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Theorem curry2f 6869
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 6420 . . . . 5  |-  ( ( F : ( A  X.  B ) --> D  /\  x  e.  A  /\  C  e.  B
)  ->  ( x F C )  e.  D
)
213com23 1197 . . . 4  |-  ( ( F : ( A  X.  B ) --> D  /\  C  e.  B  /\  x  e.  A
)  ->  ( x F C )  e.  D
)
323expa 1191 . . 3  |-  ( ( ( F : ( A  X.  B ) --> D  /\  C  e.  B )  /\  x  e.  A )  ->  (
x F C )  e.  D )
4 eqid 2460 . . 3  |-  ( x  e.  A  |->  ( x F C ) )  =  ( x  e.  A  |->  ( x F C ) )
53, 4fmptd 6036 . 2  |-  ( ( F : ( A  X.  B ) --> D  /\  C  e.  B
)  ->  ( x  e.  A  |->  ( x F C ) ) : A --> D )
6 ffn 5722 . . . 4  |-  ( F : ( A  X.  B ) --> D  ->  F  Fn  ( A  X.  B ) )
7 curry2.1 . . . . 5  |-  G  =  ( F  o.  `' ( 1st  |`  ( _V  X.  { C } ) ) )
87curry2 6868 . . . 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 5708 . 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 1374    e. wcel 1762   _Vcvv 3106   {csn 4020    |-> cmpt 4498    X. cxp 4990   `'ccnv 4991    |` cres 4994    o. ccom 4996    Fn wfn 5574   -->wf 5575  (class class class)co 6275   1stc1st 6772
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-id 4788  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-ov 6278  df-1st 6774  df-2nd 6775
This theorem is referenced by:  curry2ima  27184
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