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Theorem curry2 6833
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 5615 . . . . 5  |-  ( F  Fn  ( A  X.  B )  ->  Fun  F )
2 1stconst 6826 . . . . . 6  |-  ( C  e.  B  ->  ( 1st  |`  ( _V  X.  { C } ) ) : ( _V  X.  { C } ) -1-1-onto-> _V )
3 dff1o3 5761 . . . . . . 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 462 . . . . . 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 5563 . . . . 5  |-  ( ( Fun  F  /\  Fun  `' ( 1st  |`  ( _V  X.  { C }
) ) )  ->  Fun  ( F  o.  `' ( 1st  |`  ( _V  X.  { C } ) ) ) )
71, 5, 6syl2an 475 . . . 4  |-  ( ( F  Fn  ( A  X.  B )  /\  C  e.  B )  ->  Fun  ( F  o.  `' ( 1st  |`  ( _V  X.  { C }
) ) ) )
8 dmco 5452 . . . . 5  |-  dom  ( F  o.  `' ( 1st  |`  ( _V  X.  { C } ) ) )  =  ( `' `' ( 1st  |`  ( _V  X.  { C }
) ) " dom  F )
9 fndm 5617 . . . . . . . 8  |-  ( F  Fn  ( A  X.  B )  ->  dom  F  =  ( A  X.  B ) )
109adantr 463 . . . . . . 7  |-  ( ( F  Fn  ( A  X.  B )  /\  C  e.  B )  ->  dom  F  =  ( A  X.  B ) )
1110imaeq2d 5278 . . . . . 6  |-  ( ( F  Fn  ( A  X.  B )  /\  C  e.  B )  ->  ( `' `' ( 1st  |`  ( _V  X.  { C } ) ) " dom  F
)  =  ( `' `' ( 1st  |`  ( _V  X.  { C }
) ) " ( A  X.  B ) ) )
12 imacnvcnv 5409 . . . . . . . . 9  |-  ( `' `' ( 1st  |`  ( _V  X.  { C }
) ) " ( A  X.  B ) )  =  ( ( 1st  |`  ( _V  X.  { C } ) ) "
( A  X.  B
) )
13 df-ima 4955 . . . . . . . . 9  |-  ( ( 1st  |`  ( _V  X.  { C } ) ) " ( A  X.  B ) )  =  ran  ( ( 1st  |`  ( _V  X.  { C } ) )  |`  ( A  X.  B ) )
14 resres 5227 . . . . . . . . . 10  |-  ( ( 1st  |`  ( _V  X.  { C } ) )  |`  ( A  X.  B ) )  =  ( 1st  |`  (
( _V  X.  { C } )  i^i  ( A  X.  B ) ) )
1514rneqi 5171 . . . . . . . . 9  |-  ran  (
( 1st  |`  ( _V 
X.  { C }
) )  |`  ( A  X.  B ) )  =  ran  ( 1st  |`  ( ( _V  X.  { C } )  i^i  ( A  X.  B
) ) )
1612, 13, 153eqtri 2435 . . . . . . . 8  |-  ( `' `' ( 1st  |`  ( _V  X.  { C }
) ) " ( A  X.  B ) )  =  ran  ( 1st  |`  ( ( _V  X.  { C } )  i^i  ( A  X.  B
) ) )
17 inxp 5077 . . . . . . . . . . . . 13  |-  ( ( _V  X.  { C } )  i^i  ( A  X.  B ) )  =  ( ( _V 
i^i  A )  X.  ( { C }  i^i  B ) )
18 incom 3631 . . . . . . . . . . . . . . 15  |-  ( _V 
i^i  A )  =  ( A  i^i  _V )
19 inv1 3765 . . . . . . . . . . . . . . 15  |-  ( A  i^i  _V )  =  A
2018, 19eqtri 2431 . . . . . . . . . . . . . 14  |-  ( _V 
i^i  A )  =  A
2120xpeq1i 4962 . . . . . . . . . . . . 13  |-  ( ( _V  i^i  A )  X.  ( { C }  i^i  B ) )  =  ( A  X.  ( { C }  i^i  B ) )
2217, 21eqtri 2431 . . . . . . . . . . . 12  |-  ( ( _V  X.  { C } )  i^i  ( A  X.  B ) )  =  ( A  X.  ( { C }  i^i  B ) )
23 snssi 4115 . . . . . . . . . . . . . 14  |-  ( C  e.  B  ->  { C }  C_  B )
24 df-ss 3427 . . . . . . . . . . . . . 14  |-  ( { C }  C_  B  <->  ( { C }  i^i  B )  =  { C } )
