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Theorem curry1 6659
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 5503 . . . . 5  |-  ( F  Fn  ( A  X.  B )  ->  Fun  F )
2 2ndconst 6657 . . . . . 6  |-  ( C  e.  A  ->  ( 2nd  |`  ( { C }  X.  _V ) ) : ( { C }  X.  _V ) -1-1-onto-> _V )
3 dff1o3 5642 . . . . . . 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 464 . . . . . 6  |-  ( ( 2nd  |`  ( { C }  X.  _V )
) : ( { C }  X.  _V )
-1-1-onto-> _V  ->  Fun  `' ( 2nd  |`  ( { C }  X.  _V ) ) )
52, 4syl 16 . . . . 5  |-  ( C  e.  A  ->  Fun  `' ( 2nd  |`  ( { C }  X.  _V ) ) )
6 funco 5451 . . . . 5  |-  ( ( Fun  F  /\  Fun  `' ( 2nd  |`  ( { C }  X.  _V ) ) )  ->  Fun  ( F  o.  `' ( 2nd  |`  ( { C }  X.  _V )
) ) )
71, 5, 6syl2an 477 . . . 4  |-  ( ( F  Fn  ( A  X.  B )  /\  C  e.  A )  ->  Fun  ( F  o.  `' ( 2nd  |`  ( { C }  X.  _V ) ) ) )
8 dmco 5341 . . . . 5  |-  dom  ( F  o.  `' ( 2nd  |`  ( { C }  X.  _V ) ) )  =  ( `' `' ( 2nd  |`  ( { C }  X.  _V ) ) " dom  F )
9 fndm 5505 . . . . . . . 8  |-  ( F  Fn  ( A  X.  B )  ->  dom  F  =  ( A  X.  B ) )
109adantr 465 . . . . . . 7  |-  ( ( F  Fn  ( A  X.  B )  /\  C  e.  A )  ->  dom  F  =  ( A  X.  B ) )
1110imaeq2d 5164 . . . . . 6  |-  ( ( F  Fn  ( A  X.  B )  /\  C  e.  A )  ->  ( `' `' ( 2nd  |`  ( { C }  X.  _V )
) " dom  F
)  =  ( `' `' ( 2nd  |`  ( { C }  X.  _V ) ) " ( A  X.  B ) ) )
12 imacnvcnv 5298 . . . . . . . . 9  |-  ( `' `' ( 2nd  |`  ( { C }  X.  _V ) ) " ( A  X.  B ) )  =  ( ( 2nd  |`  ( { C }  X.  _V ) ) "
( A  X.  B
) )
13 df-ima 4848 . . . . . . . . 9  |-  ( ( 2nd  |`  ( { C }  X.  _V )
) " ( A  X.  B ) )  =  ran  ( ( 2nd  |`  ( { C }  X.  _V )
)  |`  ( A  X.  B ) )
14 resres 5118 . . . . . . . . . 10  |-  ( ( 2nd  |`  ( { C }  X.  _V )
)  |`  ( A  X.  B ) )  =  ( 2nd  |`  (
( { C }  X.  _V )  i^i  ( A  X.  B ) ) )
1514rneqi 5061 . . . . . . . . 9  |-  ran  (
( 2nd  |`  ( { C }  X.  _V ) )  |`  ( A  X.  B ) )  =  ran  ( 2nd  |`  ( ( { C }  X.  _V )  i^i  ( A  X.  B
) ) )
1612, 13, 153eqtri 2462 . . . . . . . 8  |-  ( `' `' ( 2nd  |`  ( { C }  X.  _V ) ) " ( A  X.  B ) )  =  ran  ( 2nd  |`  ( ( { C }  X.  _V )  i^i  ( A  X.  B
) ) )
17 inxp 4967 . . . . . . . . . . . . 13  |-  ( ( { C }  X.  _V )  i^i  ( A  X.  B ) )  =  ( ( { C }  i^i  A
)  X.  ( _V 
i^i  B ) )
18 incom 3538 . . . . . . . . . . . . . . 15  |-  ( _V 
i^i  B )  =  ( B  i^i  _V )
19 inv1 3659 . . . . . . . . . . . . . . 15  |-  ( B  i^i  _V )  =  B
