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Theorem curry2val 6870
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 6868 . . 3  |-  ( ( F  Fn  ( A  X.  B )  /\  C  e.  B )  ->  G  =  ( x  e.  A  |->  ( x F C ) ) )
32fveq1d 5850 . 2  |-  ( ( F  Fn  ( A  X.  B )  /\  C  e.  B )  ->  ( G `  D
)  =  ( ( x  e.  A  |->  ( x F C ) ) `  D ) )
4 eqid 2454 . . . . . . . . . . 11  |-  ( x  e.  A  |->  ( x F C ) )  =  ( x  e.  A  |->  ( x F C ) )
54dmmptss 5486 . . . . . . . . . 10  |-  dom  (
x  e.  A  |->  ( x F C ) )  C_  A
65sseli 3485 . . . . . . . . 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 5872 . . . . . . . 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 464 . . . . . 6  |-  ( ( F  Fn  ( A  X.  B )  /\  -.  D  e.  A
)  ->  ( (
x  e.  A  |->  ( x F C ) ) `  D )  =  (/) )
11 fndm 5662 . . . . . . 7  |-  ( F  Fn  ( A  X.  B )  ->  dom  F  =  ( A  X.  B ) )
12 simpl 455 . . . . . . . 8  |-  ( ( D  e.  A  /\  C  e.  B )  ->  D  e.  A )
1312con3i 135 . . . . . . 7  |-  ( -.  D  e.  A  ->  -.  ( D  e.  A  /\  C  e.  B
) )
14 ndmovg 6431 . . . . . . 7  |-  ( ( dom  F  =  ( A  X.  B )  /\  -.  ( D  e.  A  /\  C  e.  B ) )  -> 
( D F C )  =  (/) )
1511, 13, 14syl2an 475 . . . . . 6  |-  ( ( F  Fn  ( A  X.  B )  /\  -.  D  e.  A
)  ->  ( D F C )  =  (/) )
1610, 15eqtr4d 2498 . . . . 5  |-  ( ( F  Fn  ( A  X.  B )  /\  -.  D  e.  A
)  ->  ( (
x  e.  A  |->  ( x F C ) ) `  D )  =  ( D F C ) )
1716ex 432 . . . 4  |-  ( F  Fn  ( A  X.  B )  ->  ( -.  D  e.  A  ->  ( ( x  e.  A  |->  ( x F C ) ) `  D )  =  ( D F C ) ) )
1817adantr 463 . . 3  |-  ( ( F  Fn  ( A  X.  B )  /\  C  e.  B )  ->  ( -.  D  e.  A  ->  ( (
x  e.  A  |->  ( x F C ) ) `  D )  =  ( D F C ) ) )
19 oveq1 6277 . . . 4  |-  ( x  =  D  ->  (
x F C )  =  ( D F C ) )
20 ovex 6298 . . . 4  |-  ( D F C )  e. 
_V
2119, 4, 20fvmpt 5931 . . 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 2495 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 367    = wceq 1398    e. wcel 1823   _Vcvv 3106   (/)c0 3783   {csn 4016    |-> cmpt 4497    X. cxp 4986   `'ccnv 4987   dom cdm 4988    |` cres 4990    o. ccom 4992    Fn wfn 5565   ` cfv 5570  (class class class)co 6270   1stc1st 6771
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-ov 6273  df-1st 6773  df-2nd 6774
This theorem is referenced by:  curry2ima  27755
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