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Theorem curry2 6380
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 5482 . . . . 5  |-  ( F  Fn  ( A  X.  B )  ->  Fun  F )
2 1stconst 6374 . . . . . 6  |-  ( C  e.  B  ->  ( 1st  |`  ( _V  X.  { C } ) ) : ( _V  X.  { C } ) -1-1-onto-> _V )
3 dff1o3 5620 . . . . . . 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 451 . . . . . 6  |-  ( ( 1st  |`  ( _V  X.  { C } ) ) : ( _V 
X.  { C }
)
-1-1-onto-> _V  ->  Fun  `' ( 1st  |`  ( _V  X.  { C } ) ) )
52, 4syl 16 . . . . 5  |-  ( C  e.  B  ->  Fun  `' ( 1st  |`  ( _V  X.  { C }
) ) )
6 funco 5431 . . . . 5  |-  ( ( Fun  F  /\  Fun  `' ( 1st  |`  ( _V  X.  { C }
) ) )  ->  Fun  ( F  o.  `' ( 1st  |`  ( _V  X.  { C } ) ) ) )
71, 5, 6syl2an 464 . . . 4  |-  ( ( F  Fn  ( A  X.  B )  /\  C  e.  B )  ->  Fun  ( F  o.  `' ( 1st  |`  ( _V  X.  { C }
) ) ) )
8 dmco 5318 . . . . 5  |-  dom  ( F  o.  `' ( 1st  |`  ( _V  X.  { C } ) ) )  =  ( `' `' ( 1st  |`  ( _V  X.  { C }
) ) " dom  F )
9 fndm 5484 . . . . . . . 8  |-  ( F  Fn  ( A  X.  B )  ->  dom  F  =  ( A  X.  B ) )
109adantr 452 . . . . . . 7  |-  ( ( F  Fn  ( A  X.  B )  /\  C  e.  B )  ->  dom  F  =  ( A  X.  B ) )
1110imaeq2d 5143 . . . . . 6  |-  ( ( F  Fn  ( A  X.  B )  /\  C  e.  B )  ->  ( `' `' ( 1st  |`  ( _V  X.  { C } ) ) " dom  F
)  =  ( `' `' ( 1st  |`  ( _V  X.  { C }
) ) " ( A  X.  B ) ) )
12 imacnvcnv 5274 . . . . . . . . 9  |-  ( `' `' ( 1st  |`  ( _V  X.  { C }
) ) " ( A  X.  B ) )  =  ( ( 1st  |`  ( _V  X.  { C } ) ) "
( A  X.  B
) )
13 df-ima 4831 . . . . . . . . 9  |-  ( ( 1st  |`  ( _V  X.  { C } ) ) " ( A  X.  B ) )  =  ran  ( ( 1st  |`  ( _V  X.  { C } ) )  |`  ( A  X.  B ) )
14 resres 5099 . . . . . . . . . 10  |-  ( ( 1st  |`  ( _V  X.  { C } ) )  |`  ( A  X.  B ) )  =  ( 1st  |`  (
( _V  X.  { C } )  i^i  ( A  X.  B ) ) )
1514rneqi 5036 . . . . . . . . 9  |-  ran  (
( 1st  |`  ( _V 
X.  { C }
) )  |`  ( A  X.  B ) )  =  ran  ( 1st  |`  ( ( _V  X.  { C } )  i^i  ( A  X.  B
) ) )
1612, 13, 153eqtri 2411 . . . . . . . 8  |-  ( `' `' ( 1st  |`  ( _V  X.  { C }
) ) " ( A  X.  B ) )  =  ran  ( 1st  |`  ( ( _V  X.  { C } )  i^i  ( A  X.  B
) ) )
17 inxp 4947 . . . . . . . . . . . . 13  |-  ( ( _V  X.  { C } )  i^i  ( A  X.  B ) )  =  ( ( _V 
i^i  A )  X.  ( { C }  i^i  B ) )