2018, 19eqtri 2458 . . . . . . . . . . . . . 14  |-  ( _V 
i^i  B )  =  B
2120xpeq2i 4856 . . . . . . . . . . . . 13  |-  ( ( { C }  i^i  A )  X.  ( _V 
i^i  B ) )  =  ( ( { C }  i^i  A
)  X.  B )
2217, 21eqtri 2458 . . . . . . . . . . . 12  |-  ( ( { C }  X.  _V )  i^i  ( A  X.  B ) )  =  ( ( { C }  i^i  A
)  X.  B )
23 snssi 4012 . . . . . . . . . . . . . 14  |-  ( C  e.  A  ->  { C }  C_  A )
24 df-ss 3337 . . . . . . . . . . . . . 14  |-  ( { C }  C_  A  <->  ( { C }  i^i  A )  =  { C } )
2523, 24sylib 196 . . . . . . . . . . . . 13  |-  ( C  e.  A  ->  ( { C }  i^i  A
)  =  { C } )
2625xpeq1d 4858 . . . . . . . . . . . 12  |-  ( C  e.  A  ->  (
( { C }  i^i  A )  X.  B
)  =  ( { C }  X.  B
) )
2722, 26syl5eq 2482 . . . . . . . . . . 11  |-  ( C  e.  A  ->  (
( { C }  X.  _V )  i^i  ( A  X.  B ) )  =  ( { C }  X.  B ) )
2827reseq2d 5105 . . . . . . . . . 10  |-  ( C  e.  A  ->  ( 2nd  |`  ( ( { C }  X.  _V )  i^i  ( A  X.  B ) ) )  =  ( 2nd  |`  ( { C }  X.  B
) ) )
2928rneqd 5062 . . . . . . . . 9  |-  ( C  e.  A  ->  ran  ( 2nd  |`  ( ( { C }  X.  _V )  i^i  ( A  X.  B ) ) )  =  ran  ( 2nd  |`  ( { C }  X.  B ) ) )
30 2ndconst 6657 . . . . . . . . . 10  |-  ( C  e.  A  ->  ( 2nd  |`  ( { C }  X.  B ) ) : ( { C }  X.  B ) -1-1-onto-> B )
31 f1ofo 5643 . . . . . . . . . 10  |-  ( ( 2nd  |`  ( { C }  X.  B
) ) : ( { C }  X.  B ) -1-1-onto-> B  ->  ( 2nd  |`  ( { C }  X.  B ) ) : ( { C }  X.  B ) -onto-> B )
32 forn 5618 . . . . . . . . . 10  |-  ( ( 2nd  |`  ( { C }  X.  B
) ) : ( { C }  X.  B ) -onto-> B  ->  ran  ( 2nd  |`  ( { C }  X.  B
) )  =  B )
3330, 31, 323syl 20 . . . . . . . . 9  |-  ( C  e.  A  ->  ran  ( 2nd  |`  ( { C }  X.  B
) )  =  B )
3429, 33eqtrd 2470 . . . . . . . 8  |-  ( C  e.  A  ->  ran  ( 2nd  |`  ( ( { C }  X.  _V )  i^i  ( A  X.  B ) ) )  =  B )
3516, 34syl5eq 2482 . . . . . . 7  |-  ( C  e.  A  ->  ( `' `' ( 2nd  |`  ( { C }  X.  _V ) ) " ( A  X.  B ) )  =  B )
3635adantl 466 . . . . . 6  |-  ( ( F  Fn  ( A  X.  B )  /\  C  e.  A )  ->  ( `' `' ( 2nd  |`  ( { C }  X.  _V )
) " ( A  X.  B ) )  =  B )
3711, 36eqtrd 2470 . . . . 5  |-  ( ( F  Fn  ( A  X.  B )  /\  C  e.  A )  ->  ( `' `' ( 2nd  |`  ( { C }  X.  _V )
) " dom  F
)  =  B )
388, 37syl5eq 2482 . . . 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 5500 . . . . 5  |-  ( G  Fn  B  <->  ( F  o.  `' ( 2nd  |`  ( { C }  X.  _V ) ) )  Fn  B )
41 df-fn 5416 . . . . 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 249 . . . 4  |-  ( G  Fn  B  <->  ( Fun  ( F  o.  `' ( 2nd  |`  ( { C }  X.  _V )
