) /\ C e. B ) -> G = ( x e. A |-> ( x F C ) ) )

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

Proof of Theorem curry2

Step	Hyp	Ref	Expression
1		fnfun 5988	. . . . 5 |- ( F Fn ( A X. B ) -> Fun F )
2		1stconst 7265	. . . . . 6 |- ( C e. B -> ( 1st |` ( _V X. { C } ) ) : ( _V X. { C } ) -1-1-onto-> _V )
3		dff1o3 6143	. . . . . . 7 |- ( ( 1st |` ( _V X. { C } ) ) : ( _V X. { C } ) -1-1-onto-> _V <-> ( ( 1st |` ( _V X. { C } ) ) : ( _V X. { C } ) -onto-> _V /\ Fun `' ( 1st |` ( _V X. { C } ) ) ) )
4	3	simprbi 480	. . . . . 6 |- ( ( 1st |` ( _V X. { C } ) ) : ( _V X. { C } ) -1-1-onto-> _V -> Fun `' ( 1st |` ( _V X. { C } ) ) )
5	2, 4	syl 17	. . . . 5 |- ( C e. B -> Fun `' ( 1st |` ( _V X. { C } ) ) )
6		funco 5928	. . . . 5 |- ( ( Fun F /\ Fun `' ( 1st |` ( _V X. { C } ) ) ) -> Fun ( F o. `' ( 1st |` ( _V X. { C } ) ) ) )
7	1, 5, 6	syl2an 494	. . . 4 |- ( ( F Fn ( A X. B ) /\ C e. B ) -> Fun ( F o. `' ( 1st |` ( _V X. { C } ) ) ) )
8		dmco 5643	. . . . 5 |- dom ( F o. `' ( 1st |` ( _V X. { C } ) ) ) = ( `' `' ( 1st |` ( _V X. { C } ) ) " dom F )
9		fndm 5990	. . . . . . . 8 |- ( F Fn ( A X. B ) -> dom F = ( A X. B ) )
10	9	adantr 481	. . . . . . 7 |- ( ( F Fn ( A X. B ) /\ C e. B ) -> dom F = ( A X. B ) )
11	10	imaeq2d 5466	. . . . . 6 |- ( ( F Fn ( A X. B ) /\ C e. B ) -> ( `' `' ( 1st |` ( _V X. { C } ) ) " dom F ) = ( `' `' ( 1st |` ( _V X. { C } ) ) " ( A X. B ) ) )
12		imacnvcnv 5599	. . . . . . . . 9 |- ( `' `' ( 1st |` ( _V X. { C } ) ) " ( A X. B ) ) = ( ( 1st |` ( _V X. { C } ) ) " ( A X. B ) )
13		df-ima 5127	. . . . . . . . 9 |- ( ( 1st |` ( _V X. { C } ) ) " ( A X. B ) ) = ran ( ( 1st |` ( _V X. { C } ) ) |` ( A X. B ) )
14		resres 5409	. . . . . . . . . 10 |- ( ( 1st |` ( _V X. { C } ) ) |` ( A X. B ) ) = ( 1st |` ( ( _V X. { C } ) i^i ( A X. B ) ) )
15	14	rneqi 5352	. . . . . . . . 9 |- ran ( ( 1st |` ( _V X. { C } ) ) |` ( A X. B ) ) = ran ( 1st |` ( ( _V X. { C } ) i^i ( A X. B ) ) )
16	12, 13, 15	3eqtri 2648	. . . . . . . 8 |- ( `' `' ( 1st |` ( _V X. { C } ) ) " ( A X. B ) ) = ran ( 1st |` ( ( _V X. { C } ) i^i ( A X. B ) ) )
17		inxp 5254	. . . . . . . . . . . . 13 |- ( ( _V X. { C } ) i^i ( A X. B ) ) = ( ( _V i^i A ) X. ( { C } i^i B ) )
18		incom 3805	. . . . . . . . . . . . . . 15 |- ( _V i^i A ) = ( A i^i _V )
19		inv1 3970	. . . . . . . . . . . . . . 15 |- ( A i^i _V ) = A
20	18, 19	eqtri 2644	. . . . . . . . . . . . . 14 |- ( _V i^i A ) = A
