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Theorem curry2 6900
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
StepHypRef Expression
1 fnfun 5689 . . . . 5  |-  ( F  Fn  ( A  X.  B )  ->  Fun  F )
2 1stconst 6893 . . . . . 6  |-  ( C  e.  B  ->  ( 1st  |`  ( _V  X.  { C } ) ) : ( _V  X.  { C } ) -1-1-onto-> _V )
3 dff1o3 5835 . . . . . . 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 }
) ) ) )
43simprbi 466 . . . . . 6  |-  ( ( 1st  |`  ( _V  X.  { C } ) ) : ( _V 
X.  { C }
)
-1-1-onto-> _V  ->  Fun  `' ( 1st  |`  ( _V  X.  { C } ) ) )
52, 4syl 17 . . . . 5  |-  ( C  e.  B  ->  Fun  `' ( 1st  |`  ( _V  X.  { C }
) ) )
6 funco 5637 . . . . 5  |-  ( ( Fun  F  /\  Fun  `' ( 1st  |`  ( _V  X.  { C }
) ) )  ->  Fun  ( F  o.  `' ( 1st  |`  ( _V  X.  { C } ) ) ) )
71, 5, 6syl2an 480 . . . 4  |-  ( ( F  Fn  ( A  X.  B )  /\  C  e.  B )  ->  Fun  ( F  o.  `' ( 1st  |`  ( _V  X.  { C }
) ) ) )
8 dmco 5360 . . . . 5  |-  dom  ( F  o.  `' ( 1st  |`  ( _V  X.  { C } ) ) )  =  ( `' `' ( 1st  |`  ( _V  X.  { C }
) ) " dom  F )
9 fndm 5691 . . . . . . . 8  |-  ( F  Fn  ( A  X.  B )  ->  dom  F  =  ( A  X.  B ) )
109adantr 467 . . . . . . 7  |-  ( ( F  Fn  ( A  X.  B )  /\  C  e.  B )  ->  dom  F  =  ( A  X.  B ) )
1110imaeq2d 5185 . . . . . 6  |-  ( ( F  Fn  ( A  X.  B )  /\  C  e.  B )  ->  ( `' `' ( 1st  |`  ( _V  X.  { C } ) ) " dom  F
)  =  ( `' `' ( 1st  |`  ( _V  X.  { C }
) ) " ( A  X.  B ) ) )
12 imacnvcnv 5317 . . . . . . . . 9  |-  ( `' `' ( 1st  |`  ( _V  X.  { C }
) ) " ( A  X.  B ) )  =  ( ( 1st  |`  ( _V  X.  { C } ) ) "
( A  X.  B
) )
13 df-ima 4864 . . . . . . . . 9  |-  ( ( 1st  |`  ( _V  X.  { C } ) ) " ( A  X.  B ) )  =  ran  ( ( 1st  |`  ( _V  X.  { C } ) )  |`  ( A  X.  B ) )
14 resres 5134 . . . . . . . . . 10  |-  ( ( 1st  |`  ( _V  X.  { C } ) )  |`  ( A  X.  B ) )  =  ( 1st  |`  (
( _V  X.  { C } )  i^i  ( A  X.  B ) ) )
1514rneqi 5078 . . . . . . . . 9  |-  ran  (
( 1st  |`  ( _V 
X.  { C }
) )  |`  ( A  X.  B ) )  =  ran  ( 1st  |`  ( ( _V  X.  { C } )  i^i  ( A  X.  B
) ) )
1612, 13, 153eqtri 2456 . . . . . . . 8  |-  ( `' `' ( 1st  |`  ( _V  X.  { C }
) ) " ( A  X.  B ) )  =  ran  ( 1st  |`  ( ( _V  X.  { C } )  i^i  ( A  X.  B
) ) )
17 inxp 4984 . . . . . . . . . . . . 13  |-  ( ( _V  X.  { C } )  i^i  ( A  X.  B ) )  =  ( ( _V 
i^i  A )  X.  ( { C }  i^i  B ) )
18 incom 3656 . . . . . . . . . . . . . . 15  |-  ( _V 
i^i  A )  =  ( A  i^i  _V )
19 inv1 3790 . . . . . . . . . . . . . . 15  |-  ( A  i^i  _V )  =  A
2018, 19eqtri 2452 . . . . . . . . . . . . . 14  |-  ( _V 
i^i  A )  =  A
2120xpeq1i 4871 . . . . . . . . . . . . 13  |-  ( ( _V  i^i  A )  X.  ( { C }  i^i  B ) )  =  ( A  X.  ( { C }  i^i  B ) )
2217, 21eqtri 2452 . . . . . . . . . . . 12  |-  ( ( _V  X.  { C } )  i^i  ( A  X.  B ) )  =  ( A  X.  ( { C }  i^i  B ) )
23 snssi 4142 . . . . . . . . . . . . . 14  |-  ( C  e.  B  ->  { C }  C_  B )
24 df-ss 3451 . . . . . . . . . . . . . 14  |-  ( { C }  C_  B  <->  ( { C }  i^i  B )  =  { C } )
