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Theorem curry1val 6380
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 6379 . . 3  |-  ( ( F  Fn  ( A  X.  B )  /\  C  e.  A )  ->  G  =  ( x  e.  B  |->  ( C F x ) ) )
32fveq1d 5672 . 2  |-  ( ( F  Fn  ( A  X.  B )  /\  C  e.  A )  ->  ( G `  D
)  =  ( ( x  e.  B  |->  ( C F x ) ) `  D ) )
4 eqid 2389 . . . . . . . . . 10  |-  ( x  e.  B  |->  ( C F x ) )  =  ( x  e.  B  |->  ( C F x ) )
54dmmptss 5308 . . . . . . . . 9  |-  dom  (
x  e.  B  |->  ( C F x ) )  C_  B
65sseli 3289 . . . . . . . 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 5697 . . . . . . 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 5486 . . . . . . 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 6171 . . . . . 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 2424 . . . 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 6030 . . . 4  |-  ( x  =  D  ->  ( C F x )  =  ( C F D ) )
20 ovex 6047 . . . 4  |-  ( C F D )  e. 
_V
2119, 4, 20fvmpt 5747 . . 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 2421 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 1649    e. wcel 1717   _Vcvv 2901   (/)c0 3573   {csn 3759    e. cmpt 4209    X. cxp 4818   `'ccnv 4819   dom cdm 4820    |` cres 4822    o. ccom 4824    Fn wfn 5391   ` cfv 5396  (class class class)co 6022   2ndc2nd 6289
This theorem is referenced by:  nvinvfval  21971  hhssabloi  22612
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 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-sep 4273  ax-nul 4281  ax-pow 4320  ax-pr 4346  ax-un 4643
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 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-ral 2656  df-rex 2657  df-rab 2660  df-v 2903  df-sbc 3107  df-csb 3197  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-nul 3574  df-if 3685  df-sn 3765  df-pr 3766  df-op 3768  df-uni 3960  df-iun 4039  df-br 4156  df-opab 4210  df-mpt 4211  df-id 4441  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-res 4832  df-ima 4833  df-iota 5360  df-fun 5398  df-fn 5399  df-f 5400  df-f1 5401  df-fo 5402  df-f1o 5403  df-fv 5404  df-ov 6025  df-1st 6290  df-2nd 6291
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