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Theorem curry1 6777
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 5619 . . . . 5  |-  ( F  Fn  ( A  X.  B )  ->  Fun  F )
2 2ndconst 6775 . . . . . 6  |-  ( C  e.  A  ->  ( 2nd  |`  ( { C }  X.  _V ) ) : ( { C }  X.  _V ) -1-1-onto-> _V )
3 dff1o3 5758 . . . . . . 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 5567 . . . . 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 5457 . . . . 5  |-  dom  ( F  o.  `' ( 2nd  |`  ( { C }  X.  _V ) ) )  =  ( `' `' ( 2nd  |`  ( { C }  X.  _V ) ) " dom  F )
9 fndm 5621 . . . . . . . 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 5280 . . . . . 6  |-  ( ( F  Fn  ( A  X.  B )  /\  C  e.  A )  ->  ( `' `' ( 2nd  |`  ( { C }  X.  _V )
) " dom  F
)  =  ( `' `' ( 2nd  |`  ( { C }  X.  _V ) ) " ( A  X.  B ) ) )
12 imacnvcnv 5414 . . . . . . . . 9  |-  ( `' `' ( 2nd  |`  ( { C }  X.  _V ) ) " ( A  X.  B ) )  =  ( ( 2nd  |`  ( { C }  X.  _V ) ) "
( A  X.  B
) )
13 df-ima 4964 . . . . . . . . 9  |-  ( ( 2nd  |`  ( { C }  X.  _V )
) " ( A  X.  B ) )  =  ran  ( ( 2nd  |`  ( { C }  X.  _V )
)  |`  ( A  X.  B ) )
14 resres 5234 . . . . . . . . . 10  |-  ( ( 2nd  |`  ( { C }  X.  _V )
)  |`  ( A  X.  B ) )  =  ( 2nd  |`  (
( { C }  X.  _V )  i^i  ( A  X.  B ) ) )
1514rneqi 5177 . . . . . . . . 9  |-  ran  (
( 2nd  |`  ( { C }  X.  _V ) )  |`  ( A  X.  B ) )  =  ran  ( 2nd  |`  ( ( { C }  X.  _V )  i^i  ( A  X.  B
) ) )
1612, 13, 153eqtri 2487 . . . . . . . 8  |-  ( `' `' ( 2nd  |`  ( { C }  X.  _V ) ) " ( A  X.  B ) )  =  ran  ( 2nd  |`  ( ( { C }  X.  _V )  i^i  ( A  X.  B
) ) )
17 inxp 5083 . . . . . . . . . . . . 13  |-  ( ( { C }  X.  _V )  i^i  ( A  X.  B ) )  =  ( ( { C }  i^i  A
)  X.  ( _V 
i^i  B ) )
18 incom 3654 . . . . . . . . . . . . . . 15  |-  ( _V 
i^i  B )  =  ( B  i^i  _V )
19 inv1 3775 . . . . . . . . . . . . . . 15  |-  ( B  i^i  _V )  =  B
2018, 19eqtri 2483 . . . . . . . . . . . . . 14  |-  ( _V 
i^i  B )  =  B
2120xpeq2i 4972 . . . . . . . . . . . . 13  |-  ( ( { C }  i^i  A )  X.  ( _V 
i^i  B ) )  =  ( ( { C }  i^i  A
)  X.  B )
2217, 21eqtri 2483 . . . . . . . . . . . 12  |-  ( ( { C }  X.  _V )  i^i  ( A  X.  B ) )  =  ( ( { C }  i^i  A
)  X.  B )
23 snssi 4128 . . . . . . . . . . . . . 14  |-  ( C  e.  A  ->  { C }  C_  A )
24 df-ss 3453 . . . . . . . . . . . . . 14  |-  ( { C }  C_  A  <->  ( { C }  i^i  A )  =  { C } )
2523, 24sylib 196 . . . . . . . . . . . . 13  |-  ( C  e.  A  ->  ( { C }  i^i  A
)  =  { C } )
2625xpeq1d 4974 . . . . . . . . . . . 12  |-  ( C  e.  A  ->  (
( { C }  i^i  A )  X.  B
)  =  ( { C }  X.  B
) )
2722, 26syl5eq 2507 . . . . . . . . . . 11  |-  ( C  e.  A  ->  (
( { C }  X.  _V )  i^i  ( A  X.  B ) )  =  ( { C }  X.  B ) )
2827reseq2d 5221 . . . . . . . . . 10  |-  ( C  e.  A  ->  ( 2nd  |`  ( ( { C }  X.  _V )  i^i  ( A  X.  B ) ) )  =  ( 2nd  |`  ( { C }  X.  B
) ) )
2928rneqd 5178 . . . . . . . . 9  |-  ( C  e.  A  ->  ran  ( 2nd  |`  ( ( { C }  X.  _V )  i^i  ( A  X.  B ) ) )  =  ran  ( 2nd  |`  ( { C }  X.  B ) ) )
30 2ndconst 6775 . . . . . . . . . 10  |-  ( C  e.  A  ->  ( 2nd  |`  ( { C }  X.  B ) ) : ( { C }  X.  B ) -1-1-onto-> B )
