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Theorem curry1val 6439
Description: The value of a curried function with a constant first argument. (Contributed by NM, 28-Mar-2008.) (Revised by Mario Carneiro, 26-Apr-2015.)
Hypothesis
Ref Expression
curry1.1  |-  G  =  ( F  o.  `' ( 2nd  |`  ( { C }  X.  _V )
) )
Assertion
Ref Expression
curry1val  |-  ( ( F  Fn  ( A  X.  B )  /\  C  e.  A )  ->  ( G `  D
)  =  ( C F D ) )

Proof of Theorem curry1val
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 curry1.1 . . . 4  |-  G  =  ( F  o.  `' ( 2nd  |`  ( { C }  X.  _V )
) )
21curry1 6438 . . 3  |-  ( ( F  Fn  ( A  X.  B )  /\  C  e.  A )  ->  G  =  ( x  e.  B  |->  ( C F x ) ) )
32fveq1d 5730 . 2  |-  ( ( F  Fn  ( A  X.  B )  /\  C  e.  A )  ->  ( G `  D
)  =  ( ( x  e.  B  |->  ( C F x ) ) `  D ) )
4 eqid 2436 . . . . . . . . . 10  |-  ( x  e.  B  |->  ( C F x ) )  =  ( x  e.  B  |->  ( C F x ) )
54dmmptss 5366 . . . . . . . . 9  |-  dom  (
x  e.  B  |->  ( C F x ) )  C_  B
65sseli 3344 . . . . . . . 8  |-  ( D  e.  dom  ( x  e.  B  |->  ( C F x ) )  ->  D  e.  B
)
76con3i 129 . . . . . . 7  |-  ( -.  D  e.  B  ->  -.  D  e.  dom  ( x  e.  B  |->  ( C F x ) ) )
8 ndmfv 5755 . . . . . . 7  |-  ( -.  D  e.  dom  (
x  e.  B  |->  ( C F x ) )  ->  ( (
x  e.  B  |->  ( C F x ) ) `  D )  =  (/) )
97, 8syl 16 . . . . . 6  |-  ( -.  D  e.  B  -> 
( ( x  e.  B  |->  ( C F x ) ) `  D )  =  (/) )
109adantl 453 . . . . 5  |-  ( ( ( F  Fn  ( A  X.  B )  /\  C  e.  A )  /\  -.  D  e.  B
)  ->  ( (
x  e.  B  |->  ( C F x ) ) `  D )  =  (/) )
11 fndm 5544 . . . . . . 7  |-  ( F  Fn  ( A  X.  B )  ->  dom  F  =  ( A  X.  B ) )
1211adantr 452 . . . . . 6  |-  ( ( F  Fn  ( A  X.  B )  /\  C  e.  A )  ->  dom  F  =  ( A  X.  B ) )
13 simpr 448 . . . . . . 7  |-  ( ( C  e.  A  /\  D  e.  B )  ->  D  e.  B )
1413con3i 129 . . . . . 6  |-  ( -.  D  e.  B  ->  -.  ( C  e.  A  /\  D  e.  B
) )
15 ndmovg 6230 . . . . . 6  |-  ( ( dom  F  =  ( A  X.  B )  /\  -.  ( C  e.  A  /\  D  e.  B ) )  -> 
( C F D )  =  (/) )
1612, 14, 15syl2an 464 . . . . 5  |-  ( ( ( F  Fn  ( A  X.  B )  /\  C  e.  A )  /\  -.  D  e.  B
)  ->  ( C F D )  =  (/) )
1710, 16eqtr4d 2471 . . . 4  |-  ( ( ( F  Fn  ( A  X.  B )  /\  C  e.  A )  /\  -.  D  e.  B
)  ->  ( (
x  e.  B  |->  ( C F x ) ) `  D )  =  ( C F D ) )
1817ex 424 . . 3  |-  ( ( F  Fn  ( A  X.  B )  /\  C  e.  A )  ->  ( -.  D  e.  B  ->  ( (
x  e.  B  |->  ( C F x ) ) `  D )  =  ( C F D ) ) )
19 oveq2 6089 . . . 4  |-  ( x  =  D  ->  ( C F x )  =  ( C F D ) )
20 ovex 6106 . . . 4  |-  ( C F D )  e. 
_V
2119, 4, 20fvmpt 5806 . . 3  |-  ( D  e.  B  ->  (
( x  e.  B  |->  ( C F x ) ) `  D
)  =  ( C F D ) )
2218, 21pm2.61d2 154 . 2  |-  ( ( F  Fn  ( A  X.  B )  /\  C  e.  A )  ->  ( ( x  e.  B  |->  ( C F x ) ) `  D )  =  ( C F D ) )
233, 22eqtrd 2468 1  |-  ( ( F  Fn  ( A  X.  B )  /\  C  e.  A )  ->  ( G `  D
)  =  ( C F D ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   _Vcvv 2956   (/)c0 3628   {csn 3814    e. cmpt 4266    X. cxp 4876   `'ccnv 4877   dom cdm 4878    |` cres 4880    o. ccom 4882    Fn wfn 5449   ` cfv 5454  (class class class)co 6081   2ndc2nd 6348
This theorem is referenced by:  nvinvfval  22121  hhssabloi  22762
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-1st 6349  df-2nd 6350
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