) )  /\  dom  ( F  o.  `' ( 2nd  |`  ( { C }  X.  _V )
) )  =  B ) )
437, 38, 42sylanbrc 664 . . 3  |-  ( ( F  Fn  ( A  X.  B )  /\  C  e.  A )  ->  G  Fn  B )
44 dffn5 5732 . . 3  |-  ( G  Fn  B  <->  G  =  ( x  e.  B  |->  ( G `  x
) ) )
4543, 44sylib 196 . 2  |-  ( ( F  Fn  ( A  X.  B )  /\  C  e.  A )  ->  G  =  ( x  e.  B  |->  ( G `
 x ) ) )
4639fveq1i 5687 . . . . 5  |-  ( G `
 x )  =  ( ( F  o.  `' ( 2nd  |`  ( { C }  X.  _V ) ) ) `  x )
47 dff1o4 5644 . . . . . . . . 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 196 . . . . . . . 8  |-  ( C  e.  A  ->  (
( 2nd  |`  ( { C }  X.  _V ) )  Fn  ( { C }  X.  _V )  /\  `' ( 2nd  |`  ( { C }  X.  _V ) )  Fn 
_V ) )
4948simprd 463 . . . . . . 7  |-  ( C  e.  A  ->  `' ( 2nd  |`  ( { C }  X.  _V )
)  Fn  _V )
50 vex 2970 . . . . . . . 8  |-  x  e. 
_V
51 fvco2 5761 . . . . . . . 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 671 . . . . . . 7  |-  ( `' ( 2nd  |`  ( { C }  X.  _V ) )  Fn  _V  ->  ( ( F  o.  `' ( 2nd  |`  ( { C }  X.  _V ) ) ) `  x )  =  ( F `  ( `' ( 2nd  |`  ( { C }  X.  _V ) ) `  x
) ) )
5349, 52syl 16 . . . . . 6  |-  ( C  e.  A  ->  (
( F  o.  `' ( 2nd  |`  ( { C }  X.  _V )
) ) `  x
)  =  ( F `
 ( `' ( 2nd  |`  ( { C }  X.  _V )
) `  x )
) )
5453ad2antlr 726 . . . . 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 2482 . . . 4  |-  ( ( ( F  Fn  ( A  X.  B )  /\  C  e.  A )  /\  x  e.  B
)  ->  ( G `  x )  =  ( F `  ( `' ( 2nd  |`  ( { C }  X.  _V ) ) `  x
) ) )
562adantr 465 . . . . . . . . 9  |-  ( ( C  e.  A  /\  x  e.  B )  ->  ( 2nd  |`  ( { C }  X.  _V ) ) : ( { C }  X.  _V ) -1-1-onto-> _V )
57 snidg 3898 . . . . . . . . . . . 12  |-  ( C  e.  A  ->  C  e.  { C } )
5857, 50jctir 538 . . . . . . . . . . 11  |-  ( C  e.  A  ->  ( C  e.  { C }  /\  x  e.  _V ) )
59 opelxp 4864 . . . . . . . . . . 11  |-  ( <. C ,  x >.  e.  ( { C }  X.  _V )  <->  ( C  e.  { C }  /\  x  e.  _V )
)
6058, 59sylibr 212 . . . . . . . . . 10  |-  ( C  e.  A  ->  <. C ,  x >.  e.  ( { C }  X.  _V ) )
6160adantr 465 . . . . . . . . 9  |-  ( ( C  e.  A  /\  x  e.  B )  -> 
<. C ,  x >.  e.  ( { C }  X.  _V ) )
6256, 61jca 532 . . . . . . . 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 5699 . . . . . . . . . . 11  |-  ( <. C ,  x >.  e.  ( { C }  X.  _V )  ->  (
( 2nd  |`  ( { C }  X.  _V ) ) `  <. C ,  x >. )  =  ( 2nd `  <. C ,  x >. )
)
6460, 63syl 16 . . . . . . . . . 10  |-  ( C  e.  A  ->  (