2523, 24sylib 200 . . . . . . . . . . . . 13  |-  ( C  e.  B  ->  ( { C }  i^i  B
)  =  { C } )
2625xpeq2d 4875 . . . . . . . . . . . 12  |-  ( C  e.  B  ->  ( A  X.  ( { C }  i^i  B ) )  =  ( A  X.  { C } ) )
2722, 26syl5eq 2476 . . . . . . . . . . 11  |-  ( C  e.  B  ->  (
( _V  X.  { C } )  i^i  ( A  X.  B ) )  =  ( A  X.  { C } ) )
2827reseq2d 5122 . . . . . . . . . 10  |-  ( C  e.  B  ->  ( 1st  |`  ( ( _V 
X.  { C }
)  i^i  ( A  X.  B ) ) )  =  ( 1st  |`  ( A  X.  { C }
) ) )
2928rneqd 5079 . . . . . . . . 9  |-  ( C  e.  B  ->  ran  ( 1st  |`  ( ( _V  X.  { C }
)  i^i  ( A  X.  B ) ) )  =  ran  ( 1st  |`  ( A  X.  { C } ) ) )
30 1stconst 6893 . . . . . . . . . 10  |-  ( C  e.  B  ->  ( 1st  |`  ( A  X.  { C } ) ) : ( A  X.  { C } ) -1-1-onto-> A )
31 f1ofo 5836 . . . . . . . . . 10  |-  ( ( 1st  |`  ( A  X.  { C } ) ) : ( A  X.  { C }
)
-1-1-onto-> A  ->  ( 1st  |`  ( A  X.  { C }
) ) : ( A  X.  { C } ) -onto-> A )
32 forn 5811 . . . . . . . . . 10  |-  ( ( 1st  |`  ( A  X.  { C } ) ) : ( A  X.  { C }
) -onto-> A  ->  ran  ( 1st  |`  ( A  X.  { C } ) )  =  A )
3330, 31, 323syl 18 . . . . . . . . 9  |-  ( C  e.  B  ->  ran  ( 1st  |`  ( A  X.  { C } ) )  =  A )
3429, 33eqtrd 2464 . . . . . . . 8  |-  ( C  e.  B  ->  ran  ( 1st  |`  ( ( _V  X.  { C }
)  i^i  ( A  X.  B ) ) )  =  A )
3516, 34syl5eq 2476 . . . . . . 7  |-  ( C  e.  B  ->  ( `' `' ( 1st  |`  ( _V  X.  { C }
) ) " ( A  X.  B ) )  =  A )
3635adantl 468 . . . . . 6  |-  ( ( F  Fn  ( A  X.  B )  /\  C  e.  B )  ->  ( `' `' ( 1st  |`  ( _V  X.  { C } ) ) " ( A  X.  B ) )  =  A )
3711, 36eqtrd 2464 . . . . 5  |-  ( ( F  Fn  ( A  X.  B )  /\  C  e.  B )  ->  ( `' `' ( 1st  |`  ( _V  X.  { C } ) ) " dom  F
)  =  A )
388, 37syl5eq 2476 . . . 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 } ) ) )
4039fneq1i 5686 . . . . 5  |-  ( G  Fn  A  <->  ( F  o.  `' ( 1st  |`  ( _V  X.  { C }
) ) )  Fn  A )
41 df-fn 5602 . . . . 5  |-  ( ( F  o.  `' ( 1st  |`  ( _V  X.  { C } ) ) )  Fn  A  <->  ( Fun  ( F  o.  `' ( 1st  |`  ( _V  X.  { C }
) ) )  /\  dom  ( F  o.  `' ( 1st  |`  ( _V  X.  { C } ) ) )  =  A ) )
4240, 41bitri 253 . . . 4  |-  ( G  Fn  A  <->  ( Fun  ( F  o.  `' ( 1st  |`  ( _V  X.  { C } ) ) )  /\  dom  ( F  o.  `' ( 1st  |`  ( _V  X.  { C } ) ) )  =  A ) )
437, 38, 42sylanbrc 669 . . 3  |-  ( ( F  Fn  ( A  X.  B )  /\  C  e.  B )  ->  G  Fn  A )
44 dffn5 5924 . . 3  |-  ( G  Fn  A  <->  G  =  ( x  e.  A  |->  ( G `  x
) ) )
4543, 44sylib 200 . 2  |-  ( ( F  Fn  ( A  X.  B )  /\  C  e.  B )  ->  G  =  ( x  e.  A  |->  ( G `
 x ) ) )
4639fveq1i 5880 . . . . 5  |-  ( G `
 x )  =  ( ( F  o.  `' ( 1st  |`  ( _V  X.  { C }
) ) ) `  x )
47 dff1o4 5837 . . . . . . . . 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 ) )
482, 47sylib 200 . . . . . . . 8  |-  ( C  e.  B  ->  (
( 1st  |`  ( _V 
X.  { C }
) )  Fn  ( _V  X.  { C }
)  /\  `' ( 1st  |`  ( _V  X.  { C } ) )  Fn  _V ) )
4948simprd 465 . . . . . . 7  |-  ( C  e.  B  ->  `' ( 1st  |`  ( _V  X.  { C } ) )  Fn  _V )
50 vex 3085 . . . . . . 7  |-  x  e. 
