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Theorem curry2val 6402
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 6400 . . 3  |-  ( ( F  Fn  ( A  X.  B )  /\  C  e.  B )  ->  G  =  ( x  e.  A  |->  ( x F C ) ) )
32fveq1d 5689 . 2  |-  ( ( F  Fn  ( A  X.  B )  /\  C  e.  B )  ->  ( G `  D
)  =  ( ( x  e.  A  |->  ( x F C ) ) `  D ) )
4 eqid 2404 . . . . . . . . . . 11  |-  ( x  e.  A  |->  ( x F C ) )  =  ( x  e.  A  |->  ( x F C ) )
54dmmptss 5325 . . . . . . . . . 10  |-  dom  (
x  e.  A  |->  ( x F C ) )  C_  A
65sseli 3304 . . . . . . . . 9  |-  ( D  e.  dom  ( x  e.  A  |->  ( x F C ) )  ->  D  e.  A
)
76con3i 129 . . . . . . . 8  |-  ( -.  D  e.  A  ->  -.  D  e.  dom  ( x  e.  A  |->  ( x F C ) ) )
8 ndmfv 5714 . . . . . . . 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 453 . . . . . 6  |-  ( ( F  Fn  ( A  X.  B )  /\  -.  D  e.  A
)  ->  ( (
x  e.  A  |->  ( x F C ) ) `  D )  =  (/) )
11 fndm 5503 . . . . . . 7  |-  ( F  Fn  ( A  X.  B )  ->  dom  F  =  ( A  X.  B ) )
12 simpl 444 . . . . . . . 8  |-  ( ( D  e.  A  /\  C  e.  B )  ->  D  e.  A )
1312con3i 129 . . . . . . 7  |-  ( -.  D  e.  A  ->  -.  ( D  e.  A  /\  C  e.  B
) )
14 ndmovg 6189 . . . . . . 7  |-  ( ( dom  F  =  ( A  X.  B )  /\  -.  ( D  e.  A  /\  C  e.  B ) )  -> 
( D F C )  =  (/) )
1511, 13, 14syl2an 464 . . . . . 6  |-  ( ( F  Fn  ( A  X.  B )  /\  -.  D  e.  A
)  ->  ( D F C )  =  (/) )
1610, 15eqtr4d 2439 . . . . 5  |-  ( ( F  Fn  ( A  X.  B )  /\  -.  D  e.  A
)  ->  ( (
x  e.  A  |->  ( x F C ) ) `  D )  =  ( D F C ) )
1716ex 424 . . . 4  |-  ( F  Fn  ( A  X.  B )  ->  ( -.  D  e.  A  ->  ( ( x  e.  A  |->  ( x F C ) ) `  D )  =  ( D F C ) ) )
1817adantr 452 . . 3  |-  ( ( F  Fn  ( A  X.  B )  /\  C  e.  B )  ->  ( -.  D  e.  A  ->  ( (
x  e.  A  |->  ( x F C ) ) `  D )  =  ( D F C ) ) )
19 oveq1 6047 . . . 4  |-  ( x  =  D  ->  (
x F C )  =  ( D F C ) )
20 ovex 6065 . . . 4  |-  ( D F C )  e. 
_V
2119, 4, 20fvmpt 5765 . . 3  |-  ( D  e.  A  ->  (
( x  e.  A  |->  ( x F C ) ) `  D
)  =  ( D F C ) )
2218, 21pm2.61d2 154 . 2  |-  ( ( F  Fn  ( A  X.  B )  /\  C  e.  B )  ->  ( ( x  e.  A  |->  ( x F C ) ) `  D )  =  ( D F C ) )
233, 22eqtrd 2436 1  |-  ( ( F  Fn  ( A  X.  B )  /\  C  e.  B )  ->  ( G `  D
)  =  ( D F C ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1721   _Vcvv 2916   (/)c0 3588   {csn 3774    e. cmpt 4226    X. cxp 4835   `'ccnv 4836   dom cdm 4837    |` cres 4839    o. ccom 4841    Fn wfn 5408   ` cfv 5413  (class class class)co 6040   1stc1st 6306
This theorem is referenced by:  curry2ima  24050
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-1st 6308  df-2nd 6309
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