Theorem curry1 5075

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."

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, y>. | (x e. B /\ y = (CFx))})

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

Proof of Theorem curry1

Step	Hyp	Ref	Expression
1		2ndconst 5071	. . . . . . . . . 10 |- (C e. A -> (2nd |` ({C} X. _V)):({C} X. _V)-1-1-onto->_V)
2	1	adantr 425	. . . . . . . . 9 |- ((C e. A /\ z e. B) -> (2nd |` ({C} X. _V)):({C} X. _V)-1-1-onto->_V)
3		snidg 3067	. . . . . . . . . . . 12 |- (C e. A -> C e. {C})
4		visset 2295	. . . . . . . . . . . 12 |- z e. _V
5	3, 4	jctir 317	. . . . . . . . . . 11 |- (C e. A -> (C e. {C} /\ z e. _V))
6	4	opelxp 4036	. . . . . . . . . . 11 |- (<.C, z>. e. ({C} X. _V) <-> (C e. {C} /\ z e. _V))
7	5, 6	sylibr 217	. . . . . . . . . 10 |- (C e. A -> <.C, z>. e. ({C} X. _V))
8	7	adantr 425	. . . . . . . . 9 |- ((C e. A /\ z e. B) -> <.C, z>. e. ({C} X. _V))
9	2, 8	jca 310	. . . . . . . 8 |- ((C e. A /\ z e. B) -> ((2nd |` ({C} X. _V)):({C} X. _V)-1-1-onto->_V /\ <.C, z>. e. ({C} X. _V)))
10		fvres 4691	. . . . . . . . . . 11 |- (<.C, z>. e. ({C} X. _V) -> ((2nd |` ({C} X. _V))` <.C, z>.) = (2nd` <.C, z>.))
11	7, 10	syl 12	. . . . . . . . . 10 |- (C e. A -> ((2nd |` ({C} X. _V))` <.C, z>.) = (2nd` <.C, z>.))
12		op2ndg 5029	. . . . . . . . . . 11 |- ((C e. A /\ z e. _V) -> (2nd` <.C, z>.) = z)
13	4, 12	mpan2 760	. . . . . . . . . 10 |- (C e. A -> (2nd` <.C, z>.) = z)
14	11, 13	eqtrd 1925	. . . . . . . . 9 |- (C e. A -> ((2nd |` ({C} X. _V))` <.C, z>.) = z)
15	14	adantr 425	. . . . . . . 8 |- ((C e. A /\ z e. B) -> ((2nd |` ({C} X. _V))` <.C, z>.) = z)
16		f1ocnvfv 4856	. . . . . . . 8 |- (((2nd |` ({C} X. _V)):({C} X. _V)-1-1-onto->_V /\ <.C, z>. e. ({C} X. _V)) -> (((2nd |` ({C} X. _V))` <.C, z>.) = z -> (`'(2nd |` ({C} X. _V))` z) = <.C, z>.))
17	9, 15, 16	sylc 83	. . . . . . 7 |- ((C e. A /\ z e. B) -> (`'(2nd |` ({C} X. _V))` z) = <.C, z>.)
18	17	fveq2d 4685	. . . . . 6 |- ((C e. A /\ z e. B) -> (F` (`'(2nd |` ({C} X. _V))` z)) = (F` <.C, z>.))
19	18	adantll 428	. . . . 5 |- (((F Fn (A X. B) /\ C e. A) /\ z e. B) -> (F` (`'(2nd |` ({C} X. _V))` z)) = (F` <.C, z>.))
20		df-opr 4886	. . . . 5 |- (CFz) = (F` <.C, z>.)
21	19, 20	syl6eqr 1946	. . . 4 |- (((F Fn (A X. B) /\ C e. A) /\ z e. B) -> (F` (`'(2nd |` ({C} X. _V))` z)) = (CFz))
22		fvco2 4737	. . . . . . . 8 |- ((Fun F /\ `'(2nd |` ({C} X. _V)) Fn _V /\ z e. _V) -> ((F o. `'(2nd |` ({C} X. _V)))` z) = (F` (`'(2nd |` ({C} X. _V))` z)))