31 f1ofo 5759 . . . . . . . . . 10  |-  ( ( 2nd  |`  ( { C }  X.  B
) ) : ( { C }  X.  B ) -1-1-onto-> B  ->  ( 2nd  |`  ( { C }  X.  B ) ) : ( { C }  X.  B ) -onto-> B )
32 forn 5734 . . . . . . . . . 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 2495 . . . . . . . 8  |-  ( C  e.  A  ->  ran  ( 2nd  |`  ( ( { C }  X.  _V )  i^i  ( A  X.  B ) ) )  =  B )
3516, 34syl5eq 2507 . . . . . . 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 2495 . . . . 5  |-  ( ( F  Fn  ( A  X.  B )  /\  C  e.  A )  ->  ( `' `' ( 2nd  |`  ( { C }  X.  _V )
) " dom  F
)  =  B )
388, 37syl5eq 2507 . . . 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 5616 . . . . 5  |-  ( G  Fn  B  <->  ( F  o.  `' ( 2nd  |`  ( { C }  X.  _V ) ) )  Fn  B )
41 df-fn 5532 . . . . 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 5849 . . 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 5803 . . . . 5  |-  ( G `
 x )  =  ( ( F  o.  `' ( 2nd  |`  ( { C }  X.  _V ) ) ) `  x )
47 dff1o4 5760 . . . . . . . . 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 3081 . . . . . . . 8  |-  x  e. 
_V
51 fvco2 5878 . . . . . . . 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 2507 . . . 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 4014 . . . . . . . . . . . 12  |-  ( C  e.  A  ->  C  e.  { C } )
5857, 50jctir 538 . . . . . . . . . . 11  |-  ( C  e.  A  ->  ( C  e.  { C }  /\  x  e.  _V ) )
59 opelxp 4980 . . . . . . . . . . 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 5816 . . . . . . . . . . 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 6703 . . . . . . . . . . 11  |-  ( ( C  e.  A  /\  x  e.  _V )  ->  ( 2nd `  <. C ,  x >. )  =  x )
6650, 65mpan2 671 . . . . . . . . . 10  |-  ( C  e.  A  ->  ( 2nd `  <. C ,  x >. )  =  x )
6764, 66eqtrd 2495 . . . . . . . . 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 6097 . . . . . . . 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 5806 . . . . . 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 6206 . . . . 5  |-  ( C F x )  =  ( F `  <. C ,  x >. )
7472, 73syl6eqr 2513 . . . 4  |-  ( ( ( F  Fn  ( A  X.  B )  /\  C  e.  A )  /\  x  e.  B
)  ->  ( F `  ( `' ( 2nd  |`  ( { C }  X.  _V ) ) `  x ) )  =  ( C F x ) )
7555, 74eqtrd 2495 . . 3  |-  ( ( ( F  Fn  ( A  X.  B )  /\  C  e.  A )  /\  x  e.  B
)  ->  ( G `  x )  =  ( C F x ) )
7675mpteq2dva 4489 . 2  |-  ( ( F  Fn  ( A  X.  B )  /\  C  e.  A )  ->  ( x  e.  B  |->  ( G `  x
) )  =  ( x  e.  B  |->  ( C F x ) ) )
7745, 76eqtrd 2495 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 1370    e. wcel 1758   _Vcvv 3078    i^i cin 3438    C_ wss 3439   {csn 3988   <.cop 3994    |-> cmpt 4461    X. cxp 4949   `'ccnv 4950   dom cdm 4951   ran crn 4952    |` cres 4953   "cima 4954    o. ccom 4955   Fun wfun 5523    Fn wfn 5524   -onto->wfo 5527   -1-1-onto->wf1o 5528   ` cfv 5529  (class class class)co 6203   2ndc2nd 6689
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-id 4747  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-ov 6206  df-1st 6690  df-2nd 6691
This theorem is referenced by:  curry1val  6778  curry1f  6779
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