21	20	xpeq1i 5135	. . . . . . . . . . . . 13 |- ( ( _V i^i A ) X. ( { C } i^i B ) ) = ( A X. ( { C } i^i B ) )
22	17, 21	eqtri 2644	. . . . . . . . . . . 12 |- ( ( _V X. { C } ) i^i ( A X. B ) ) = ( A X. ( { C } i^i B ) )
23		snssi 4339	. . . . . . . . . . . . . 14 |- ( C e. B -> { C } C_ B )
24		df-ss 3588	. . . . . . . . . . . . . 14 |- ( { C } C_ B <-> ( { C } i^i B ) = { C } )
25	23, 24	sylib 208	. . . . . . . . . . . . 13 |- ( C e. B -> ( { C } i^i B ) = { C } )
26	25	xpeq2d 5139	. . . . . . . . . . . 12 |- ( C e. B -> ( A X. ( { C } i^i B ) ) = ( A X. { C } ) )
27	22, 26	syl5eq 2668	. . . . . . . . . . 11 |- ( C e. B -> ( ( _V X. { C } ) i^i ( A X. B ) ) = ( A X. { C } ) )
28	27	reseq2d 5396	. . . . . . . . . 10 |- ( C e. B -> ( 1st |` ( ( _V X. { C } ) i^i ( A X. B ) ) ) = ( 1st |` ( A X. { C } ) ) )
29	28	rneqd 5353	. . . . . . . . 9 |- ( C e. B -> ran ( 1st |` ( ( _V X. { C } ) i^i ( A X. B ) ) ) = ran ( 1st |` ( A X. { C } ) ) )
30		1stconst 7265	. . . . . . . . . 10 |- ( C e. B -> ( 1st |` ( A X. { C } ) ) : ( A X. { C } ) -1-1-onto-> A )
31		f1ofo 6144	. . . . . . . . . 10 |- ( ( 1st |` ( A X. { C } ) ) : ( A X. { C } ) -1-1-onto-> A -> ( 1st |` ( A X. { C } ) ) : ( A X. { C } ) -onto-> A )
32		forn 6118	. . . . . . . . . 10 |- ( ( 1st |` ( A X. { C } ) ) : ( A X. { C } ) -onto-> A -> ran ( 1st |` ( A X. { C } ) ) = A )
33	30, 31, 32	3syl 18	. . . . . . . . 9 |- ( C e. B -> ran ( 1st |` ( A X. { C } ) ) = A )
34	29, 33	eqtrd 2656	. . . . . . . 8 |- ( C e. B -> ran ( 1st |` ( ( _V X. { C } ) i^i ( A X. B ) ) ) = A )
35	16, 34	syl5eq 2668	. . . . . . 7 |- ( C e. B -> ( `' `' ( 1st |` ( _V X. { C } ) ) " ( A X. B ) ) = A )
36	35	adantl 482	. . . . . 6 |- ( ( F Fn ( A X. B ) /\ C e. B ) -> ( `' `' ( 1st |` ( _V X. { C } ) ) " ( A X. B ) ) = A )
37	11, 36	eqtrd 2656	. . . . 5 |- ( ( F Fn ( A X. B ) /\ C e. B ) -> ( `' `' ( 1st |` ( _V X. { C } ) ) " dom F ) = A )
38	8, 37	syl5eq 2668	. . . 4 |- ( ( F Fn ( A X. B ) /\ C e. B ) -> dom ( F o. `' ( 1st |` ( _V X. { C } ) ) ) = A )
39		curry2.1	. . . . . 6 |- G = ( F o. `' ( 1st |` ( _V X. { C } ) ) )
40	39	fneq1i 5985	. . . . 5 |- ( G Fn A <-> ( F o. `' ( 1st |` ( _V X. { C } ) ) ) Fn A )
41		df-fn 5891	. . . . 5 |- ( ( F o. `' ( 1st |` ( _V X. { C } ) ) ) Fn A <-> ( Fun ( F o. `' ( 1st |` ( _V X. { C } ) ) ) /\ dom ( F o. `' ( 1st |` ( _V X. { C } ) ) ) = A ) )
