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Theorem curry1 6885
Description: Composition with  `' ( 2nd  |`  ( { C }  X.  _V )
) turns any binary operation  F with a constant first operand into a function  G of the second operand only. This transformation is called "currying." (Contributed by NM, 28-Mar-2008.) (Revised by Mario Carneiro, 26-Dec-2014.)
Hypothesis
Ref Expression
curry1.1  |-  G  =  ( F  o.  `' ( 2nd  |`  ( { C }  X.  _V )
) )
Assertion
Ref Expression
curry1  |-  ( ( F  Fn  ( A  X.  B )  /\  C  e.  A )  ->  G  =  ( x  e.  B  |->  ( C F x ) ) )
Distinct variable groups:    x, A    x, B    x, C    x, F    x, G

Proof of Theorem curry1
StepHypRef Expression
1 fnfun 5671 . . . . 5  |-  ( F  Fn  ( A  X.  B )  ->  Fun  F )
2 2ndconst 6882 . . . . . 6  |-  ( C  e.  A  ->  ( 2nd  |`  ( { C }  X.  _V ) ) : ( { C }  X.  _V ) -1-1-onto-> _V )
3 dff1o3 5818 . . . . . . 7  |-  ( ( 2nd  |`  ( { C }  X.  _V )
) : ( { C }  X.  _V )
-1-1-onto-> _V 
<->  ( ( 2nd  |`  ( { C }  X.  _V ) ) : ( { C }  X.  _V ) -onto-> _V  /\  Fun  `' ( 2nd  |`  ( { C }  X.  _V )
) ) )
43simprbi 466 . . . . . 6  |-  ( ( 2nd  |`  ( { C }  X.  _V )
) : ( { C }  X.  _V )
-1-1-onto-> _V  ->  Fun  `' ( 2nd  |`  ( { C }  X.  _V ) ) )
52, 4syl 17 . . . . 5  |-  ( C  e.  A  ->  Fun  `' ( 2nd  |`  ( { C }  X.  _V ) ) )
6 funco 5619 . . . . 5  |-  ( ( Fun  F  /\  Fun  `' ( 2nd  |`  ( { C }  X.  _V ) ) )  ->  Fun  ( F  o.  `' ( 2nd  |`  ( { C }  X.  _V )
) ) )
71, 5, 6syl2an 480 . . . 4  |-  ( ( F  Fn  ( A  X.  B )  /\  C  e.  A )  ->  Fun  ( F  o.  `' ( 2nd  |`  ( { C }  X.  _V ) ) ) )
8 dmco 5342 . . . . 5  |-  dom  ( F  o.  `' ( 2nd  |`  ( { C }  X.  _V ) ) )  =  ( `' `' ( 2nd  |`  ( { C }  X.  _V ) ) " dom  F )
9 fndm 5673 . . . . . . . 8  |-  ( F  Fn  ( A  X.  B )  ->  dom  F  =  ( A  X.  B ) )
109adantr 467 . . . . . . 7  |-  ( ( F  Fn  ( A  X.  B )  /\  C  e.  A )  ->  dom  F  =  ( A  X.  B ) )
1110imaeq2d 5167 . . . . . 6  |-  ( ( F  Fn  ( A  X.  B )  /\  C  e.  A )  ->  ( `' `' ( 2nd  |`  ( { C }  X.  _V )
) " dom  F
)  =  ( `' `' ( 2nd  |`  ( { C }  X.  _V ) ) " ( A  X.  B ) ) )
12 imacnvcnv 5299 . . . . . . . . 9  |-  ( `' `' ( 2nd  |`  ( { C }  X.  _V ) ) " ( A  X.  B ) )  =  ( ( 2nd  |`  ( { C }  X.  _V ) ) "
( A  X.  B
) )
13 df-ima 4846 . . . . . . . . 9  |-  ( ( 2nd  |`  ( { C }  X.  _V )
