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Theorem curry1 6895
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 5687 . . . . 5  |-  ( F  Fn  ( A  X.  B )  ->  Fun  F )
2 2ndconst 6892 . . . . . 6  |-  ( C  e.  A  ->  ( 2nd  |`  ( { C }  X.  _V ) ) : ( { C }  X.  _V ) -1-1-onto-> _V )
3 dff1o3 5833 . . . . . . 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 465 . . . . . 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 5635 . . . . 5  |-  ( ( Fun  F  /\  Fun  `' ( 2nd  |`  ( { C }  X.  _V ) ) )  ->  Fun  ( F  o.  `' ( 2nd  |`  ( { C }  X.  _V )
) ) )
71, 5, 6syl2an 479 . . . 4  |-  ( ( F  Fn  ( A  X.  B )  /\  C  e.  A )  ->  Fun  ( F  o.  `' ( 2nd  |`  ( { C }  X.  _V ) ) ) )
8 dmco 5358 . . . . 5  |-  dom  ( F  o.  `' ( 2nd  |`  ( { C }  X.  _V ) ) )  =  ( `' `' ( 2nd  |`  ( { C }  X.  _V ) ) " dom  F )
9 fndm 5689 . . . . . . . 8  |-  ( F  Fn  ( A  X.  B )  ->  dom  F  =  ( A  X.  B ) )
109adantr 466 . . . . . . 7  |-  ( ( F  Fn  ( A  X.  B )  /\  C  e.  A )  ->  dom  F  =  ( A  X.  B ) )
1110imaeq2d 5183 . . . . . 6  |-  ( ( F  Fn  ( A  X.  B )  /\  C  e.  A )  ->  ( `' `' ( 2nd  |`  ( { C }  X.  _V )
) " dom  F
)  =  ( `' `' ( 2nd  |`  ( { C }  X.  _V ) ) " ( A  X.  B ) ) )
12 imacnvcnv 5315 . . . . . . . . 9  |-  ( `' `' ( 2nd  |`  ( { C }  X.  _V ) ) " ( A  X.  B ) )  =  ( ( 2nd  |`  ( { C }  X.  _V ) ) "
( A  X.  B
) )
13 df-ima 4862 . . . . . . . . 9  |-  ( ( 2nd  |`  ( { C }  X.  _V )
) " ( A  X.  B ) )  =  ran  ( ( 2nd  |`  ( { C }  X.  _V )
)  |`  ( A  X.  B ) )
14 resres 5132 . . . . . . . . . 10  |-  ( ( 2nd  |`  ( { C }  X.  _V )
)  |`  ( A  X.  B ) )  =  ( 2nd  |`  (
( { C }  X.  _V )  i^i  ( A  X.  B ) ) )
1514rneqi 5076 . . . . . . . . 9  |-  ran  (
( 2nd  |`  ( { C }  X.  _V ) )  |`  ( A  X.  B ) )  =  ran  ( 2nd  |`  ( ( { C }  X.  _V )  i^i  ( A  X.  B
) ) )
1612, 13, 153eqtri 2455 . . . . . . . 8  |-  ( `' `' ( 2nd  |`  ( { C }  X.  _V ) ) " ( A  X.  B ) )  =  ran  ( 2nd  |`  ( ( { C }  X.  _V )  i^i  ( A  X.  B
) ) )
17 inxp 4982 . . . . . . . . . . . . 13  |-  ( ( { C }  X.  _V )  i^i  ( A  X.  B ) )  =  ( ( { C }  i^i  A
)  X.  ( _V 
i^i  B ) )
18 incom 3655 . . . . . . . . . . . . . . 15  |-  ( _V 
i^i  B )  =  ( B  i^i  _V )
19 inv1 3789 . . . . . . . . . . . . . . 15  |-  ( B  i^i  _V )  =  B
2018, 19eqtri 2451 . . . . . . . . . . . . . 14  |-  ( _V 
i^i  B )  =  B
2120xpeq2i 4870 . . . . . . . . . . . . 13  |-  ( ( { C }  i^i  A )  X.  ( _V 
i^i  B ) )  =  ( ( { C }  i^i  A
)  X.  B )
2217, 21eqtri 2451 . . . . . . . . . . . 12  |-  ( ( { C }  X.  _V )  i^i  ( A  X.  B ) )  =  ( ( { C }  i^i  A
)  X.  B )
23 snssi 4141 . . . . . . . . . . . . . 14  |-  ( C  e.  A  ->  { C }  C_  A )