18 incom 3476 . . . . . . . . . . . . . . 15  |-  ( _V 
i^i  A )  =  ( A  i^i  _V )
19 inv1 3597 . . . . . . . . . . . . . . 15  |-  ( A  i^i  _V )  =  A
2018, 19eqtri 2407 . . . . . . . . . . . . . 14  |-  ( _V 
i^i  A )  =  A
2120xpeq1i 4838 . . . . . . . . . . . . 13  |-  ( ( _V  i^i  A )  X.  ( { C }  i^i  B ) )  =  ( A  X.  ( { C }  i^i  B ) )
2217, 21eqtri 2407 . . . . . . . . . . . 12  |-  ( ( _V  X.  { C } )  i^i  ( A  X.  B ) )  =  ( A  X.  ( { C }  i^i  B ) )
23 snssi 3885 . . . . . . . . . . . . . 14  |-  ( C  e.  B  ->  { C }  C_  B )
24 df-ss 3277 . . . . . . . . . . . . . 14  |-  ( { C }  C_  B  <->  ( { C }  i^i  B )  =  { C } )
2523, 24sylib 189 . . . . . . . . . . . . 13  |-  ( C  e.  B  ->  ( { C }  i^i  B
)  =  { C } )
2625xpeq2d 4842 . . . . . . . . . . . 12  |-  ( C  e.  B  ->  ( A  X.  ( { C }  i^i  B ) )  =  ( A  X.  { C } ) )
2722, 26syl5eq 2431 . . . . . . . . . . 11  |-  ( C  e.  B  ->  (
( _V  X.  { C } )  i^i  ( A  X.  B ) )  =  ( A  X.  { C } ) )
2827reseq2d 5086 . . . . . . . . . 10  |-  ( C  e.  B  ->  ( 1st  |`  ( ( _V 
X.  { C }
)  i^i  ( A  X.  B ) ) )  =  ( 1st  |`  ( A  X.  { C }
) ) )
2928rneqd 5037 . . . . . . . . 9  |-  ( C  e.  B  ->  ran  ( 1st  |`  ( ( _V  X.  { C }
)  i^i  ( A  X.  B ) ) )  =  ran  ( 1st  |`  ( A  X.  { C } ) ) )
30 1stconst 6374 . . . . . . . . . 10  |-  ( C  e.  B  ->  ( 1st  |`  ( A  X.  { C } ) ) : ( A  X.  { C } ) -1-1-onto-> A )
31 f1ofo 5621 . . . . . . . . . 10  |-  ( ( 1st  |`  ( A  X.  { C } ) ) : ( A  X.  { C }
)
-1-1-onto-> A  ->  ( 1st  |`  ( A  X.  { C }
) ) : ( A  X.  { C } ) -onto-> A )
32 forn 5596 . . . . . . . . . 10  |-  ( ( 1st  |`  ( A  X.  { C } ) ) : ( A  X.  { C }
) -onto-> A  ->  ran  ( 1st  |`  ( A  X.  { C } ) )  =  A )
3330, 31, 323syl 19 . . . . . . . . 9  |-  ( C  e.  B  ->  ran  ( 1st  |`  ( A  X.  { C } ) )  =  A )
3429, 33eqtrd 2419 . . . . . . . 8  |-  ( C  e.  B  ->  ran  ( 1st  |`  ( ( _V  X.  { C }
)  i^i  ( A  X.  B ) ) )  =  A )
3516, 34syl5eq 2431 . . . . . . 7  |-  ( C  e.  B  ->  ( `' `' ( 1st  |`  ( _V  X.  { C }
) ) " ( A  X.  B ) )  =  A )
3635adantl 453 . . . . . 6  |-  ( ( F  Fn  ( A  X.  B )  /\  C  e.  B )  ->  ( `' `' ( 1st  |`  ( _V  X.  { C } ) ) " ( A  X.  B ) )  =  A )
3711, 36eqtrd 2419 . . . . 5  |-  ( ( F  Fn  ( A  X.  B )  /\  C  e.  B )  ->  ( `' `' ( 1st  |`  ( _V  X.  { C } ) ) " dom  F