) " ( A  X.  B ) )  =  ran  ( ( 2nd  |`  ( { C }  X.  _V )
)  |`  ( A  X.  B ) )
14 resres 5116 . . . . . . . . . 10  |-  ( ( 2nd  |`  ( { C }  X.  _V )
)  |`  ( A  X.  B ) )  =  ( 2nd  |`  (
( { C }  X.  _V )  i^i  ( A  X.  B ) ) )
1514rneqi 5060 . . . . . . . . 9  |-  ran  (
( 2nd  |`  ( { C }  X.  _V ) )  |`  ( A  X.  B ) )  =  ran  ( 2nd  |`  ( ( { C }  X.  _V )  i^i  ( A  X.  B
) ) )
1612, 13, 153eqtri 2476 . . . . . . . 8  |-  ( `' `' ( 2nd  |`  ( { C }  X.  _V ) ) " ( A  X.  B ) )  =  ran  ( 2nd  |`  ( ( { C }  X.  _V )  i^i  ( A  X.  B
) ) )
17 inxp 4966 . . . . . . . . . . . . 13  |-  ( ( { C }  X.  _V )  i^i  ( A  X.  B ) )  =  ( ( { C }  i^i  A
)  X.  ( _V 
i^i  B ) )
18 incom 3624 . . . . . . . . . . . . . . 15  |-  ( _V 
i^i  B )  =  ( B  i^i  _V )
19 inv1 3760 . . . . . . . . . . . . . . 15  |-  ( B  i^i  _V )  =  B
2018, 19eqtri 2472 . . . . . . . . . . . . . 14  |-  ( _V 
i^i  B )  =  B
2120xpeq2i 4854 . . . . . . . . . . . . 13  |-  ( ( { C }  i^i  A )  X.  ( _V 
i^i  B ) )  =  ( ( { C }  i^i  A
)  X.  B )
2217, 21eqtri 2472 . . . . . . . . . . . 12  |-  ( ( { C }  X.  _V )  i^i  ( A  X.  B ) )  =  ( ( { C }  i^i  A
)  X.  B )
23 snssi 4115 . . . . . . . . . . . . . 14  |-  ( C  e.  A  ->  { C }  C_  A )
24 df-ss 3417 . . . . . . . . . . . . . 14  |-  ( { C }  C_  A  <->  ( { C }  i^i  A )  =  { C } )
2523, 24sylib 200 . . . . . . . . . . . . 13  |-  ( C  e.  A  ->  ( { C }  i^i  A
)  =  { C } )
2625xpeq1d 4856 . . . . . . . . . . . 12  |-  ( C  e.  A  ->  (
( { C }  i^i  A )  X.  B
)  =  ( { C }  X.  B
) )
2722, 26syl5eq 2496 . . . . . . . . . . 11  |-  ( C  e.  A  ->  (
( { C }  X.  _V )  i^i  ( A  X.  B ) )  =  ( { C }  X.  B ) )
2827reseq2d 5104 . . . . . . . . . 10  |-  ( C  e.  A  ->  ( 2nd  |`  ( ( { C }  X.  _V )  i^i  ( A  X.  B ) ) )  =  ( 2nd  |`  ( { C }  X.  B
) ) )
2928rneqd 5061 . . . . . . . . 9  |-  ( C  e.  A  ->  ran  ( 2nd  |`  ( ( { C }  X.  _V )  i^i  ( A  X.  B ) ) )  =  ran  ( 2nd  |`  ( { C }  X.  B ) ) )
30 2ndconst 6882 . . . . . . . . . 10  |-  ( C  e.  A  ->  ( 2nd  |`  ( { C }  X.  B ) ) : ( { C }  X.  B ) -1-1-onto-> B )
31 f1ofo 5819 . . . . . . . . . 10  |-  ( ( 2nd  |`  ( { C }  X.  B
) ) : ( { C }  X.  B ) -1-1-onto-> B  ->  ( 2nd  |`  ( { C }  X.  B ) ) : ( { C }  X.  B ) -onto-> B )
32 forn 5794 . . . . . . . . . 10  |-  ( ( 2nd  |`  ( { C }  X.  B