2523, 24sylib 196 . . . . . . . . . . . . 13  |-  ( C  e.  B  ->  ( { C }  i^i  B
)  =  { C } )
2625xpeq2d 4966 . . . . . . . . . . . 12  |-  ( C  e.  B  ->  ( A  X.  ( { C }  i^i  B ) )  =  ( A  X.  { C } ) )
2722, 26syl5eq 2455 . . . . . . . . . . 11  |-  ( C  e.  B  ->  (
( _V  X.  { C } )  i^i  ( A  X.  B ) )  =  ( A  X.  { C } ) )
2827reseq2d 5215 . . . . . . . . . 10  |-  ( C  e.  B  ->  ( 1st  |`  ( ( _V 
X.  { C }
)  i^i  ( A  X.  B ) ) )  =  ( 1st  |`  ( A  X.  { C }
) ) )
2928rneqd 5172 . . . . . . . . 9  |-  ( C  e.  B  ->  ran  ( 1st  |`  ( ( _V  X.  { C }
)  i^i  ( A  X.  B ) ) )  =  ran  ( 1st  |`  ( A  X.  { C } ) ) )
30 1stconst 6826 . . . . . . . . . 10  |-  ( C  e.  B  ->  ( 1st  |`  ( A  X.  { C } ) ) : ( A  X.  { C } ) -1-1-onto-> A )
31 f1ofo 5762 . . . . . . . . . 10  |-  ( ( 1st  |`  ( A  X.  { C } ) ) : ( A  X.  { C }
)
-1-1-onto-> A  ->  ( 1st  |`  ( A  X.  { C }
) ) : ( A  X.  { C } ) -onto-> A )
32 forn 5737 . . . . . . . . . 10  |-  ( ( 1st  |`  ( A  X.  { C } ) ) : ( A  X.  { C }
) -onto-> A  ->  ran  ( 1st  |`  ( A  X.  { C } ) )  =  A )
3330, 31, 323syl 20 . . . . . . . . 9  |-  ( C  e.  B  ->  ran  ( 1st  |`  ( A  X.  { C } ) )  =  A )
3429, 33eqtrd 2443 . . . . . . . 8  |-  ( C  e.  B  ->  ran  ( 1st  |`  ( ( _V  X.  { C }
)  i^i  ( A  X.  B ) ) )  =  A )
3516, 34syl5eq 2455 . . . . . . 7  |-  ( C  e.  B  ->  ( `' `' ( 1st  |`  ( _V  X.  { C }
) ) " ( A  X.  B ) )  =  A )
3635adantl 464 . . . . . 6  |-  ( ( F  Fn  ( A  X.  B )  /\  C  e.  B )  ->  ( `' `' ( 1st  |`  ( _V  X.  { C } ) ) " ( A  X.  B ) )  =  A )
3711, 36eqtrd 2443 . . . . 5  |-  ( ( F  Fn  ( A  X.  B )  /\  C  e.  B )  ->  ( `' `' ( 1st  |`  ( _V  X.  { C } ) ) " dom  F
)  =  A )
388, 37syl5eq 2455 . . . 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 5612 . . . . 5  |-  ( G  Fn  A  <->  ( F  o.  `' ( 1st  |`  ( _V  X.  { C }
) ) )  Fn  A )
41 df-fn 5528 . . . . 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 249 . . . 4  |-  ( G  Fn  A  <->  ( Fun  ( F  o.  `' ( 1st  |`  ( _V  X.  { C } ) ) )  /\  dom  ( F  o.  `' ( 1st  |`  ( _V  X.  { C } ) ) )  =  A ) )
437, 38, 42sylanbrc 662 . . 3  |-  ( ( F  Fn  ( A  X.  B )  /\  C  e.  B )  ->  G  Fn  A )
44 dffn5 5850 . . 3  |-  ( G  Fn  A  <->  G  =  ( x  e.  A  |->  ( G `  x
) ) )
4543, 44sylib 196 . 2  |-  ( ( F  Fn  ( A  X.  B )  /\  C  e.  B )  ->  G  =  ( x  e.  A  |->  ( G `
 x ) ) )
4639fveq1i 5806 . . . . 5  |-  ( G `
 x )  =  ( ( F  o.  `' ( 1st  |`  ( _V  X.  { C }
) ) ) `  x )
47 dff1o4 5763 . . . . . . . . 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 196 . . . . . . . 8  |-  ( C  e.  B  ->  (
( 1st  |`  ( _V 
X.  { C }
) )  Fn  ( _V  X.  { C }
)  /\  `' ( 1st  |`  ( _V  X.  { C } ) )  Fn  _V ) )
4948simprd 461 . . . . . . 7  |-  ( C  e.  B  ->  `' ( 1st  |`  ( _V  X.  { C } ) )  Fn  _V )
50 vex 3061 . . . . . . 7  |-  x  e. 
