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Theorem curry1f 6892
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 6444 . . . 4  |-  ( ( F : ( A  X.  B ) --> D  /\  C  e.  A  /\  x  e.  B
)  ->  ( C F x )  e.  D )
213expa 1205 . . 3  |-  ( ( ( F : ( A  X.  B ) --> D  /\  C  e.  A )  /\  x  e.  B )  ->  ( C F x )  e.  D )
3 eqid 2420 . . 3  |-  ( x  e.  B  |->  ( C F x ) )  =  ( x  e.  B  |->  ( C F x ) )
42, 3fmptd 6052 . 2  |-  ( ( F : ( A  X.  B ) --> D  /\  C  e.  A
)  ->  ( x  e.  B  |->  ( C F x ) ) : B --> D )
5 ffn 5737 . . . 4  |-  ( F : ( A  X.  B ) --> D  ->  F  Fn  ( A  X.  B ) )
6 curry1.1 . . . . 5  |-  G  =  ( F  o.  `' ( 2nd  |`  ( { C }  X.  _V )
) )
76curry1 6890 . . . 4  |-  ( ( F  Fn  ( A  X.  B )  /\  C  e.  A )  ->  G  =  ( x  e.  B  |->  ( C F x ) ) )
85, 7sylan 473 . . 3  |-  ( ( F : ( A  X.  B ) --> D  /\  C  e.  A
)  ->  G  =  ( x  e.  B  |->  ( C F x ) ) )
98feq1d 5723 . 2  |-  ( ( F : ( A  X.  B ) --> D  /\  C  e.  A
)  ->  ( G : B --> D  <->  ( x  e.  B  |->  ( C F x ) ) : B --> D ) )
104, 9mpbird 235 1  |-  ( ( F : ( A  X.  B ) --> D  /\  C  e.  A
)  ->  G : B
--> D )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370    = wceq 1437    e. wcel 1867   _Vcvv 3078   {csn 3993    |-> cmpt 4475    X. cxp 4843   `'ccnv 4844    |` cres 4847    o. ccom 4849    Fn wfn 5587   -->wf 5588  (class class class)co 6296   2ndc2nd 6797
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 1838  ax-8 1869  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-sep 4539  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6588
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 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-ral 2778  df-rex 2779  df-rab 2782  df-v 3080  df-sbc 3297  df-csb 3393  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-nul 3759  df-if 3907  df-sn 3994  df-pr 3996  df-op 4000  df-uni 4214  df-iun 4295  df-br 4418  df-opab 4476  df-mpt 4477  df-id 4760  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5556  df-fun 5594  df-fn 5595  df-f 5596  df-f1 5597  df-fo 5598  df-f1o 5599  df-fv 5600  df-ov 6299  df-1st 6798  df-2nd 6799
This theorem is referenced by:  nvinvfval  26147
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