24 df-ss 3450 . . . . . . . . . . . . . 14  |-  ( { C }  C_  A  <->  ( { C }  i^i  A )  =  { C } )
2523, 24sylib 199 . . . . . . . . . . . . 13  |-  ( C  e.  A  ->  ( { C }  i^i  A
)  =  { C } )
2625xpeq1d 4872 . . . . . . . . . . . 12  |-  ( C  e.  A  ->  (
( { C }  i^i  A )  X.  B
)  =  ( { C }  X.  B
) )
2722, 26syl5eq 2475 . . . . . . . . . . 11  |-  ( C  e.  A  ->  (
( { C }  X.  _V )  i^i  ( A  X.  B ) )  =  ( { C }  X.  B ) )
2827reseq2d 5120 . . . . . . . . . 10  |-  ( C  e.  A  ->  ( 2nd  |`  ( ( { C }  X.  _V )  i^i  ( A  X.  B ) ) )  =  ( 2nd  |`  ( { C }  X.  B
) ) )
2928rneqd 5077 . . . . . . . . 9  |-  ( C  e.  A  ->  ran  ( 2nd  |`  ( ( { C }  X.  _V )  i^i  ( A  X.  B ) ) )  =  ran  ( 2nd  |`  ( { C }  X.  B ) ) )
30 2ndconst 6892 . . . . . . . . . 10  |-  ( C  e.  A  ->  ( 2nd  |`  ( { C }  X.  B ) ) : ( { C }  X.  B ) -1-1-onto-> B )
31 f1ofo 5834 . . . . . . . . . 10  |-  ( ( 2nd  |`  ( { C }  X.  B
) ) : ( { C }  X.  B ) -1-1-onto-> B  ->  ( 2nd  |`  ( { C }  X.  B ) ) : ( { C }  X.  B ) -onto-> B )
32 forn 5809 . . . . . . . . . 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 2463 . . . . . . . 8  |-  ( C  e.  A  ->  ran  ( 2nd  |`  ( ( { C }  X.  _V )  i^i  ( A  X.  B ) ) )  =  B )
3516, 34syl5eq 2475 . . . . . . 7  |-  ( C  e.  A  ->  ( `' `' ( 2nd  |`  ( { C }  X.  _V ) ) " ( A  X.  B ) )  =  B )
3635adantl 467 . . . . . 6  |-  ( ( F  Fn  ( A  X.  B )  /\  C  e.  A )  ->  ( `' `' ( 2nd  |`  ( { C }  X.  _V )
) " ( A  X.  B ) )  =  B )
3711, 36eqtrd 2463 . . . . 5  |-  ( ( F  Fn  ( A  X.  B )  /\  C  e.  A )  ->  ( `' `' ( 2nd  |`  ( { C }  X.  _V )
) " dom  F
)  =  B )
388, 37syl5eq 2475 . . . 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 5684 . . . . 5  |-  ( G  Fn  B  <->  ( F  o.  `' ( 2nd  |`  ( { C }  X.  _V ) ) )  Fn  B )
41 df-fn 5600 . . . . 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 252 . . . 4  |-  ( G  Fn  B  <->  ( Fun  ( F  o.  `' ( 2nd  |`  ( { C }  X.  _V )
) )  /\  dom  ( F  o.  `' ( 2nd  |`  ( { C }  X.  _V )
) )  =  B ) )
437, 38, 42sylanbrc 668 . . 3  |-  ( ( F  Fn  ( A  X.  B )  /\  C  e.  A )  ->  G  Fn  B )
44 dffn5 5922 . . 3  |-  ( G  Fn  B  <->  G  =  ( x  e.  B  |->  ( G `  x
) ) )
4543, 44sylib 199 . 2  |-  ( ( F  Fn  ( A  X.  B )  /\  C  e.  A )  ->  G  =  ( x  e.  B  |->  ( G `
 x ) ) )
4639fveq1i 5878 . . . . 5  |-  ( G `
 x )  =  ( ( F  o.  `' ( 2nd  |`  ( { C }  X.  _V ) ) ) `  x )
47 dff1o4 5835 . . . . . . . . 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 199 . . . . . . . 8  |-  ( C  e.  A  ->  (
( 2nd  |`  ( { C }  X.  _V ) )  Fn  ( { C }  X.  _V )  /\  `' ( 2nd  |`  ( { C }  X.  _V ) )  Fn 
_V ) )
4948simprd 464 . . . . . . 7  |-  ( C  e.  A  ->  `' ( 2nd  |`  ( { C }  X.  _V )
)  Fn  _V )
50 vex 3084 . . . . . . . 8  |-  x  e. 