) ) : ( { C }  X.  B ) -onto-> B  ->  ran  ( 2nd  |`  ( { C }  X.  B
) )  =  B )
3330, 31, 323syl 18 . . . . . . . . 9  |-  ( C  e.  A  ->  ran  ( 2nd  |`  ( { C }  X.  B
) )  =  B )
3429, 33eqtrd 2484 . . . . . . . 8  |-  ( C  e.  A  ->  ran  ( 2nd  |`  ( ( { C }  X.  _V )  i^i  ( A  X.  B ) ) )  =  B )
3516, 34syl5eq 2496 . . . . . . 7  |-  ( C  e.  A  ->  ( `' `' ( 2nd  |`  ( { C }  X.  _V ) ) " ( A  X.  B ) )  =  B )
3635adantl 468 . . . . . 6  |-  ( ( F  Fn  ( A  X.  B )  /\  C  e.  A )  ->  ( `' `' ( 2nd  |`  ( { C }  X.  _V )
) " ( A  X.  B ) )  =  B )
3711, 36eqtrd 2484 . . . . 5  |-  ( ( F  Fn  ( A  X.  B )  /\  C  e.  A )  ->  ( `' `' ( 2nd  |`  ( { C }  X.  _V )
) " dom  F
)  =  B )
388, 37syl5eq 2496 . . . 4  |-  ( ( F  Fn  ( A  X.  B )  /\  C  e.  A )  ->  dom  ( F  o.  `' ( 2nd  |`  ( { C }  X.  _V ) ) )  =  B )
39 curry1.1 . . . . . 6  |-  G  =  ( F  o.  `' ( 2nd  |`  ( { C }  X.  _V )
) )
4039fneq1i 5668 . . . . 5  |-  ( G  Fn  B  <->  ( F  o.  `' ( 2nd  |`  ( { C }  X.  _V ) ) )  Fn  B )
41 df-fn 5584 . . . . 5  |-  ( ( F  o.  `' ( 2nd  |`  ( { C }  X.  _V )
) )  Fn  B  <->  ( Fun  ( F  o.  `' ( 2nd  |`  ( { C }  X.  _V ) ) )  /\  dom  ( F  o.  `' ( 2nd  |`  ( { C }  X.  _V )
) )  =  B ) )
4240, 41bitri 253 . . . 4  |-  ( G  Fn  B  <->  ( Fun  ( F  o.  `' ( 2nd  |`  ( { C }  X.  _V )
) )  /\  dom  ( F  o.  `' ( 2nd  |`  ( { C }  X.  _V )
) )  =  B ) )
437, 38, 42sylanbrc 669 . . 3  |-  ( ( F  Fn  ( A  X.  B )  /\  C  e.  A )  ->  G  Fn  B )
44 dffn5 5908 . . 3  |-  ( G  Fn  B  <->  G  =  ( x  e.  B  |->  ( G `  x
) ) )
4543, 44sylib 200 . 2  |-  ( ( F  Fn  ( A  X.  B )  /\  C  e.  A )  ->  G  =  ( x  e.  B  |->  ( G `
 x ) ) )
4639fveq1i 5864 . . . . 5  |-  ( G `
 x )  =  ( ( F  o.  `' ( 2nd  |`  ( { C }  X.  _V ) ) ) `  x )
47 dff1o4 5820 . . . . . . . . 9  |-  ( ( 2nd  |`  ( { C }  X.  _V )
) : ( { C }  X.  _V )
-1-1-onto-> _V 
<->  ( ( 2nd  |`  ( { C }  X.  _V ) )  Fn  ( { C }  X.  _V )  /\  `' ( 2nd  |`  ( { C }  X.  _V ) )  Fn 
_V ) )
482, 47sylib 200 . . . . . . . 8  |-  ( C  e.  A  ->  (
( 2nd  |`  ( { C }  X.  _V ) )  Fn  ( { C }  X.  _V )  /\  `' ( 2nd  |`  ( { C }  X.  _V ) )  Fn 
_V ) )
4948simprd 465 . . . . . . 7  |-  ( C  e.  A  ->  `' ( 2nd  |`  ( { C }  X.  _V )
)  Fn  _V )
50 vex 3047 . . . . . . . 8  |-  x  e. 
