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Theorem curry2val 6660
Description: The value 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
curry2val  |-  ( ( F  Fn  ( A  X.  B )  /\  C  e.  B )  ->  ( G `  D
)  =  ( D F C ) )

Proof of Theorem curry2val
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 curry2.1 . . . 4  |-  G  =  ( F  o.  `' ( 1st  |`  ( _V  X.  { C } ) ) )
21curry2 6658 . . 3  |-  ( ( F  Fn  ( A  X.  B )  /\  C  e.  B )  ->  G  =  ( x  e.  A  |->  ( x F C ) ) )
32fveq1d 5683 . 2  |-  ( ( F  Fn  ( A  X.  B )  /\  C  e.  B )  ->  ( G `  D
)  =  ( ( x  e.  A  |->  ( x F C ) ) `  D ) )
4 eqid 2435 . . . . . . . . . . 11  |-  ( x  e.  A  |->  ( x F C ) )  =  ( x  e.  A  |->  ( x F C ) )
54dmmptss 5324 . . . . . . . . . 10  |-  dom  (
x  e.  A  |->  ( x F C ) )  C_  A
65sseli 3342 . . . . . . . . 9  |-  ( D  e.  dom  ( x  e.  A  |->  ( x F C ) )  ->  D  e.  A
)
76con3i 135 . . . . . . . 8  |-  ( -.  D  e.  A  ->  -.  D  e.  dom  ( x  e.  A  |->  ( x F C ) ) )
8 ndmfv 5704 . . . . . . . 8  |-  ( -.  D  e.  dom  (
x  e.  A  |->  ( x F C ) )  ->  ( (
x  e.  A  |->  ( x F C ) ) `  D )  =  (/) )
97, 8syl 16 . . . . . . 7  |-  ( -.  D  e.  A  -> 
( ( x  e.  A  |->  ( x F C ) ) `  D )  =  (/) )
109adantl 463 . . . . . 6  |-  ( ( F  Fn  ( A  X.  B )  /\  -.  D  e.  A
)  ->  ( (
x  e.  A  |->  ( x F C ) ) `  D )  =  (/) )
11 fndm 5500 . . . . . . 7  |-  ( F  Fn  ( A  X.  B )  ->  dom  F  =  ( A  X.  B ) )
12 simpl 454 . . . . . . . 8  |-  ( ( D  e.  A  /\  C  e.  B )  ->  D  e.  A )
1312con3i 135 . . . . . . 7  |-  ( -.  D  e.  A  ->  -.  ( D  e.  A  /\  C  e.  B
) )
14 ndmovg 6237 . . . . . . 7  |-  ( ( dom  F  =  ( A  X.  B )  /\  -.  ( D  e.  A  /\  C  e.  B ) )  -> 
( D F C )  =  (/) )
1511, 13, 14syl2an 474 . . . . . 6  |-  ( ( F  Fn  ( A  X.  B )  /\  -.  D  e.  A
)  ->  ( D F C )  =  (/) )
1610, 15eqtr4d 2470 . . . . 5  |-  ( ( F  Fn  ( A  X.  B )  /\  -.  D  e.  A
)  ->  ( (
x  e.  A  |->  ( x F C ) ) `  D )  =  ( D F C ) )
1716ex 434 . . . 4  |-  ( F  Fn  ( A  X.  B )  ->  ( -.  D  e.  A  ->  ( ( x  e.  A  |->  ( x F C ) ) `  D )  =  ( D F C ) ) )
1817adantr 462 . . 3  |-  ( ( F  Fn  ( A  X.  B )  /\  C  e.  B )  ->  ( -.  D  e.  A  ->  ( (
x  e.  A  |->  ( x F C ) ) `  D )  =  ( D F C ) ) )
19 oveq1 6089 . . . 4  |-  ( x  =  D  ->  (
x F C )  =  ( D F C ) )
20 ovex 6107 . . . 4  |-  ( D F C )  e. 
_V
2119, 4, 20fvmpt 5764 . . 3  |-  ( D  e.  A  ->  (
( x  e.  A  |->  ( x F C ) ) `  D
)  =  ( D F C ) )
2218, 21pm2.61d2 160 . 2  |-  ( ( F  Fn  ( A  X.  B )  /\  C  e.  B )  ->  ( ( x  e.  A  |->  ( x F C ) ) `  D )  =  ( D F C ) )
233, 22eqtrd 2467 1  |-  ( ( F  Fn  ( A  X.  B )  /\  C  e.  B )  ->  ( G `  D
)  =  ( D F C ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    = wceq 1364    e. wcel 1757   _Vcvv 2964   (/)c0 3627   {csn 3867    e. cmpt 4340    X. cxp 4827   `'ccnv 4828   dom cdm 4829    |` cres 4831    o. ccom 4833    Fn wfn 5403   ` cfv 5408  (class class class)co 6082   1stc1st 6566
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 1671  ax-6 1709  ax-7 1729  ax-8 1759  ax-9 1761  ax-10 1776  ax-11 1781  ax-12 1793  ax-13 1944  ax-ext 2416  ax-sep 4403  ax-nul 4411  ax-pow 4460  ax-pr 4521  ax-un 6363
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1702  df-eu 2260  df-mo 2261  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2966  df-sbc 3178  df-csb 3279  df-dif 3321  df-un 3323  df-in 3325  df-ss 3332  df-nul 3628  df-if 3782  df-sn 3868  df-pr 3870  df-op 3874  df-uni 4082  df-iun 4163  df-br 4283  df-opab 4341  df-mpt 4342  df-id 4625  df-xp 4835  df-rel 4836  df-cnv 4837  df-co 4838  df-dm 4839  df-rn 4840  df-res 4841  df-ima 4842  df-iota 5371  df-fun 5410  df-fn 5411  df-f 5412  df-f1 5413  df-fo 5414  df-f1o 5415  df-fv 5416  df-ov 6085  df-1st 6568  df-2nd 6569
This theorem is referenced by:  curry2ima  25829
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