_V
51 fvco2 5952 . . . . . . . 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 675 . . . . . . 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 731 . . . . 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 2475 . . . 4  |-  ( ( ( F  Fn  ( A  X.  B )  /\  C  e.  A )  /\  x  e.  B
)  ->  ( G `  x )  =  ( F `  ( `' ( 2nd  |`  ( { C }  X.  _V ) ) `  x
) ) )
562adantr 466 . . . . . . . . 9  |-  ( ( C  e.  A  /\  x  e.  B )  ->  ( 2nd  |`  ( { C }  X.  _V ) ) : ( { C }  X.  _V ) -1-1-onto-> _V )
57 snidg 4022 . . . . . . . . . . . 12  |-  ( C  e.  A  ->  C  e.  { C } )
5857, 50jctir 540 . . . . . . . . . . 11  |-  ( C  e.  A  ->  ( C  e.  { C }  /\  x  e.  _V ) )
59 opelxp 4879 . . . . . . . . . . 11  |-  ( <. C ,  x >.  e.  ( { C }  X.  _V )  <->  ( C  e.  { C }  /\  x  e.  _V )
)
6058, 59sylibr 215 . . . . . . . . . 10  |-  ( C  e.  A  ->  <. C ,  x >.  e.  ( { C }  X.  _V ) )
6160adantr 466 . . . . . . . . 9  |-  ( ( C  e.  A  /\  x  e.  B )  -> 
<. C ,  x >.  e.  ( { C }  X.  _V ) )
6256, 61jca 534 . . . . . . . 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 5891 . . . . . . . . . . 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 6816 . . . . . . . . . . 11  |-  ( ( C  e.  A  /\  x  e.  _V )  ->  ( 2nd `  <. C ,  x >. )  =  x )
6650, 65mpan2 675 . . . . . . . . . 10  |-  ( C  e.  A  ->  ( 2nd `  <. C ,  x >. )  =  x )
6764, 66eqtrd 2463 . . . . . . . . 9  |-  ( C  e.  A  ->  (
( 2nd  |`  ( { C }  X.  _V ) ) `  <. C ,  x >. )  =  x )
6867adantr 466 . . . . . . . 8  |-  ( ( C  e.  A  /\  x  e.  B )  ->  ( ( 2nd  |`  ( { C }  X.  _V ) ) `  <. C ,  x >. )  =  x )
69 f1ocnvfv 6188 . . . . . . . 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 5881 . . . . . 6  |-  ( ( C  e.  A  /\  x  e.  B )  ->  ( F `  ( `' ( 2nd  |`  ( { C }  X.  _V ) ) `  x
) )  =  ( F `  <. C ,  x >. ) )
7271adantll 718 . . . . 5  |-  ( ( ( F  Fn  ( A  X.  B )  /\  C  e.  A )  /\  x  e.  B
)  ->  ( F `  ( `' ( 2nd  |`  ( { C }  X.  _V ) ) `  x ) )  =  ( F `  <. C ,  x >. )
)
73 df-ov 6304 . . . . 5  |-  ( C F x )  =  ( F `  <. C ,  x >. )
7472, 73syl6eqr 2481 . . . 4  |-  ( ( ( F  Fn  ( A  X.  B )  /\  C  e.  A )  /\  x  e.  B
)  ->  ( F `  ( `' ( 2nd  |`  ( { C }  X.  _V ) ) `  x ) )  =  ( C F x ) )
7555, 74eqtrd 2463 . . 3  |-  ( ( ( F  Fn  ( A  X.  B )  /\  C  e.  A )  /\  x  e.  B
)  ->  ( G `  x )  =  ( C F x ) )
7675mpteq2dva 4507 . 2  |-  ( ( F  Fn  ( A  X.  B )  /\  C  e.  A )  ->  ( x  e.  B  |->  ( G `  x
) )  =  ( x  e.  B  |->  ( C F x ) ) )
7745, 76eqtrd 2463 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 370    = wceq 1437    e. wcel 1868   _Vcvv 3081    i^i cin 3435    C_ wss 3436   {csn 3996   <.cop 4002    |-> cmpt 4479    X. cxp 4847   `'ccnv 4848   dom cdm 4849   ran crn 4850    |` cres 4851   "cima 4852    o. ccom 4853   Fun wfun 5591    Fn wfn 5592   -onto->wfo 5595   -1-1-onto->wf1o 5596   ` cfv 5597  (class class class)co 6301   2ndc2nd 6802
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-8 1870  ax-9 1872  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400  ax-sep 4543  ax-nul 4551  ax-pow 4598  ax-pr 4656  ax-un 6593
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2269  df-mo 2270  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-ne 2620  df-ral 2780  df-rex 2781  df-rab 2784  df-v 3083  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-nul 3762  df-if 3910  df-sn 3997  df-pr 3999  df-op 4003  df-uni 4217  df-iun 4298  df-br 4421  df-opab 4480  df-mpt 4481  df-id 4764  df-xp 4855  df-rel 4856  df-cnv 4857  df-co 4858  df-dm 4859  df-rn 4860  df-res 4861  df-ima 4862  df-iota 5561  df-fun 5599  df-fn 5600  df-f 5601  df-f1 5602  df-fo 5603  df-f1o 5604  df-fv 5605  df-ov 6304  df-1st 6803  df-2nd 6804
This theorem is referenced by:  curry1val  6896  curry1f  6897
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