_V
51 fvco2 5954 . . . . . . 7  |-  ( ( `' ( 1st  |`  ( _V  X.  { C }
) )  Fn  _V  /\  x  e.  _V )  ->  ( ( F  o.  `' ( 1st  |`  ( _V  X.  { C }
) ) ) `  x )  =  ( F `  ( `' ( 1st  |`  ( _V  X.  { C }
) ) `  x
) ) )
5249, 50, 51sylancl 667 . . . . . 6  |-  ( C  e.  B  ->  (
( F  o.  `' ( 1st  |`  ( _V  X.  { C } ) ) ) `  x
)  =  ( F `
 ( `' ( 1st  |`  ( _V  X.  { C } ) ) `  x ) ) )
5352ad2antlr 732 . . . . 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
) ) )
5446, 53syl5eq 2476 . . . 4  |-  ( ( ( F  Fn  ( A  X.  B )  /\  C  e.  B )  /\  x  e.  A
)  ->  ( G `  x )  =  ( F `  ( `' ( 1st  |`  ( _V  X.  { C }
) ) `  x
) ) )
552adantr 467 . . . . . . . . 9  |-  ( ( C  e.  B  /\  x  e.  A )  ->  ( 1st  |`  ( _V  X.  { C }
) ) : ( _V  X.  { C } ) -1-1-onto-> _V )
5650a1i 11 . . . . . . . . . 10  |-  ( ( C  e.  B  /\  x  e.  A )  ->  x  e.  _V )
57 snidg 4023 . . . . . . . . . . 11  |-  ( C  e.  B  ->  C  e.  { C } )
5857adantr 467 . . . . . . . . . 10  |-  ( ( C  e.  B  /\  x  e.  A )  ->  C  e.  { C } )
59 opelxp 4881 . . . . . . . . . 10  |-  ( <.
x ,  C >.  e.  ( _V  X.  { C } )  <->  ( x  e.  _V  /\  C  e. 
{ C } ) )
6056, 58, 59sylanbrc 669 . . . . . . . . 9  |-  ( ( C  e.  B  /\  x  e.  A )  -> 
<. x ,  C >.  e.  ( _V  X.  { C } ) )
6155, 60jca 535 . . . . . . . 8  |-  ( ( C  e.  B  /\  x  e.  A )  ->  ( ( 1st  |`  ( _V  X.  { C }
) ) : ( _V  X.  { C } ) -1-1-onto-> _V  /\  <. x ,  C >.  e.  ( _V  X.  { C }
) ) )
6250a1i 11 . . . . . . . . . . . 12  |-  ( C  e.  B  ->  x  e.  _V )
6362, 57, 59sylanbrc 669 . . . . . . . . . . 11  |-  ( C  e.  B  ->  <. x ,  C >.  e.  ( _V  X.  { C }
) )
64 fvres 5893 . . . . . . . . . . 11  |-  ( <.
