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Theorem curry2val 6890
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 6888 . . 3  |-  ( ( F  Fn  ( A  X.  B )  /\  C  e.  B )  ->  G  =  ( x  e.  A  |->  ( x F C ) ) )
32fveq1d 5873 . 2  |-  ( ( F  Fn  ( A  X.  B )  /\  C  e.  B )  ->  ( G `  D
)  =  ( ( x  e.  A  |->  ( x F C ) ) `  D ) )
4 eqid 2467 . . . . . . . . . . 11  |-  ( x  e.  A  |->  ( x F C ) )  =  ( x  e.  A  |->  ( x F C ) )
54dmmptss 5508 . . . . . . . . . 10  |-  dom  (
x  e.  A  |->  ( x F C ) )  C_  A
65sseli 3505 . . . . . . . . 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 5895 . . . . . . . 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 466 . . . . . 6  |-  ( ( F  Fn  ( A  X.  B )  /\  -.  D  e.  A
)  ->  ( (
x  e.  A  |->  ( x F C ) ) `  D )  =  (/) )
11 fndm 5685 . . . . . . 7  |-  ( F  Fn  ( A  X.  B )  ->  dom  F  =  ( A  X.  B ) )
12 simpl 457 . . . . . . . 8  |-  ( ( D  e.  A  /\  C  e.  B )  ->  D  e.  A )
1312con3i 135 . . . . . . 7  |-  ( -.  D  e.  A  ->  -.  ( D  e.  A  /\  C  e.  B
) )
14 ndmovg 6452 . . . . . . 7  |-  ( ( dom  F  =  ( A  X.  B )  /\  -.  ( D  e.  A  /\  C  e.  B ) )  -> 
( D F C )  =  (/) )
1511, 13, 14syl2an 477 . . . . . 6  |-  ( ( F  Fn  ( A  X.  B )  /\  -.  D  e.  A
)  ->  ( D F C )  =  (/) )
1610, 15eqtr4d 2511 . . . . 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 465 . . 3  |-  ( ( F  Fn  ( A  X.  B )  /\  C  e.  B )  ->  ( -.  D  e.  A  ->  ( (
x  e.  A  |->  ( x F C ) ) `  D )  =  ( D F C ) ) )
19 oveq1 6301 . . . 4  |-  ( x  =  D  ->  (
x F C )  =  ( D F C ) )
20 ovex 6319 . . . 4  |-  ( D F C )  e. 
_V
2119, 4, 20fvmpt 5956 . . 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 2508 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 1379    e. wcel 1767   _Vcvv 3118   (/)c0 3790   {csn 4032    |-> cmpt 4510    X. cxp 5002   `'ccnv 5003   dom cdm 5004    |` cres 5006    o. ccom 5008    Fn wfn 5588   ` cfv 5593  (class class class)co 6294   1stc1st 6792
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4573  ax-nul 4581  ax-pow 4630  ax-pr 4691  ax-un 6586
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4251  df-iun 4332  df-br 4453  df-opab 4511  df-mpt 4512  df-id 4800  df-xp 5010  df-rel 5011  df-cnv 5012  df-co 5013  df-dm 5014  df-rn 5015  df-res 5016  df-ima 5017  df-iota 5556  df-fun 5595  df-fn 5596  df-f 5597  df-f1 5598  df-fo 5599  df-f1o 5600  df-fv 5601  df-ov 6297  df-1st 6794  df-2nd 6795
This theorem is referenced by:  curry2ima  27317
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