)  =  A )
388, 37syl5eq 2431 . . . 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 5479 . . . . 5  |-  ( G  Fn  A  <->  ( F  o.  `' ( 1st  |`  ( _V  X.  { C }
) ) )  Fn  A )
41 df-fn 5397 . . . . 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 241 . . . 4  |-  ( G  Fn  A  <->  ( Fun  ( F  o.  `' ( 1st  |`  ( _V  X.  { C } ) ) )  /\  dom  ( F  o.  `' ( 1st  |`  ( _V  X.  { C } ) ) )  =  A ) )
437, 38, 42sylanbrc 646 . . 3  |-  ( ( F  Fn  ( A  X.  B )  /\  C  e.  B )  ->  G  Fn  A )
44 dffn5 5711 . . 3  |-  ( G  Fn  A  <->  G  =  ( x  e.  A  |->  ( G `  x
) ) )
4543, 44sylib 189 . 2  |-  ( ( F  Fn  ( A  X.  B )  /\  C  e.  B )  ->  G  =  ( x  e.  A  |->  ( G `
 x ) ) )
4639fveq1i 5669 . . . . 5  |-  ( G `
 x )  =  ( ( F  o.  `' ( 1st  |`  ( _V  X.  { C }
) ) ) `  x )
47 dff1o4 5622 . . . . . . . . 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 189 . . . . . . . 8  |-  ( C  e.  B  ->  (
( 1st  |`  ( _V 
X.  { C }
) )  Fn  ( _V  X.  { C }
)  /\  `' ( 1st  |`  ( _V  X.  { C } ) )  Fn  _V ) )
4948simprd 450 . . . . . . 7  |-  ( C  e.  B  ->  `' ( 1st  |`  ( _V  X.  { C } ) )  Fn  _V )
50 vex 2902 . . . . . . 7  |-  x  e. 
_V
51 fvco2 5737 . . . . . . 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 644 . . . . . 6  |-  ( C  e.  B  ->  (
( F  o.  `' ( 1st  |`  ( _V  X.  { C } ) ) ) `  x
)  =  ( F `
 ( `' ( 1st  |`  ( _V  X.  { C } ) ) `  x ) ) )
5352ad2antlr 708 . . . . 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 2431 . . . 4  |-  ( ( ( F  Fn  ( A  X.  B )  /\  C  e.  B )  /\  x  e.  A
)  ->  ( G `  x )  =  ( F `  ( `' ( 1st  |`  ( _V  X.  { C }
) ) `  x
) ) )
552adantr 452 . . . . . . . . 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 3782 . . . . . . . . . . 11  |-  ( C  e.  B  ->  C  e.  { C } )
5857adantr 452 . . . . . . . . . 10  |-  ( ( C  e.  B  /\  x  e.  A )  ->  C  e.  { C } )
59 opelxp 4848 . . . . . . . . . 10  |-  ( <.
x ,  C >.  e.  ( _V  X.  { C } )  <->  ( x  e.  _V  /\  C  e. 
{ C } ) )
6056, 58, 59sylanbrc 646 . . . . . . . . 9  |-  ( ( C  e.  B  /\  x  e.  A )  -> 
<. x ,  C >.  e.  ( _V  X.  { C } ) )
6155, 60jca 519 . . . . . . . 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 646 . . . . . . . . . . 11  |-  ( C  e.  B  ->  <. x ,  C >.  e.  ( _V  X.  { C }
) )
64 fvres 5685 . . . . . . . . . . 11  |-  ( <.
