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Theorem curry1f 6878
Description: Functionality of a curried function with a constant first argument. (Contributed by NM, 29-Mar-2008.)
Hypothesis
Ref Expression
curry1.1  |-  G  =  ( F  o.  `' ( 2nd  |`  ( { C }  X.  _V )
) )
Assertion
Ref Expression
curry1f  |-  ( ( F : ( A  X.  B ) --> D  /\  C  e.  A
)  ->  G : B
--> D )

Proof of Theorem curry1f
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 fovrn 6430 . . . 4  |-  ( ( F : ( A  X.  B ) --> D  /\  C  e.  A  /\  x  e.  B
)  ->  ( C F x )  e.  D )
213expa 1196 . . 3  |-  ( ( ( F : ( A  X.  B ) --> D  /\  C  e.  A )  /\  x  e.  B )  ->  ( C F x )  e.  D )
3 eqid 2467 . . 3  |-  ( x  e.  B  |->  ( C F x ) )  =  ( x  e.  B  |->  ( C F x ) )
42, 3fmptd 6046 . 2  |-  ( ( F : ( A  X.  B ) --> D  /\  C  e.  A
)  ->  ( x  e.  B  |->  ( C F x ) ) : B --> D )
5 ffn 5731 . . . 4  |-  ( F : ( A  X.  B ) --> D  ->  F  Fn  ( A  X.  B ) )
6 curry1.1 . . . . 5  |-  G  =  ( F  o.  `' ( 2nd  |`  ( { C }  X.  _V )
) )
76curry1 6876 . . . 4  |-  ( ( F  Fn  ( A  X.  B )  /\  C  e.  A )  ->  G  =  ( x  e.  B  |->  ( C F x ) ) )
85, 7sylan 471 . . 3  |-  ( ( F : ( A  X.  B ) --> D  /\  C  e.  A
)  ->  G  =  ( x  e.  B  |->  ( C F x ) ) )
98feq1d 5717 . 2  |-  ( ( F : ( A  X.  B ) --> D  /\  C  e.  A
)  ->  ( G : B --> D  <->  ( x  e.  B  |->  ( C F x ) ) : B --> D ) )
104, 9mpbird 232 1  |-  ( ( F : ( A  X.  B ) --> D  /\  C  e.  A
)  ->  G : B
--> D )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767   _Vcvv 3113   {csn 4027    |-> cmpt 4505    X. cxp 4997   `'ccnv 4998    |` cres 5001    o. ccom 5003    Fn wfn 5583   -->wf 5584  (class class class)co 6285   2ndc2nd 6784
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6577
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-ov 6288  df-1st 6785  df-2nd 6786
This theorem is referenced by:  nvinvfval  25308
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