42	40, 41	bitri 264	. . . 4 |- ( G Fn A <-> ( Fun ( F o. `' ( 1st |` ( _V X. { C } ) ) ) /\ dom ( F o. `' ( 1st |` ( _V X. { C } ) ) ) = A ) )
43	7, 38, 42	sylanbrc 698	. . 3 |- ( ( F Fn ( A X. B ) /\ C e. B ) -> G Fn A )
44		dffn5 6241	. . 3 |- ( G Fn A <-> G = ( x e. A |-> ( G ` x ) ) )
45	43, 44	sylib 208	. 2 |- ( ( F Fn ( A X. B ) /\ C e. B ) -> G = ( x e. A |-> ( G ` x ) ) )
46	39	fveq1i 6192	. . . . 5 |- ( G ` x ) = ( ( F o. `' ( 1st |` ( _V X. { C } ) ) ) ` x )
47		dff1o4 6145	. . . . . . . . 9 |- ( ( 1st |` ( _V X. { C } ) ) : ( _V X. { C } ) -1-1-onto-> _V <-> ( ( 1st |` ( _V X. { C } ) ) Fn ( _V X. { C } ) /\ `' ( 1st |` ( _V X. { C } ) ) Fn _V ) )
48	2, 47	sylib 208	. . . . . . . 8 |- ( C e. B -> ( ( 1st |` ( _V X. { C } ) ) Fn ( _V X. { C } ) /\ `' ( 1st |` ( _V X. { C } ) ) Fn _V ) )
49	48	simprd 479	. . . . . . 7 |- ( C e. B -> `' ( 1st |` ( _V X. { C } ) ) Fn _V )
50		vex 3203	. . . . . . 7 |- x e. _V
51		fvco2 6273	. . . . . . 7 |- ( ( `' ( 1st |` ( _V X. { C } ) ) Fn _V /\ x e. _V ) -> ( ( F o. `' ( 1st |` ( _V X. { C } ) ) ) ` x ) = ( F ` ( `' ( 1st |` ( _V X. { C } ) ) ` x ) ) )
52	49, 50, 51	sylancl 694	. . . . . 6 |- ( C e. B -> ( ( F o. `' ( 1st |` ( _V X. { C } ) ) ) ` x ) = ( F ` ( `' ( 1st |` ( _V X. { C } ) ) ` x ) ) )
53	52	ad2antlr 763	. . . . 5 |- ( ( ( F Fn ( A X. B ) /\ C e. B ) /\ x e. A ) -> ( ( F o. `' ( 1st |` ( _V X. { C } ) ) ) ` x ) = ( F ` ( `' ( 1st |` ( _V X. { C } ) ) ` x ) ) )
54	46, 53	syl5eq 2668	. . . 4 |- ( ( ( F Fn ( A X. B ) /\ C e. B ) /\ x e. A ) -> ( G ` x ) = ( F ` ( `' ( 1st |` ( _V X. { C } ) ) ` x ) ) )
55	2	adantr 481	. . . . . . . . 9 |- ( ( C e. B /\ x e. A ) -> ( 1st |` ( _V X. { C } ) ) : ( _V X. { C } ) -1-1-onto-> _V )
56	50	a1i 11	. . . . . . . . . 10 |- ( ( C e. B /\ x e. A ) -> x e. _V )
57		snidg 4206	. . . . . . . . . . 11 |- ( C e. B -> C e. { C } )
58	57	adantr 481	. . . . . . . . . 10 |- ( ( C e. B /\ x e. A ) -> C e. { C } )
59		opelxp 5146	. . . . . . . . . 10 |- ( <. x , C >. e. ( _V X. { C } ) <-> ( x e. _V /\ C e. { C } ) )
60	56, 58, 59	sylanbrc 698	. . . . . . . . 9 |- ( ( C e. B /\ x e. A ) -> <. x , C >. e. ( _V X. { C } ) )
61	55, 60	jca 554	. . . . . . . 8 |- ( ( C e. B /\ x e. A ) -> ( ( 1st |` ( _V X. { C } ) ) : ( _V X. { C } ) -1-1-onto-> _V /\ <. x , C >. e. ( _V X. { C } ) ) )
62	50	a1i 11	. . . . . . . . . . . 12 |- ( C e. B -> x e. _V )
63	62, 57, 59	sylanbrc 698	. . . . . . . . . . 11 |- ( C e. B -> <. x , C >. e. ( _V X. { C } ) )
