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Theorem curry1val 6227
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 6226 . . 3  |-  ( ( F  Fn  ( A  X.  B )  /\  C  e.  A )  ->  G  =  ( x  e.  B  |->  ( C F x ) ) )
32fveq1d 5543 . 2  |-  ( ( F  Fn  ( A  X.  B )  /\  C  e.  A )  ->  ( G `  D
)  =  ( ( x  e.  B  |->  ( C F x ) ) `  D ) )
4 eqid 2296 . . . . . . . . . 10  |-  ( x  e.  B  |->  ( C F x ) )  =  ( x  e.  B  |->  ( C F x ) )
54dmmptss 5185 . . . . . . . . 9  |-  dom  (
x  e.  B  |->  ( C F x ) )  C_  B
65sseli 3189 . . . . . . . 8  |-  ( D  e.  dom  ( x  e.  B  |->  ( C F x ) )  ->  D  e.  B
)
76con3i 127 . . . . . . 7  |-  ( -.  D  e.  B  ->  -.  D  e.  dom  ( x  e.  B  |->  ( C F x ) ) )
8 ndmfv 5568 . . . . . . 7  |-  ( -.  D  e.  dom  (
x  e.  B  |->  ( C F x ) )  ->  ( (
x  e.  B  |->  ( C F x ) ) `  D )  =  (/) )
97, 8syl 15 . . . . . 6  |-  ( -.  D  e.  B  -> 
( ( x  e.  B  |->  ( C F x ) ) `  D )  =  (/) )
109adantl 452 . . . . 5  |-  ( ( ( F  Fn  ( A  X.  B )  /\  C  e.  A )  /\  -.  D  e.  B
)  ->  ( (
x  e.  B  |->  ( C F x ) ) `  D )  =  (/) )
11 fndm 5359 . . . . . . 7  |-  ( F  Fn  ( A  X.  B )  ->  dom  F  =  ( A  X.  B ) )
1211adantr 451 . . . . . 6  |-  ( ( F  Fn  ( A  X.  B )  /\  C  e.  A )  ->  dom  F  =  ( A  X.  B ) )
13 simpr 447 . . . . . . 7  |-  ( ( C  e.  A  /\  D  e.  B )  ->  D  e.  B )
1413con3i 127 . . . . . 6  |-  ( -.  D  e.  B  ->  -.  ( C  e.  A  /\  D  e.  B
) )
15 ndmovg 6019 . . . . . 6  |-  ( ( dom  F  =  ( A  X.  B )  /\  -.  ( C  e.  A  /\  D  e.  B ) )  -> 
( C F D )  =  (/) )
1612, 14, 15syl2an 463 . . . . 5  |-  ( ( ( F  Fn  ( A  X.  B )  /\  C  e.  A )  /\  -.  D  e.  B
)  ->  ( C F D )  =  (/) )
1710, 16eqtr4d 2331 . . . 4  |-  ( ( ( F  Fn  ( A  X.  B )  /\  C  e.  A )  /\  -.  D  e.  B
)  ->  ( (
x  e.  B  |->  ( C F x ) ) `  D )  =  ( C F D ) )
1817ex 423 . . 3  |-  ( ( F  Fn  ( A  X.  B )  /\  C  e.  A )  ->  ( -.  D  e.  B  ->  ( (
x  e.  B  |->  ( C F x ) ) `  D )  =  ( C F D ) ) )
19 oveq2 5882 . . . 4  |-  ( x  =  D  ->  ( C F x )  =  ( C F D ) )
20 ovex 5899 . . . 4  |-  ( C F D )  e. 
_V
2119, 4, 20fvmpt 5618 . . 3  |-  ( D  e.  B  ->  (
( x  e.  B  |->  ( C F x ) ) `  D
)  =  ( C F D ) )
2218, 21pm2.61d2 152 . 2  |-  ( ( F  Fn  ( A  X.  B )  /\  C  e.  A )  ->  ( ( x  e.  B  |->  ( C F x ) ) `  D )  =  ( C F D ) )
233, 22eqtrd 2328 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 358    = wceq 1632    e. wcel 1696   _Vcvv 2801   (/)c0 3468   {csn 3653    e. cmpt 4093    X. cxp 4703   `'ccnv 4704   dom cdm 4705    |` cres 4707    o. ccom 4709    Fn wfn 5266   ` cfv 5271  (class class class)co 5874   2ndc2nd 6137
This theorem is referenced by:  nvinvfval  21214  hhssabloi  21855
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-1st 6138  df-2nd 6139
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