_V
51 fvco2 5880 . . . . . . 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 660 . . . . . 6  |-  ( C  e.  B  ->  (
( F  o.  `' ( 1st  |`  ( _V  X.  { C } ) ) ) `  x
)  =  ( F `
 ( `' ( 1st  |`  ( _V  X.  { C } ) ) `  x ) ) )
5352ad2antlr 725 . . . . 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 2455 . . . 4  |-  ( ( ( F  Fn  ( A  X.  B )  /\  C  e.  B )  /\  x  e.  A
)  ->  ( G `  x )  =  ( F `  ( `' ( 1st  |`  ( _V  X.  { C }
) ) `  x
) ) )
552adantr 463 . . . . . . . . 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 3997 . . . . . . . . . . 11  |-  ( C  e.  B  ->  C  e.  { C } )
5857adantr 463 . . . . . . . . . 10  |-  ( ( C  e.  B  /\  x  e.  A )  ->  C  e.  { C } )
59 opelxp 4972 . . . . . . . . . 10  |-  ( <.
x ,  C >.  e.  ( _V  X.  { C } )  <->  ( x  e.  _V  /\  C  e. 
{ C } ) )
6056, 58, 59sylanbrc 662 . . . . . . . . 9  |-  ( ( C  e.  B  /\  x  e.  A )  -> 
<. x ,  C >.  e.  ( _V  X.  { C } ) )
6155, 60jca 530 . . . . . . . 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 662 . . . . . . . . . . 11  |-  ( C  e.  B  ->  <. x ,  C >.  e.  ( _V  X.  { C }
) )
64 fvres 5819 . . . . . . . . . . 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 463 . . . . . . . . 9  |-  ( ( C  e.  B  /\  x  e.  A )  ->  ( ( 1st  |`  ( _V  X.  { C }
) ) `  <. x ,  C >. )  =  ( 1st `  <. x ,  C >. )
)
67 op1stg 6750 . . . . . . . . . 10  |-  ( ( x  e.  A  /\  C  e.  B )  ->  ( 1st `  <. x ,  C >. )  =  x )
6867ancoms 451 . . . . . . . . 9  |-  ( ( C  e.  B  /\  x  e.  A )  ->  ( 1st `  <. x ,  C >. )  =  x )
6966, 68eqtrd 2443 . . . . . . . 8  |-  ( ( C  e.  B  /\  x  e.  A )  ->  ( ( 1st  |`  ( _V  X.  { C }
) ) `  <. x ,  C >. )  =  x )
70 f1ocnvfv 6121 . . . . . . . 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 59 . . . . . . 7  |-  ( ( C  e.  B  /\  x  e.  A )  ->  ( `' ( 1st  |`  ( _V  X.  { C } ) ) `  x )  =  <. x ,  C >. )
7271fveq2d 5809 . . . . . 6  |-  ( ( C  e.  B  /\  x  e.  A )  ->  ( F `  ( `' ( 1st  |`  ( _V  X.  { C }
) ) `  x
) )  =  ( F `  <. x ,  C >. ) )
7372adantll 712 . . . . 5  |-  ( ( ( F  Fn  ( A  X.  B )  /\  C  e.  B )  /\  x  e.  A
)  ->  ( F `  ( `' ( 1st  |`  ( _V  X.  { C } ) ) `  x ) )  =  ( F `  <. x ,  C >. )
)
74 df-ov 6237 . . . . 5  |-  ( x F C )  =  ( F `  <. x ,  C >. )
7573, 74syl6eqr 2461 . . . 4  |-  ( ( ( F  Fn  ( A  X.  B )  /\  C  e.  B )  /\  x  e.  A
)  ->  ( F `  ( `' ( 1st  |`  ( _V  X.  { C } ) ) `  x ) )  =  ( x F C ) )
7654, 75eqtrd 2443 . . 3  |-  ( ( ( F  Fn  ( A  X.  B )  /\  C  e.  B )  /\  x  e.  A
)  ->  ( G `  x )  =  ( x F C ) )
7776mpteq2dva 4480 . 2  |-  ( ( F  Fn  ( A  X.  B )  /\  C  e.  B )  ->  ( x  e.  A  |->  ( G `  x
) )  =  ( x  e.  A  |->  ( x F C ) ) )
7845, 77eqtrd 2443 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 367    = wceq 1405    e. wcel 1842   _Vcvv 3058    i^i cin 3412    C_ wss 3413   {csn 3971   <.cop 3977    |-> cmpt 4452    X. cxp 4940   `'ccnv 4941   dom cdm 4942   ran crn 4943    |` cres 4944   "cima 4945    o. ccom 4946   Fun wfun 5519    Fn wfn 5520   -onto->wfo 5523   -1-1-onto->wf1o 5524   ` cfv 5525  (class class class)co 6234   1stc1st 6736
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6530
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-sn 3972  df-pr 3974  df-op 3978  df-uni 4191  df-iun 4272  df-br 4395  df-opab 4453  df-mpt 4454  df-id 4737  df-xp 4948  df-rel 4949  df-cnv 4950  df-co 4951  df-dm 4952  df-rn 4953  df-res 4954  df-ima 4955  df-iota 5489  df-fun 5527  df-fn 5528  df-f 5529  df-f1 5530  df-fo 5531  df-f1o 5532  df-fv 5533  df-ov 6237  df-1st 6738  df-2nd 6739
This theorem is referenced by:  curry2f  6834  curry2val  6835
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