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Theorem curry1val 6908
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 6907 . . 3  |-  ( ( F  Fn  ( A  X.  B )  /\  C  e.  A )  ->  G  =  ( x  e.  B  |->  ( C F x ) ) )
32fveq1d 5881 . 2  |-  ( ( F  Fn  ( A  X.  B )  /\  C  e.  A )  ->  ( G `  D
)  =  ( ( x  e.  B  |->  ( C F x ) ) `  D ) )
4 eqid 2471 . . . . . . . . . 10  |-  ( x  e.  B  |->  ( C F x ) )  =  ( x  e.  B  |->  ( C F x ) )
54dmmptss 5338 . . . . . . . . 9  |-  dom  (
x  e.  B  |->  ( C F x ) )  C_  B
65sseli 3414 . . . . . . . 8  |-  ( D  e.  dom  ( x  e.  B  |->  ( C F x ) )  ->  D  e.  B
)
76con3i 142 . . . . . . 7  |-  ( -.  D  e.  B  ->  -.  D  e.  dom  ( x  e.  B  |->  ( C F x ) ) )
8 ndmfv 5903 . . . . . . 7  |-  ( -.  D  e.  dom  (
x  e.  B  |->  ( C F x ) )  ->  ( (
x  e.  B  |->  ( C F x ) ) `  D )  =  (/) )
97, 8syl 17 . . . . . 6  |-  ( -.  D  e.  B  -> 
( ( x  e.  B  |->  ( C F x ) ) `  D )  =  (/) )
109adantl 473 . . . . 5  |-  ( ( ( F  Fn  ( A  X.  B )  /\  C  e.  A )  /\  -.  D  e.  B
)  ->  ( (
x  e.  B  |->  ( C F x ) ) `  D )  =  (/) )
11 fndm 5685 . . . . . . 7  |-  ( F  Fn  ( A  X.  B )  ->  dom  F  =  ( A  X.  B ) )
1211adantr 472 . . . . . 6  |-  ( ( F  Fn  ( A  X.  B )  /\  C  e.  A )  ->  dom  F  =  ( A  X.  B ) )
13 simpr 468 . . . . . . 7  |-  ( ( C  e.  A  /\  D  e.  B )  ->  D  e.  B )
1413con3i 142 . . . . . 6  |-  ( -.  D  e.  B  ->  -.  ( C  e.  A  /\  D  e.  B
) )
15 ndmovg 6471 . . . . . 6  |-  ( ( dom  F  =  ( A  X.  B )  /\  -.  ( C  e.  A  /\  D  e.  B ) )  -> 
( C F D )  =  (/) )
1612, 14, 15syl2an 485 . . . . 5  |-  ( ( ( F  Fn  ( A  X.  B )  /\  C  e.  A )  /\  -.  D  e.  B
)  ->  ( C F D )  =  (/) )
1710, 16eqtr4d 2508 . . . 4  |-  ( ( ( F  Fn  ( A  X.  B )  /\  C  e.  A )  /\  -.  D  e.  B
)  ->  ( (
x  e.  B  |->  ( C F x ) ) `  D )  =  ( C F D ) )
1817ex 441 . . 3  |-  ( ( F  Fn  ( A  X.  B )  /\  C  e.  A )  ->  ( -.  D  e.  B  ->  ( (
x  e.  B  |->  ( C F x ) ) `  D )  =  ( C F D ) ) )
19 oveq2 6316 . . . 4  |-  ( x  =  D  ->  ( C F x )  =  ( C F D ) )
20 ovex 6336 . . . 4  |-  ( C F D )  e. 
_V
2119, 4, 20fvmpt 5963 . . 3  |-  ( D  e.  B  ->  (
( x  e.  B  |->  ( C F x ) ) `  D
)  =  ( C F D ) )
2218, 21pm2.61d2 165 . 2  |-  ( ( F  Fn  ( A  X.  B )  /\  C  e.  A )  ->  ( ( x  e.  B  |->  ( C F x ) ) `  D )  =  ( C F D ) )
233, 22eqtrd 2505 1  |-  ( ( F  Fn  ( A  X.  B )  /\  C  e.  A )  ->  ( G `  D
)  =  ( C F D ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 376    = wceq 1452    e. wcel 1904   _Vcvv 3031   (/)c0 3722   {csn 3959    |-> cmpt 4454    X. cxp 4837   `'ccnv 4838   dom cdm 4839    |` cres 4841    o. ccom 4843    Fn wfn 5584   ` cfv 5589  (class class class)co 6308   2ndc2nd 6811
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-id 4754  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-ov 6311  df-1st 6812  df-2nd 6813
This theorem is referenced by:  nvinvfval  26342  hhssabloi  26994
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