_V
51 fvco2 5938 . . . . . . . 8  |-  ( ( `' ( 2nd  |`  ( { C }  X.  _V ) )  Fn  _V  /\  x  e.  _V )  ->  ( ( F  o.  `' ( 2nd  |`  ( { C }  X.  _V ) ) ) `  x )  =  ( F `  ( `' ( 2nd  |`  ( { C }  X.  _V ) ) `  x
) ) )
5250, 51mpan2 676 . . . . . . 7  |-  ( `' ( 2nd  |`  ( { C }  X.  _V ) )  Fn  _V  ->  ( ( F  o.  `' ( 2nd  |`  ( { C }  X.  _V ) ) ) `  x )  =  ( F `  ( `' ( 2nd  |`  ( { C }  X.  _V ) ) `  x
) ) )
5349, 52syl 17 . . . . . 6  |-  ( C  e.  A  ->  (
( F  o.  `' ( 2nd  |`  ( { C }  X.  _V )
) ) `  x
)  =  ( F `
 ( `' ( 2nd  |`  ( { C }  X.  _V )
) `  x )
) )
5453ad2antlr 732 . . . . 5  |-  ( ( ( F  Fn  ( A  X.  B )  /\  C  e.  A )  /\  x  e.  B
)  ->  ( ( F  o.  `' ( 2nd  |`  ( { C }  X.  _V ) ) ) `  x )  =  ( F `  ( `' ( 2nd  |`  ( { C }  X.  _V ) ) `  x
) ) )
5546, 54syl5eq 2496 . . . 4  |-  ( ( ( F  Fn  ( A  X.  B )  /\  C  e.  A )  /\  x  e.  B
)  ->  ( G `  x )  =  ( F `  ( `' ( 2nd  |`  ( { C }  X.  _V ) ) `  x
) ) )
562adantr 467 . . . . . . . . 9  |-  ( ( C  e.  A  /\  x  e.  B )  ->  ( 2nd  |`  ( { C }  X.  _V ) ) : ( { C }  X.  _V ) -1-1-onto-> _V )
57 snidg 3993 . . . . . . . . . . . 12  |-  ( C  e.  A  ->  C  e.  { C } )
5857, 50jctir 541 . . . . . . . . . . 11  |-  ( C  e.  A  ->  ( C  e.  { C }  /\  x  e.  _V ) )
59 opelxp 4863 . . . . . . . . . . 11  |-  ( <. C ,  x >.  e.  ( { C }  X.  _V )  <->  ( C  e.  { C }  /\  x  e.  _V )
)
6058, 59sylibr 216 . . . . . . . . . 10  |-  ( C  e.  A  ->  <. C ,  x >.  e.  ( { C }  X.  _V ) )
6160adantr 467 . . . . . . . . 9  |-  ( ( C  e.  A  /\  x  e.  B )  -> 
<. C ,  x >.  e.  ( { C }  X.  _V ) )
6256, 61jca 535 . . . . . . . 8  |-  ( ( C  e.  A  /\  x  e.  B )  ->  ( ( 2nd  |`  ( { C }  X.  _V ) ) : ( { C }  X.  _V ) -1-1-onto-> _V  /\  <. C ,  x >.  e.  ( { C }  X.  _V ) ) )
63 fvres 5877 . . . . . . . . . . 11  |-  ( <. C ,  x >.  e.  ( { C }  X.  _V )  ->  (
( 2nd  |`  ( { C }  X.  _V ) ) `  <. C ,  x >. )  =  ( 2nd `  <. C ,  x >. )
)
6460, 63syl 17 . . . . . . . . . 10  |-  ( C  e.  A  ->  (
( 2nd  |`  ( { C }  X.  _V ) ) `  <. C ,  x >. )  =  ( 2nd `  <. C ,  x >. )
)
65 op2ndg 6803 . . . . . . . . . . 11  |-  ( ( C  e.  A  /\  x  e.  _V )  ->  ( 2nd `  <. C ,  x >. )  =  x )
6650, 65mpan2 676 . . . . . . . . . 10  |-  ( C  e.  A  ->  ( 2nd `  <. C ,  x >. )  =  x )
6764, 66eqtrd 2484 . . . . . . . . 9  |-  ( C  e.  A  ->  (
( 2nd  |`  ( { C }  X.  _V ) ) `  <. C ,  x >. )  =  x )
6867adantr 467 . . . . . . . 8  |-  ( ( C  e.  A  /\  x  e.  B )  ->  ( ( 2nd  |`  ( { C }  X.  _V ) ) `  <. C ,  x >. )  =  x )
69 f1ocnvfv 6175 . . . . . . . 8  |-  ( ( ( 2nd  |`  ( { C }  X.  _V ) ) : ( { C }  X.  _V ) -1-1-onto-> _V  /\  <. C ,  x >.  e.  ( { C }  X.  _V ) )  ->  (