( 2nd  |`  ( { C }  X.  _V ) ) `  <. C ,  x >. )  =  ( 2nd `  <. C ,  x >. )
)
65 op2ndg 6585 . . . . . . . . . . 11  |-  ( ( C  e.  A  /\  x  e.  _V )  ->  ( 2nd `  <. C ,  x >. )  =  x )
6650, 65mpan2 671 . . . . . . . . . 10  |-  ( C  e.  A  ->  ( 2nd `  <. C ,  x >. )  =  x )
6764, 66eqtrd 2470 . . . . . . . . 9  |-  ( C  e.  A  ->  (
( 2nd  |`  ( { C }  X.  _V ) ) `  <. C ,  x >. )  =  x )
6867adantr 465 . . . . . . . 8  |-  ( ( C  e.  A  /\  x  e.  B )  ->  ( ( 2nd  |`  ( { C }  X.  _V ) ) `  <. C ,  x >. )  =  x )
69 f1ocnvfv 5980 . . . . . . . 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 60 . . . . . . 7  |-  ( ( C  e.  A  /\  x  e.  B )  ->  ( `' ( 2nd  |`  ( { C }  X.  _V ) ) `  x )  =  <. C ,  x >. )
7170fveq2d 5690 . . . . . 6  |-  ( ( C  e.  A  /\  x  e.  B )  ->  ( F `  ( `' ( 2nd  |`  ( { C }  X.  _V ) ) `  x
) )  =  ( F `  <. C ,  x >. ) )
7271adantll 713 . . . . 5  |-  ( ( ( F  Fn  ( A  X.  B )  /\  C  e.  A )  /\  x  e.  B
)  ->  ( F `  ( `' ( 2nd  |`  ( { C }  X.  _V ) ) `  x ) )  =  ( F `  <. C ,  x >. )
)
73 df-ov 6089 . . . . 5  |-  ( C F x )  =  ( F `  <. C ,  x >. )
7472, 73syl6eqr 2488 . . . 4  |-  ( ( ( F  Fn  ( A  X.  B )  /\  C  e.  A )  /\  x  e.  B
)  ->  ( F `  ( `' ( 2nd  |`  ( { C }  X.  _V ) ) `  x ) )  =  ( C F x ) )
7555, 74eqtrd 2470 . . 3  |-  ( ( ( F  Fn  ( A  X.  B )  /\  C  e.  A )  /\  x  e.  B
)  ->  ( G `  x )  =  ( C F x ) )
7675mpteq2dva 4373 . 2  |-  ( ( F  Fn  ( A  X.  B )  /\  C  e.  A )  ->  ( x  e.  B  |->  ( G `  x
) )  =  ( x  e.  B  |->  ( C F x ) ) )
7745, 76eqtrd 2470 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 369    = wceq 1369    e. wcel 1756   _Vcvv 2967    i^i cin 3322    C_ wss 3323   {csn 3872   <.cop 3878    e. cmpt 4345    X. cxp 4833   `'ccnv 4834   dom cdm 4835   ran crn 4836    |` cres 4837   "cima 4838    o. ccom 4839   Fun wfun 5407    Fn wfn 5408   -onto->wfo 5411   -1-1-onto->wf1o 5412   ` cfv 5413  (class class class)co 6086   2ndc2nd 6571
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2715  df-rex 2716  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-nul 3633  df-if 3787  df-sn 3873  df-pr 3875  df-op 3879  df-uni 4087  df-iun 4168  df-br 4288  df-opab 4346  df-mpt 4347  df-id 4631  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6089  df-1st 6572  df-2nd 6573
This theorem is referenced by:  curry1val  6660  curry1f  6661
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