MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  curry2 Unicode version

Theorem curry2 6400
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 5501 . . . . 5  |-  ( F  Fn  ( A  X.  B )  ->  Fun  F )
2 1stconst 6394 . . . . . 6  |-  ( C  e.  B  ->  ( 1st  |`  ( _V  X.  { C } ) ) : ( _V  X.  { C } ) -1-1-onto-> _V )
3 dff1o3 5639 . . . . . . 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 5450 . . . . 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 5337 . . . . 5  |-  dom  ( F  o.  `' ( 1st  |`  ( _V  X.  { C } ) ) )  =  ( `' `' ( 1st  |`  ( _V  X.  { C }
) ) " dom  F )
9 fndm 5503 . . . . . . . 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 5162 . . . . . 6  |-  ( ( F  Fn  ( A  X.  B )  /\  C  e.  B )  ->  ( `' `' ( 1st  |`  ( _V  X.  { C } ) ) " dom  F
)  =  ( `' `' ( 1st  |`  ( _V  X.  { C }
) ) " ( A  X.  B ) ) )
12 imacnvcnv 5293 . . . . . . . . 9  |-  ( `' `' ( 1st  |`  ( _V  X.  { C }
) ) " ( A  X.  B ) )  =  ( ( 1st  |`  ( _V  X.  { C } ) ) "
( A  X.  B
) )
13 df-ima 4850 . . . . . . . . 9  |-  ( ( 1st  |`  ( _V  X.  { C } ) ) " ( A  X.  B ) )  =  ran  ( ( 1st  |`  ( _V  X.  { C } ) )  |`  ( A  X.  B ) )
14 resres 5118 . . . . . . . . . 10  |-  ( ( 1st  |`  ( _V  X.  { C } ) )  |`  ( A  X.  B ) )  =  ( 1st  |`  (
( _V  X.  { C } )  i^i  ( A  X.  B ) ) )
1514rneqi 5055 . . . . . . . . 9  |-  ran  (
( 1st  |`  ( _V 
X.  { C }
) )  |`  ( A  X.  B ) )  =  ran  ( 1st  |`  ( ( _V  X.  { C } )  i^i  ( A  X.  B
) ) )
1612, 13, 153eqtri 2428 . . . . . . . 8  |-  ( `' `' ( 1st  |`  ( _V  X.  { C }
) ) " ( A  X.  B ) )  =  ran  ( 1st  |`  ( ( _V  X.  { C } )  i^i  ( A  X.  B
) ) )
17 inxp 4966 . . . . . . . . . . . . 13  |-  ( ( _V  X.  { C } )  i^i  ( A  X.  B ) )  =  ( ( _V 
i^i  A )  X.  ( { C }  i^i  B ) )
18 incom 3493 . . . . . . . . . . . . . . 15  |-  ( _V 
i^i  A )  =  ( A  i^i  _V )
19 inv1 3614 . . . . . . . . . . . . . . 15  |-  ( A  i^i  _V )  =  A
2018, 19eqtri 2424 . . . . . . . . . . . . . 14  |-  ( _V 
i^i  A )  =  A
2120xpeq1i 4857 . . . . . . . . . . . . 13  |-  ( ( _V  i^i  A )  X.  ( { C }  i^i  B ) )  =  ( A  X.  ( { C }  i^i  B ) )
2217, 21eqtri 2424 . . . . . . . . . . . 12  |-  ( ( _V  X.  { C } )  i^i  ( A  X.  B ) )  =  ( A  X.  ( { C }  i^i  B ) )
23 snssi 3902 . . . . . . . . . . . . . 14  |-  ( C  e.  B  ->  { C }  C_  B )
24 df-ss 3294 . . . . . . . . . . . . . 14  |-  ( { C }  C_  B  <->  ( { C }  i^i  B )  =  { C } )
2523, 24sylib 189 . . . . . . . . . . . . 13  |-  ( C  e.  B  ->  ( { C }  i^i  B
)  =  { C } )
2625xpeq2d 4861 . . . . . . . . . . . 12  |-  ( C  e.  B  ->  ( A  X.  ( { C }  i^i  B ) )  =  ( A  X.  { C } ) )
2722, 26syl5eq 2448 . . . . . . . . . . 11  |-  ( C  e.  B  ->  (
( _V  X.  { C } )  i^i  ( A  X.  B ) )  =  ( A  X.  { C } ) )
2827reseq2d 5105 . . . . . . . . . 10  |-  ( C  e.  B  ->  ( 1st  |`  ( ( _V 
X.  { C }
)  i^i  ( A  X.  B ) ) )  =  ( 1st  |`  ( A  X.  { C }
) ) )
2928rneqd 5056 . . . . . . . . 9  |-  ( C  e.  B  ->  ran  ( 1st  |`  ( ( _V  X.  { C }
)  i^i  ( A  X.  B ) ) )  =  ran  ( 1st  |`  ( A  X.  { C } ) ) )
30 1stconst 6394 . . . . . . . . . 10  |-  ( C  e.  B  ->  ( 1st  |`  ( A  X.  { C } ) ) : ( A  X.  { C } ) -1-1-onto-> A )