23	4, 22	mp3an3 1180	. . . . . . 7 |- ((Fun F /\ `'(2nd |` ({C} X. _V)) Fn _V) -> ((F o. `'(2nd |` ({C} X. _V)))` z) = (F` (`'(2nd |` ({C} X. _V))` z)))
24		fnfun 4510	. . . . . . 7 |- (F Fn (A X. B) -> Fun F)
25		dff1o4 4644	. . . . . . . . 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))
26	1, 25	sylib 215	. . . . . . . 8 |- (C e. A -> ((2nd |` ({C} X. _V)) Fn ({C} X. _V) /\ `'(2nd |` ({C} X. _V)) Fn _V))
27	26	simprd 352	. . . . . . 7 |- (C e. A -> `'(2nd |` ({C} X. _V)) Fn _V)
28	23, 24, 27	syl2an 503	. . . . . 6 |- ((F Fn (A X. B) /\ C e. A) -> ((F o. `'(2nd |` ({C} X. _V)))` z) = (F` (`'(2nd |` ({C} X. _V))` z)))
29	28	adantr 425	. . . . 5 |- (((F Fn (A X. B) /\ C e. A) /\ z e. B) -> ((F o. `'(2nd |` ({C} X. _V)))` z) = (F` (`'(2nd |` ({C} X. _V))` z)))
30		curry1.1	. . . . . 6 |- G = (F o. `'(2nd |` ({C} X. _V)))
31	30	fveq1i 4682	. . . . 5 |- (G` z) = ((F o. `'(2nd |` ({C} X. _V)))` z)
32	29, 31	syl5eq 1940	. . . 4 |- (((F Fn (A X. B) /\ C e. A) /\ z e. B) -> (G` z) = (F` (`'(2nd |` ({C} X. _V))` z)))
33		opreq2 4890	. . . . . 6 |- (x = z -> (CFx) = (CFz))
34		eqid 1884	. . . . . 6 |- {<.x, y>. | (x e. B /\ y = (CFx))} = {<.x, y>. | (x e. B /\ y = (CFx))}
35		oprex 4907	. . . . . 6 |- (CFz) e. _V
36	33, 34, 35	fvopab4 4743	. . . . 5 |- (z e. B -> ({<.x, y>. | (x e. B /\ y = (CFx))}` z) = (CFz))
37	36	adantl 424	. . . 4 |- (((F Fn (A X. B) /\ C e. A) /\ z e. B) -> ({<.x, y>. | (x e. B /\ y = (CFx))}` z) = (CFz))
38	21, 32, 37	3eqtr4d 1937	. . 3 |- (((F Fn (A X. B) /\ C e. A) /\ z e. B) -> (G` z) = ({<.x, y>. | (x e. B /\ y = (CFx))}` z))
39	38	r19.21aiva 2176	. 2 |- ((F Fn (A X. B) /\ C e. A) -> A.z e. B (G` z) = ({<.x, y>. | (x e. B /\ y = (CFx))}` z))
40		eqfnfv2 4767	. . 3 |- ((G Fn B /\ {<.x, y>. | (x e. B /\ y = (CFx))} Fn B) -> (G = {<.x, y>. | (x e. B /\ y = (CFx))} <-> A.z e. B (G` z) = ({<.x, y>. | (x e. B /\ y = (CFx))}` z)))
41	30	fneq1i 4507	. . . . 5 |- (G Fn B <-> (F o. `'(2nd |` ({C} X. _V))) Fn B)
42		df-fn 4009	. . . . 5 |- ((F o. `'(2nd |` ({C} X. _V))) Fn B <-> (Fun (F o. `'(2nd |` ({C} X. _V))) /\ dom ( F o. `'(2nd |` ({C} X. _V))) = B))
43	41, 42	bitri 190	. . . 4 |- (G Fn B <-> (Fun (F o. `'(2nd |` ({C} X. _V))) /\ dom ( F o. `'(2nd |` ({C} X. _V))) = B))
44		funco 4457	. . . . 5 |- ((Fun F /\ Fun `'(2nd |` ({C} X. _V))) -> Fun (F o. `'(2nd |` ({C} X. _V))))
45		dff1o3 4641	. . . . . . 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))))