64		fvres 6207	. . . . . . . . . . 11 |- ( <. x , C >. e. ( _V X. { C } ) -> ( ( 1st |` ( _V X. { C } ) ) ` <. x , C >. ) = ( 1st ` <. x , C >. ) )
65	63, 64	syl 17	. . . . . . . . . 10 |- ( C e. B -> ( ( 1st |` ( _V X. { C } ) ) ` <. x , C >. ) = ( 1st ` <. x , C >. ) )
66	65	adantr 481	. . . . . . . . 9 |- ( ( C e. B /\ x e. A ) -> ( ( 1st |` ( _V X. { C } ) ) ` <. x , C >. ) = ( 1st ` <. x , C >. ) )
67		op1stg 7180	. . . . . . . . . 10 |- ( ( x e. A /\ C e. B ) -> ( 1st ` <. x , C >. ) = x )
68	67	ancoms 469	. . . . . . . . 9 |- ( ( C e. B /\ x e. A ) -> ( 1st ` <. x , C >. ) = x )
69	66, 68	eqtrd 2656	. . . . . . . 8 |- ( ( C e. B /\ x e. A ) -> ( ( 1st |` ( _V X. { C } ) ) ` <. x , C >. ) = x )
70		f1ocnvfv 6534	. . . . . . . 8 |- ( ( ( 1st |` ( _V X. { C } ) ) : ( _V X. { C } ) -1-1-onto-> _V /\ <. x , C >. e. ( _V X. { C } ) ) -> ( ( ( 1st |` ( _V X. { C } ) ) ` <. x , C >. ) = x -> ( `' ( 1st |` ( _V X. { C } ) ) ` x ) = <. x , C >. ) )
71	61, 69, 70	sylc 65	. . . . . . 7 |- ( ( C e. B /\ x e. A ) -> ( `' ( 1st |` ( _V X. { C } ) ) ` x ) = <. x , C >. )
72	71	fveq2d 6195	. . . . . 6 |- ( ( C e. B /\ x e. A ) -> ( F ` ( `' ( 1st |` ( _V X. { C } ) ) ` x ) ) = ( F ` <. x , C >. ) )
73	72	adantll 750	. . . . 5 |- ( ( ( F Fn ( A X. B ) /\ C e. B ) /\ x e. A ) -> ( F ` ( `' ( 1st |` ( _V X. { C } ) ) ` x ) ) = ( F ` <. x , C >. ) )
74		df-ov 6653	. . . . 5 |- ( x F C ) = ( F ` <. x , C >. )
75	73, 74	syl6eqr 2674	. . . 4 |- ( ( ( F Fn ( A X. B ) /\ C e. B ) /\ x e. A ) -> ( F ` ( `' ( 1st |` ( _V X. { C } ) ) ` x ) ) = ( x F C ) )
76	54, 75	eqtrd 2656	. . 3 |- ( ( ( F Fn ( A X. B ) /\ C e. B ) /\ x e. A ) -> ( G ` x ) = ( x F C ) )
77	76	mpteq2dva 4744	. 2 |- ( ( F Fn ( A X. B ) /\ C e. B ) -> ( x e. A |-> ( G ` x ) ) = ( x e. A |-> ( x F C ) ) )
78	45, 77	eqtrd 2656	1 |- ( ( F Fn ( A X. B ) /\ C e. B ) -> G = ( x e. A |-> ( x F C ) ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   i^i cin 3573   C_ wss 3574   {csn 4177   <.cop 4183   |-> cmpt 4729   X. cxp 5112   `'ccnv 5113   dom cdm 5114   ran crn 5115   |` cres 5116   "cima 5117   o. ccom 5118   Fun wfun 5882   Fn wfn 5883   -onto->wfo 5886   -1-1-onto->wf1o 5887   ` cfv 5888  (class class class)co 6650   1stc1st 7166

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-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-1st 7168  df-2nd 7169

This theorem is referenced by:  curry2f  7273  curry2val  7274