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Theorem curry2f 6881
Description: Functionality of a curried function with a constant second argument. (Contributed by NM, 16-Dec-2008.)
Hypothesis
Ref Expression
curry2.1  |-  G  =  ( F  o.  `' ( 1st  |`  ( _V  X.  { C } ) ) )
Assertion
Ref Expression
curry2f  |-  ( ( F : ( A  X.  B ) --> D  /\  C  e.  B
)  ->  G : A
--> D )

Proof of Theorem curry2f
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 fovrn 6430 . . . . 5  |-  ( ( F : ( A  X.  B ) --> D  /\  x  e.  A  /\  C  e.  B
)  ->  ( x F C )  e.  D
)
213com23 1203 . . . 4  |-  ( ( F : ( A  X.  B ) --> D  /\  C  e.  B  /\  x  e.  A
)  ->  ( x F C )  e.  D
)
323expa 1197 . . 3  |-  ( ( ( F : ( A  X.  B ) --> D  /\  C  e.  B )  /\  x  e.  A )  ->  (
x F C )  e.  D )
4 eqid 2443 . . 3  |-  ( x  e.  A  |->  ( x F C ) )  =  ( x  e.  A  |->  ( x F C ) )
53, 4fmptd 6040 . 2  |-  ( ( F : ( A  X.  B ) --> D  /\  C  e.  B
)  ->  ( x  e.  A  |->  ( x F C ) ) : A --> D )
6 ffn 5721 . . . 4  |-  ( F : ( A  X.  B ) --> D  ->  F  Fn  ( A  X.  B ) )
7 curry2.1 . . . . 5  |-  G  =  ( F  o.  `' ( 1st  |`  ( _V  X.  { C } ) ) )
87curry2 6880 . . . 4  |-  ( ( F  Fn  ( A  X.  B )  /\  C  e.  B )  ->  G  =  ( x  e.  A  |->  ( x F C ) ) )
96, 8sylan 471 . . 3  |-  ( ( F : ( A  X.  B ) --> D  /\  C  e.  B
)  ->  G  =  ( x  e.  A  |->  ( x F C ) ) )
109feq1d 5707 . 2  |-  ( ( F : ( A  X.  B ) --> D  /\  C  e.  B
)  ->  ( G : A --> D  <->  ( x  e.  A  |->  ( x F C ) ) : A --> D ) )
115, 10mpbird 232 1  |-  ( ( F : ( A  X.  B ) --> D  /\  C  e.  B
)  ->  G : A
--> D )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1383    e. wcel 1804   _Vcvv 3095   {csn 4014    |-> cmpt 4495    X. cxp 4987   `'ccnv 4988    |` cres 4991    o. ccom 4993    Fn wfn 5573   -->wf 5574  (class class class)co 6281   1stc1st 6783
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rex 2799  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3771  df-if 3927  df-sn 4015  df-pr 4017  df-op 4021  df-uni 4235  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-id 4785  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-ov 6284  df-1st 6785  df-2nd 6786
This theorem is referenced by:  curry2ima  27502
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