Theorem curry2 5078

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.)

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, y>. | (x e. A /\ y = (xFC))})

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

Proof of Theorem curry2

Step	Hyp	Ref	Expression
1		1stconst 5070	. . . . . . . . . 10 |- (C e. B -> (1st |` (_V X. {C})):(_V X. {C})-1-1-onto->_V)
2	1	adantr 425	. . . . . . . . 9 |- ((C e. B /\ z e. A) -> (1st |` (_V X. {C})):(_V X. {C})-1-1-onto->_V)
3		snidg 3067	. . . . . . . . . . . 12 |- (C e. B -> C e. {C})
4		visset 2295	. . . . . . . . . . . 12 |- z e. _V
5	3, 4	jctil 316	. . . . . . . . . . 11 |- (C e. B -> (z e. _V /\ C e. {C}))
6		opelxpg 4039	. . . . . . . . . . 11 |- (C e. B -> (<.z, C>. e. (_V X. {C}) <-> (z e. _V /\ C e. {C})))
7	5, 6	mpbird 213	. . . . . . . . . 10 |- (C e. B -> <.z, C>. e. (_V X. {C}))
8	7	adantr 425	. . . . . . . . 9 |- ((C e. B /\ z e. A) -> <.z, C>. e. (_V X. {C}))
9	2, 8	jca 310	. . . . . . . 8 |- ((C e. B /\ z e. A) -> ((1st |` (_V X. {C})):(_V X. {C})-1-1-onto->_V /\ <.z, C>. e. (_V X. {C})))
10		fvres 4691	. . . . . . . . . . 11 |- (<.z, C>. e. (_V X. {C}) -> ((1st |` (_V X. {C}))` <.z, C>.) = (1st` <.z, C>.))
11	7, 10	syl 12	. . . . . . . . . 10 |- (C e. B -> ((1st |` (_V X. {C}))` <.z, C>.) = (1st` <.z, C>.))
12	4	op1st 5026	. . . . . . . . . 10 |- (1st` <.z, C>.) = z
13	11, 12	syl6eq 1944	. . . . . . . . 9 |- (C e. B -> ((1st |` (_V X. {C}))` <.z, C>.) = z)
14	13	adantr 425	. . . . . . . 8 |- ((C e. B /\ z e. A) -> ((1st |` (_V X. {C}))` <.z, C>.) = z)
15		f1ocnvfv 4856	. . . . . . . 8 |- (((1st |` (_V X. {C})):(_V X. {C})-1-1-onto->_V /\ <.z, C>. e. (_V X. {C})) -> (((1st |` (_V X. {C}))` <.z, C>.) = z -> (`'(1st |` (_V X. {C}))` z) = <.z, C>.))
16	9, 14, 15	sylc 83	. . . . . . 7 |- ((C e. B /\ z e. A) -> (`'(1st |` (_V X. {C}))` z) = <.z, C>.)
17	16	fveq2d 4685	. . . . . 6 |- ((C e. B /\ z e. A) -> (F` (`'(1st |` (_V X. {C}))` z)) = (F` <.z, C>.))
18	17	adantll 428	. . . . 5 |- (((F Fn (A X. B) /\ C e. B) /\ z e. A) -> (F` (`'(1st |` (_V X. {C}))` z)) = (F` <.z, C>.))
19		df-opr 4886	. . . . 5 |- (zFC) = (F` <.z, C>.)
20	18, 19	syl6eqr 1946	. . . 4 |- (((F Fn (A X. B) /\ C e. B) /\ z e. A) -> (F` (`'(1st |` (_V X. {C}))` z)) = (zFC))
21		fvco2 4737	. . . . . . . 8 |- ((Fun F /\ `'(1st |` (_V X. {C})) Fn _V /\ z e. _V) -> ((F o. `'(1st |` (_V X. {C})))` z) = (F` (`'(1st |` (_V X. {C}))` z)))
22	4, 21	mp3an3 1180	. . . . . . 7 |- ((Fun F /\ `'(1st |` (_V X. {C})) Fn _V) -> ((F o. `'(1st |` (_V X. {C})))` z) = (F` (`'(1st |` (_V X. {C}))` z)))
23		fnfun 4510	. . . . . . 7 |- (F Fn (A X. B) -> Fun F)
24		dff1o4 4644	. . . . . . . . 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))
25	1, 24	sylib 215	. . . . . . . 8 |- (C e. B -> ((1st |` (_V X. {C})) Fn (_V X. {C}) /\ `'(1st |` (_V X. {C})) Fn _V))
26	25	simprd 352	. . . . . . 7 |- (C e. B -> `'(1st |` (_V X. {C})) Fn _V)
27	22, 23, 26	syl2an 503	. . . . . 6 |- ((F Fn (A X. B) /\ C e. B) -> ((F o. `'(1st |` (_V X. {C})))` z) = (F` (`'(1st |` (_V X. {C}))` z)))