31 f1ofo 5640 . . . . . . . . . 10  |-  ( ( 1st  |`  ( A  X.  { C } ) ) : ( A  X.  { C }
)
-1-1-onto-> A  ->  ( 1st  |`  ( A  X.  { C }
) ) : ( A  X.  { C } ) -onto-> A )
32 forn 5615 . . . . . . . . . 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 2436 . . . . . . . 8  |-  ( C  e.  B  ->  ran  ( 1st  |`  ( ( _V  X.  { C }
)  i^i  ( A  X.  B ) ) )  =  A )
3516, 34syl5eq 2448 . . . . . . 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 2436 . . . . 5  |-  ( ( F  Fn  ( A  X.  B )  /\  C  e.  B )  ->  ( `' `' ( 1st  |`  ( _V  X.  { C } ) ) " dom  F
)  =  A )
388, 37syl5eq 2448 . . . 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 5498 . . . . 5  |-  ( G  Fn  A  <->  ( F  o.  `' ( 1st  |`  ( _V  X.  { C }
) ) )  Fn  A )
41 df-fn 5416 . . . . 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 5731 . . 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 5688 . . . . 5  |-  ( G `
 x )  =  ( ( F  o.  `' ( 1st  |`  ( _V  X.  { C }
) ) ) `  x )
47 dff1o4 5641 . . . . . . . . 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 2919 . . . . . . 7  |-  x  e. 
_V
51 fvco2 5757 . . . . . . 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 2448 . . . 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 3799 . . . . . . . . . . 11  |-  ( C  e.  B  ->  C  e.  { C } )
5857adantr 452 . . . . . . . . . 10  |-  ( ( C  e.  B  /\  x  e.  A )  ->  C  e.  { C } )
59 opelxp 4867 . . . . . . . . . 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 5704 . . . . . . . . . . 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 6318 . . . . . . . . . 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 2436 . . . . . . . 8  |-  ( ( C  e.  B  /\  x  e.  A )  ->  ( ( 1st  |`  ( _V  X.  { C }
) ) `  <. x ,  C >. )  =  x )
70 f1ocnvfv 5975 . . . . . . . 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 5691 . . . . . 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 6043 . . . . 5  |-  ( x F C )  =  ( F `  <. x ,  C >. )
7573, 74syl6eqr 2454 . . . 4  |-  ( ( ( F  Fn  ( A  X.  B )  /\  C  e.  B )  /\  x  e.  A
)  ->  ( F `  ( `' ( 1st  |`  ( _V  X.  { C } ) ) `  x ) )  =  ( x F C ) )
7654, 75eqtrd 2436 . . 3  |-  ( ( ( F  Fn  ( A  X.  B )  /\  C  e.  B )  /\  x  e.  A
)  ->  ( G `  x )  =  ( x F C ) )
7776mpteq2dva 4255 . 2  |-  ( ( F  Fn  ( A  X.  B )  /\  C  e.  B )  ->  ( x  e.  A  |->  ( G `  x
) )  =  ( x  e.  A  |->  ( x F C ) ) )
7845, 77eqtrd 2436 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 1721   _Vcvv 2916    i^i cin 3279    C_ wss 3280   {csn 3774   <.cop 3777    e. cmpt 4226    X. cxp 4835   `'ccnv 4836   dom cdm 4837   ran crn 4838    |` cres 4839   "cima 4840    o. ccom 4841   Fun wfun 5407    Fn wfn 5408   -onto->wfo 5411   -1-1-onto->wf1o 5412   ` cfv 5413  (class class class)co 6040   1stc1st 6306
This theorem is referenced by:  curry2f  6401  curry2val  6402
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 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660
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 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-1st 6308  df-2nd 6309
  Copyright terms: Public domain W3C validator