x ,  C >.  e.  ( _V  X.  { C } )  ->  (
( 1st  |`  ( _V 
X.  { C }
) ) `  <. x ,  C >. )  =  ( 1st `  <. x ,  C >. )
)
6563, 64syl 16 . . . . . . . . . 10  |-  ( C  e.  B  ->  (
( 1st  |`  ( _V 
X.  { C }
) ) `  <. x ,  C >. )  =  ( 1st `  <. x ,  C >. )
)
6665adantr 452 . . . . . . . . 9  |-  ( ( C  e.  B  /\  x  e.  A )  ->  ( ( 1st  |`  ( _V  X.  { C }
) ) `  <. x ,  C >. )  =  ( 1st `  <. x ,  C >. )
)
67 op1stg 6298 . . . . . . . . . 10  |-  ( ( x  e.  A  /\  C  e.  B )  ->  ( 1st `  <. x ,  C >. )  =  x )
6867ancoms 440 . . . . . . . . 9  |-  ( ( C  e.  B  /\  x  e.  A )  ->  ( 1st `  <. x ,  C >. )  =  x )
6966, 68eqtrd 2419 . . . . . . . 8  |-  ( ( C  e.  B  /\  x  e.  A )  ->  ( ( 1st  |`  ( _V  X.  { C }
) ) `  <. x ,  C >. )  =  x )
70 f1ocnvfv 5955 . . . . . . . 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 58 . . . . . . 7  |-  ( ( C  e.  B  /\  x  e.  A )  ->  ( `' ( 1st  |`  ( _V  X.  { C } ) ) `  x )  =  <. x ,  C >. )
7271fveq2d 5672 . . . . . 6  |-  ( ( C  e.  B  /\  x  e.  A )  ->  ( F `  ( `' ( 1st  |`  ( _V  X.  { C }
) ) `  x
) )  =  ( F `  <. x ,  C >. ) )
7372adantll 695 . . . . 5  |-  ( ( ( F  Fn  ( A  X.  B )  /\  C  e.  B )  /\  x  e.  A
)  ->  ( F `  ( `' ( 1st  |`  ( _V  X.  { C } ) ) `  x ) )  =  ( F `  <. x ,  C >. )
)
74 df-ov 6023 . . . . 5  |-  ( x F C )  =  ( F `  <. x ,  C >. )
7573, 74syl6eqr 2437 . . . 4  |-  ( ( ( F  Fn  ( A  X.  B )  /\  C  e.  B )  /\  x  e.  A
)  ->  ( F `  ( `' ( 1st  |`  ( _V  X.  { C } ) ) `  x ) )  =  ( x F C ) )
7654, 75eqtrd 2419 . . 3  |-  ( ( ( F  Fn  ( A  X.  B )  /\  C  e.  B )  /\  x  e.  A
)  ->  ( G `  x )  =  ( x F C ) )
7776mpteq2dva 4236 . 2  |-  ( ( F  Fn  ( A  X.  B )  /\  C  e.  B )  ->  ( x  e.  A  |->  ( G `  x
) )  =  ( x  e.  A  |->  ( x F C ) ) )
7845, 77eqtrd 2419 1  |-  ( ( F  Fn  ( A  X.  B )  /\  C  e.  B )  ->  G  =  ( x  e.  A  |->  ( x F C ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1717   _Vcvv 2899    i^i cin 3262    C_ wss 3263   {csn 3757   <.cop 3760    e. cmpt 4207    X. cxp 4816   `'ccnv 4817   dom cdm 4818   ran crn 4819    |` cres 4820   "cima 4821    o. ccom 4822   Fun wfun 5388    Fn wfn 5389   -onto->wfo 5392   -1-1-onto->wf1o 5393   ` cfv 5394  (class class class)co 6020   1stc1st 6286
This theorem is referenced by:  curry2f  6381  curry2val  6382
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-ral 2654  df-rex 2655  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-sn 3763  df-pr 3764  df-op 3766  df-uni 3958  df-iun 4037  df-br 4154  df-opab 4208  df-mpt 4209  df-id 4439  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-ov 6023  df-1st 6288  df-2nd 6289
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