28	27	adantr 425	. . . . 5 |- (((F Fn (A X. B) /\ C e. B) /\ z e. A) -> ((F o. `'(1st |` (_V X. {C})))` z) = (F` (`'(1st |` (_V X. {C}))` z)))
29		curry2.1	. . . . . 6 |- G = (F o. `'(1st |` (_V X. {C})))
30	29	fveq1i 4682	. . . . 5 |- (G` z) = ((F o. `'(1st |` (_V X. {C})))` z)
31	28, 30	syl5eq 1940	. . . 4 |- (((F Fn (A X. B) /\ C e. B) /\ z e. A) -> (G` z) = (F` (`'(1st |` (_V X. {C}))` z)))
32		opreq1 4889	. . . . . 6 |- (x = z -> (xFC) = (zFC))
33		eqid 1884	. . . . . 6 |- {<.x, y>. | (x e. A /\ y = (xFC))} = {<.x, y>. | (x e. A /\ y = (xFC))}
34		oprex 4907	. . . . . 6 |- (zFC) e. _V
35	32, 33, 34	fvopab4 4743	. . . . 5 |- (z e. A -> ({<.x, y>. | (x e. A /\ y = (xFC))}` z) = (zFC))
36	35	adantl 424	. . . 4 |- (((F Fn (A X. B) /\ C e. B) /\ z e. A) -> ({<.x, y>. | (x e. A /\ y = (xFC))}` z) = (zFC))
37	20, 31, 36	3eqtr4d 1937	. . 3 |- (((F Fn (A X. B) /\ C e. B) /\ z e. A) -> (G` z) = ({<.x, y>. | (x e. A /\ y = (xFC))}` z))
38	37	r19.21aiva 2176	. 2 |- ((F Fn (A X. B) /\ C e. B) -> A.z e. A (G` z) = ({<.x, y>. | (x e. A /\ y = (xFC))}` z))
39		eqfnfv2 4767	. . 3 |- ((G Fn A /\ {<.x, y>. | (x e. A /\ y = (xFC))} Fn A) -> (G = {<.x, y>. | (x e. A /\ y = (xFC))} <-> A.z e. A (G` z) = ({<.x, y>. | (x e. A /\ y = (xFC))}` z)))
40	29	fneq1i 4507	. . . . 5 |- (G Fn A <-> (F o. `'(1st |` (_V X. {C}))) Fn A)
41		df-fn 4009	. . . . 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 190	. . . 4 |- (G Fn A <-> (Fun (F o. `'(1st |` (_V X. {C}))) /\ dom ( F o. `'(1st |` (_V X. {C}))) = A))
43		funco 4457	. . . . 5 |- ((Fun F /\ Fun `'(1st |` (_V X. {C}))) -> Fun (F o. `'(1st |` (_V X. {C}))))
44		dff1o3 4641	. . . . . . 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}))))
45	44	simprbi 353	. . . . . 6 |- ((1st |` (_V X. {C})):(_V X. {C})-1-1-onto->_V -> Fun `'(1st |` (_V X. {C})))
46	1, 45	syl 12	. . . . 5 |- (C e. B -> Fun `'(1st |` (_V X. {C})))
47	43, 23, 46	syl2an 503	. . . 4 |- ((F Fn (A X. B) /\ C e. B) -> Fun (F o. `'(1st |` (_V X. {C}))))
48		fndm 4512	. . . . . . . 8 |- (F Fn (A X. B) -> dom F = (A X. B))
49	48	adantr 425	. . . . . . 7 |- ((F Fn (A X. B) /\ C e. B) -> dom F = (A X. B))
50	49	imaeq2d 4264	. . . . . 6 |- ((F Fn (A X. B) /\ C e. B) -> (`'`'(1st |` (_V X. {C}))"dom F) = (`'`'(1st |` (_V X. {C}))"(A X. B)))
51		snssi 3129	. . . . . . . . . . . . . 14 |- (C e. B -> {C} C_ B)
52		df-ss 2605	. . . . . . . . . . . . . 14 |- ({C} C_ B <-> ({C} i^i B) = {C})
53	51, 52	sylib 215	. . . . . . . . . . . . 13 |- (C e. B -> ({C} i^i B) = {C})
54		xpeq2 4017	. . . . . . . . . . . . 13 |- (({C} i^i B) = {C} -> (A X. ({C} i^i B)) = (A X. {C}))
55	53, 54	syl 12	. . . . . . . . . . . 12 |- (C e. B -> (A X. ({C} i^i B)) = (A X. {C}))
56		inxp 4109	. . . . . . . . . . . . 13 |- ((_V X. {C}) i^i (A X. B)) = ((_V i^i A) X. ({C} i^i B))
57		incom 2787	. . . . . . . . . . . . . . 15 |- (_V i^i A) = (A i^i _V)
58		inv1 2898	. . . . . . . . . . . . . . 15 |- (A i^i _V) = A
59	57, 58	eqtri 1908	. . . . . . . . . . . . . 14 |- (_V i^i A) = A