x ,  C >.  e.  ( _V  X.  { C } )  ->  (
( 1st  |`  ( _V 
X.  { C }
) ) `  <. x ,  C >. )  =  ( 1st `  <. x ,  C >. )
)
6563, 64syl 17 . . . . . . . . . 10  |-  ( C  e.  B  ->  (
( 1st  |`  ( _V 
X.  { C }
) ) `  <. x ,  C >. )  =  ( 1st `  <. x ,  C >. )
)
6665adantr 467 . . . . . . . . 9  |-  ( ( C  e.  B  /\  x  e.  A )  ->  ( ( 1st  |`  ( _V  X.  { C }
) ) `  <. x ,  C >. )  =  ( 1st `  <. x ,  C >. )
)
67 op1stg 6817 . . . . . . . . . 10  |-  ( ( x  e.  A  /\  C  e.  B )  ->  ( 1st `  <. x ,  C >. )  =  x )
6867ancoms 455 . . . . . . . . 9  |-  ( ( C  e.  B  /\  x  e.  A )  ->  ( 1st `  <. x ,  C >. )  =  x )
6966, 68eqtrd 2464 . . . . . . . 8  |-  ( ( C  e.  B  /\  x  e.  A )  ->  ( ( 1st  |`  ( _V  X.  { C }
) ) `  <. x ,  C >. )  =  x )
70 f1ocnvfv 6190 . . . . . . . 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 >. ) )
7161, 69, 70sylc 63 . . . . . . 7  |-  ( ( C  e.  B  /\  x  e.  A )  ->  ( `' ( 1st  |`  ( _V  X.  { C } ) ) `  x )  =  <. x ,  C >. )
7271fveq2d 5883 . . . . . 6  |-  ( ( C  e.  B  /\  x  e.  A )  ->  ( F `  ( `' ( 1st  |`  ( _V  X.  { C }
) ) `  x
) )  =  ( F `  <. x ,  C >. ) )
7372adantll 719 . . . . 5  |-  ( ( ( F  Fn  ( A  X.  B )  /\  C  e.  B )  /\  x  e.  A
)  ->  ( F `  ( `' ( 1st  |`  ( _V  X.  { C } ) ) `  x ) )  =  ( F `  <. x ,  C >. )
)
74 df-ov 6306 . . . . 5  |-  ( x F C )  =  ( F `  <. x ,  C >. )
7573, 74syl6eqr 2482 . . . 4  |-  ( ( ( F  Fn  ( A  X.  B )  /\  C  e.  B )  /\  x  e.  A
)  ->  ( F `  ( `' ( 1st  |`  ( _V  X.  { C } ) ) `  x ) )  =  ( x F C ) )
7654, 75eqtrd 2464 . . 3  |-  ( ( ( F  Fn  ( A  X.  B )  /\  C  e.  B )  /\  x  e.  A
)  ->  ( G `  x )  =  ( x F C ) )
7776mpteq2dva 4508 . 2  |-  ( ( F  Fn  ( A  X.  B )  /\  C  e.  B )  ->  ( x  e.  A  |->  ( G `  x
) )  =  ( x  e.  A  |->  ( x F C ) ) )
7845, 77eqtrd 2464 1  |-  ( ( F  Fn  ( A  X.  B )  /\  C  e.  B )  ->  G  =  ( x  e.  A  |->  ( x F C ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 371    = wceq 1438    e. wcel 1869   _Vcvv 3082    i^i cin 3436    C_ wss 3437   {csn 3997   <.cop 4003    |-> cmpt 4480    X. cxp 4849   `'ccnv 4850   dom cdm 4851   ran crn 4852    |` cres 4853   "cima 4854    o. ccom 4855   Fun wfun 5593    Fn wfn 5594   -onto->wfo 5597   -1-1-onto->wf1o 5598   ` cfv 5599  (class class class)co 6303   1stc1st 6803
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1666  ax-4 1679  ax-5 1749  ax-6 1795  ax-7 1840  ax-8 1871  ax-9 1873  ax-10 1888  ax-11 1893  ax-12 1906  ax-13 2054  ax-ext 2401  ax-sep 4544  ax-nul 4553  ax-pow 4600  ax-pr 4658  ax-un 6595
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 985  df-tru 1441  df-ex 1661  df-nf 1665  df-sb 1788  df-eu 2270  df-mo 2271  df-clab 2409  df-cleq 2415  df-clel 2418  df-nfc 2573  df-ne 2621  df-ral 2781  df-rex 2782  df-rab 2785  df-v 3084  df-sbc 3301  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3763  df-if 3911  df-sn 3998  df-pr 4000  df-op 4004  df-uni 4218  df-iun 4299  df-br 4422  df-opab 4481  df-mpt 4482  df-id 4766  df-xp 4857  df-rel 4858  df-cnv 4859  df-co 4860  df-dm 4861  df-rn 4862  df-res 4863  df-ima 4864  df-iota 5563  df-fun 5601  df-fn 5602  df-f 5603  df-f1 5604  df-fo 5605  df-f1o 5606  df-fv 5607  df-ov 6306  df-1st 6805  df-2nd 6806
This theorem is referenced by:  curry2f  6901  curry2val  6902
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