46	45	simprbi 353	. . . . . 6 |- ((2nd |` ({C} X. _V)):({C} X. _V)-1-1-onto->_V -> Fun `'(2nd |` ({C} X. _V)))
47	1, 46	syl 12	. . . . 5 |- (C e. A -> Fun `'(2nd |` ({C} X. _V)))
48	44, 24, 47	syl2an 503	. . . 4 |- ((F Fn (A X. B) /\ C e. A) -> Fun (F o. `'(2nd |` ({C} X. _V))))
49		fndm 4512	. . . . . . . 8 |- (F Fn (A X. B) -> dom F = (A X. B))
50	49	adantr 425	. . . . . . 7 |- ((F Fn (A X. B) /\ C e. A) -> dom F = (A X. B))
51	50	imaeq2d 4264	. . . . . 6 |- ((F Fn (A X. B) /\ C e. A) -> (`'`'(2nd |` ({C} X. _V))"dom F) = (`'`'(2nd |` ({C} X. _V))"(A X. B)))
52		snssi 3129	. . . . . . . . . . . . . 14 |- (C e. A -> {C} C_ A)
53		df-ss 2605	. . . . . . . . . . . . . 14 |- ({C} C_ A <-> ({C} i^i A) = {C})
54	52, 53	sylib 215	. . . . . . . . . . . . 13 |- (C e. A -> ({C} i^i A) = {C})
55		xpeq1 4016	. . . . . . . . . . . . 13 |- (({C} i^i A) = {C} -> (({C} i^i A) X. B) = ({C} X. B))
56	54, 55	syl 12	. . . . . . . . . . . 12 |- (C e. A -> (({C} i^i A) X. B) = ({C} X. B))
57		inxp 4109	. . . . . . . . . . . . 13 |- (({C} X. _V) i^i (A X. B)) = (({C} i^i A) X. (_V i^i B))
58		incom 2787	. . . . . . . . . . . . . . 15 |- (_V i^i B) = (B i^i _V)
59		inv1 2898	. . . . . . . . . . . . . . 15 |- (B i^i _V) = B
60	58, 59	eqtri 1908	. . . . . . . . . . . . . 14 |- (_V i^i B) = B
61	60	xpeq2i 4022	. . . . . . . . . . . . 13 |- (({C} i^i A) X. (_V i^i B)) = (({C} i^i A) X. B)
62	57, 61	eqtri 1908	. . . . . . . . . . . 12 |- (({C} X. _V) i^i (A X. B)) = (({C} i^i A) X. B)
63	56, 62	syl5eq 1940	. . . . . . . . . . 11 |- (C e. A -> (({C} X. _V) i^i (A X. B)) = ({C} X. B))
64		reseq2 4219	. . . . . . . . . . 11 |- ((({C} X. _V) i^i (A X. B)) = ({C} X. B) -> (2nd |` (({C} X. _V) i^i (A X. B))) = (2nd |` ({C} X. B)))
65	63, 64	syl 12	. . . . . . . . . 10 |- (C e. A -> (2nd |` (({C} X. _V) i^i (A X. B))) = (2nd |` ({C} X. B)))
66	65	rneqd 4188	. . . . . . . . 9 |- (C e. A -> ran (2nd |` (({C} X. _V) i^i (A X. B))) = ran (2nd |` ({C} X. B)))
67		2ndconst 5071	. . . . . . . . . . 11 |- (C e. A -> (2nd |` ({C} X. B)):({C} X. B)-1-1-onto->B)
68		f1ofo 4643	. . . . . . . . . . 11 |- ((2nd |` ({C} X. B)):({C} X. B)-1-1-onto->B -> (2nd |` ({C} X. B)):({C} X. B)-onto->B)
69	67, 68	syl 12	. . . . . . . . . 10 |- (C e. A -> (2nd |` ({C} X. B)):({C} X. B)-onto->B)
70		forn 4620	. . . . . . . . . 10 |- ((2nd |` ({C} X. B)):({C} X. B)-onto->B -> ran (2nd |` ({C} X. B)) = B)