60	59	xpeq1i 4021	. . . . . . . . . . . . 13 |- ((_V i^i A) X. ({C} i^i B)) = (A X. ({C} i^i B))
61	56, 60	eqtri 1908	. . . . . . . . . . . 12 |- ((_V X. {C}) i^i (A X. B)) = (A X. ({C} i^i B))
62	55, 61	syl5eq 1940	. . . . . . . . . . 11 |- (C e. B -> ((_V X. {C}) i^i (A X. B)) = (A X. {C}))
63		reseq2 4219	. . . . . . . . . . 11 |- (((_V X. {C}) i^i (A X. B)) = (A X. {C}) -> (1st |` ((_V X. {C}) i^i (A X. B))) = (1st |` (A X. {C})))
64	62, 63	syl 12	. . . . . . . . . 10 |- (C e. B -> (1st |` ((_V X. {C}) i^i (A X. B))) = (1st |` (A X. {C})))
65	64	rneqd 4188	. . . . . . . . 9 |- (C e. B -> ran (1st |` ((_V X. {C}) i^i (A X. B))) = ran (1st |` (A X. {C})))
66		1stconst 5070	. . . . . . . . . . 11 |- (C e. B -> (1st |` (A X. {C})):(A X. {C})-1-1-onto->A)
67		f1ofo 4643	. . . . . . . . . . 11 |- ((1st |` (A X. {C})):(A X. {C})-1-1-onto->A -> (1st |` (A X. {C})):(A X. {C})-onto->A)
68	66, 67	syl 12	. . . . . . . . . 10 |- (C e. B -> (1st |` (A X. {C})):(A X. {C})-onto->A)
69		forn 4620	. . . . . . . . . 10 |- ((1st |` (A X. {C})):(A X. {C})-onto->A -> ran (1st |` (A X. {C})) = A)
70	68, 69	syl 12	. . . . . . . . 9 |- (C e. B -> ran (1st |` (A X. {C})) = A)
71	65, 70	eqtrd 1925	. . . . . . . 8 |- (C e. B -> ran (1st |` ((_V X. {C}) i^i (A X. B))) = A)
72		imacnvcnv 4388	. . . . . . . . 9 |- (`'`'(1st |` (_V X. {C}))"(A X. B)) = ((1st |` (_V X. {C}))"(A X. B))
73		df-ima 4007	. . . . . . . . 9 |- ((1st |` (_V X. {C}))"(A X. B)) = ran ((1st |` (_V X. {C})) |` (A X. B))
74		resres 4228	. . . . . . . . . 10 |- ((1st |` (_V X. {C})) |` (A X. B)) = (1st |` ((_V X. {C}) i^i (A X. B)))
75	74	rneqi 4187	. . . . . . . . 9 |- ran ((1st |` (_V X. {C})) |` (A X. B)) = ran (1st |` ((_V X. {C}) i^i (A X. B)))
76	72, 73, 75	3eqtri 1912	. . . . . . . 8 |- (`'`'(1st |` (_V X. {C}))"(A X. B)) = ran (1st |` ((_V X. {C}) i^i (A X. B)))
77	71, 76	syl5eq 1940	. . . . . . 7 |- (C e. B -> (`'`'(1st |` (_V X. {C}))"(A X. B)) = A)
78	77	adantl 424	. . . . . 6 |- ((F Fn (A X. B) /\ C e. B) -> (`'`'(1st |` (_V X. {C}))"(A X. B)) = A)
79	50, 78	eqtrd 1925	. . . . 5 |- ((F Fn (A X. B) /\ C e. B) -> (`'`'(1st |` (_V X. {C}))"dom F) = A)
80		dmco 4406	. . . . 5 |- dom ( F o. `'(1st |` (_V X. {C}))) = (`'`'(1st |` (_V X. {C}))"dom F)
81	79, 80	syl5eq 1940	. . . 4 |- ((F Fn (A X. B) /\ C e. B) -> dom ( F o. `'(1st |` (_V X. {C}))) = A)
82	42, 47, 81	sylanbrc 527	. . 3 |- ((F Fn (A X. B) /\ C e. B) -> G Fn A)
83		oprex 4907	. . . 4 |- (xFC) e. _V
84	83, 33	fnopab2 4549	. . 3 |- {<.x, y>. | (x e. A /\ y = (xFC))} Fn A
85	39, 82, 84	sylancl 525	. 2 |- ((F Fn (A X. B) /\ C e. B) -> (G = {<.x, y>. | (x e. A /\ y = (xFC))} <-> A.z e. A (G` z) = ({<.x, y>. | (x e. A /\ y = (xFC))}` z)))
86	38, 85	mpbird 213	1 |- ((F Fn (A X. B) /\ C e. B) -> G = {<.x, y>. | (x e. A /\ y = (xFC))})

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  1stc1st 5018

This theorem is referenced by:  curry2f 5079  curry2val 5080

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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