( ( 2nd  |`  ( { C }  X.  _V ) ) `  <. C ,  x >. )  =  x  ->  ( `' ( 2nd  |`  ( { C }  X.  _V ) ) `  x
)  =  <. C ,  x >. ) )
7062, 68, 69sylc 62 . . . . . . 7  |-  ( ( C  e.  A  /\  x  e.  B )  ->  ( `' ( 2nd  |`  ( { C }  X.  _V ) ) `  x )  =  <. C ,  x >. )
7170fveq2d 5867 . . . . . 6  |-  ( ( C  e.  A  /\  x  e.  B )  ->  ( F `  ( `' ( 2nd  |`  ( { C }  X.  _V ) ) `  x
) )  =  ( F `  <. C ,  x >. ) )
7271adantll 719 . . . . 5  |-  ( ( ( F  Fn  ( A  X.  B )  /\  C  e.  A )  /\  x  e.  B
)  ->  ( F `  ( `' ( 2nd  |`  ( { C }  X.  _V ) ) `  x ) )  =  ( F `  <. C ,  x >. )
)
73 df-ov 6291 . . . . 5  |-  ( C F x )  =  ( F `  <. C ,  x >. )
7472, 73syl6eqr 2502 . . . 4  |-  ( ( ( F  Fn  ( A  X.  B )  /\  C  e.  A )  /\  x  e.  B
)  ->  ( F `  ( `' ( 2nd  |`  ( { C }  X.  _V ) ) `  x ) )  =  ( C F x ) )
7555, 74eqtrd 2484 . . 3  |-  ( ( ( F  Fn  ( A  X.  B )  /\  C  e.  A )  /\  x  e.  B
)  ->  ( G `  x )  =  ( C F x ) )
7675mpteq2dva 4488 . 2  |-  ( ( F  Fn  ( A  X.  B )  /\  C  e.  A )  ->  ( x  e.  B  |->  ( G `  x
) )  =  ( x  e.  B  |->  ( C F x ) ) )
7745, 76eqtrd 2484 1  |-  ( ( F  Fn  ( A  X.  B )  /\  C  e.  A )  ->  G  =  ( x  e.  B  |->  ( C F x ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 371    = wceq 1443    e. wcel 1886   _Vcvv 3044    i^i cin 3402    C_ wss 3403   {csn 3967   <.cop 3973    |-> cmpt 4460    X. cxp 4831   `'ccnv 4832   dom cdm 4833   ran crn 4834    |` cres 4835   "cima 4836    o. ccom 4837   Fun wfun 5575    Fn wfn 5576   -onto->wfo 5579   -1-1-onto->wf1o 5580   ` cfv 5581  (class class class)co 6288   2ndc2nd 6789
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1668  ax-4 1681  ax-5 1757  ax-6 1804  ax-7 1850  ax-8 1888  ax-9 1895  ax-10 1914  ax-11 1919  ax-12 1932  ax-13 2090  ax-ext 2430  ax-sep 4524  ax-nul 4533  ax-pow 4580  ax-pr 4638  ax-un 6580
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 986  df-tru 1446  df-ex 1663  df-nf 1667  df-sb 1797  df-eu 2302  df-mo 2303  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2580  df-ne 2623  df-ral 2741  df-rex 2742  df-rab 2745  df-v 3046  df-sbc 3267  df-csb 3363  df-dif 3406  df-un 3408  df-in 3410  df-ss 3417  df-nul 3731  df-if 3881  df-sn 3968  df-pr 3970  df-op 3974  df-uni 4198  df-iun 4279  df-br 4402  df-opab 4461  df-mpt 4462  df-id 4748  df-xp 4839  df-rel 4840  df-cnv 4841  df-co 4842  df-dm 4843  df-rn 4844  df-res 4845  df-ima 4846  df-iota 5545  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-ov 6291  df-1st 6790  df-2nd 6791
This theorem is referenced by:  curry1val  6886  curry1f  6887
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