71	69, 70	syl 12	. . . . . . . . 9 |- (C e. A -> ran (2nd |` ({C} X. B)) = B)
72	66, 71	eqtrd 1925	. . . . . . . 8 |- (C e. A -> ran (2nd |` (({C} X. _V) i^i (A X. B))) = B)
73		imacnvcnv 4388	. . . . . . . . 9 |- (`'`'(2nd |` ({C} X. _V))"(A X. B)) = ((2nd |` ({C} X. _V))"(A X. B))
74		df-ima 4007	. . . . . . . . 9 |- ((2nd |` ({C} X. _V))"(A X. B)) = ran ((2nd |` ({C} X. _V)) |` (A X. B))
75		resres 4228	. . . . . . . . . 10 |- ((2nd |` ({C} X. _V)) |` (A X. B)) = (2nd |` (({C} X. _V) i^i (A X. B)))
76	75	rneqi 4187	. . . . . . . . 9 |- ran ((2nd |` ({C} X. _V)) |` (A X. B)) = ran (2nd |` (({C} X. _V) i^i (A X. B)))
77	73, 74, 76	3eqtri 1912	. . . . . . . 8 |- (`'`'(2nd |` ({C} X. _V))"(A X. B)) = ran (2nd |` (({C} X. _V) i^i (A X. B)))
78	72, 77	syl5eq 1940	. . . . . . 7 |- (C e. A -> (`'`'(2nd |` ({C} X. _V))"(A X. B)) = B)
79	78	adantl 424	. . . . . 6 |- ((F Fn (A X. B) /\ C e. A) -> (`'`'(2nd |` ({C} X. _V))"(A X. B)) = B)
80	51, 79	eqtrd 1925	. . . . 5 |- ((F Fn (A X. B) /\ C e. A) -> (`'`'(2nd |` ({C} X. _V))"dom F) = B)
81		dmco 4406	. . . . 5 |- dom ( F o. `'(2nd |` ({C} X. _V))) = (`'`'(2nd |` ({C} X. _V))"dom F)
82	80, 81	syl5eq 1940	. . . 4 |- ((F Fn (A X. B) /\ C e. A) -> dom ( F o. `'(2nd |` ({C} X. _V))) = B)
83	43, 48, 82	sylanbrc 527	. . 3 |- ((F Fn (A X. B) /\ C e. A) -> G Fn B)
84		oprex 4907	. . . 4 |- (CFx) e. _V
85	84, 34	fnopab2 4549	. . 3 |- {<.x, y>. | (x e. B /\ y = (CFx))} Fn B
86	40, 83, 85	sylancl 525	. 2 |- ((F Fn (A X. B) /\ C e. A) -> (G = {<.x, y>. | (x e. B /\ y = (CFx))} <-> A.z e. B (G` z) = ({<.x, y>. | (x e. B /\ y = (CFx))}` z)))
87	39, 86	mpbird 213	1 |- ((F Fn (A X. B) /\ C e. A) -> G = {<.x, y>. | (x e. B /\ y = (CFx))})

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   = wceq 1298   e. wcel 1300  A.wral 2105  _Vcvv 2292   i^i cin 2592   C_ wss 2593  {csn 3044  <.cop 3046  {copab 3395   X. cxp 3984  `'ccnv 3985  dom cdm 3986  ran crn 3987   |` cres 3988  "cima 3989   o. ccom 3990  Fun wfun 3992   Fn wfn 3993  -onto->wfo 3996  -1-1-onto->wf1o 3997  ` cfv 3998  (class class class)co 4884  2ndc2nd 5019

This theorem is referenced by:  curry1f 5076  curry1val 5077  valcurfn2 14553

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-uni 3178  df-br 3339  df-opab 3396  df-id 3586  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-f1 4011  df-fo 4012  df-f1o 4013  df-fv 4014  df-opr 4886  df-1st 5020  df-2nd 5021

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