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Theorem curry2val 6895
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 6893 . . 3  |-  ( ( F  Fn  ( A  X.  B )  /\  C  e.  B )  ->  G  =  ( x  e.  A  |->  ( x F C ) ) )
32fveq1d 5874 . 2  |-  ( ( F  Fn  ( A  X.  B )  /\  C  e.  B )  ->  ( G `  D
)  =  ( ( x  e.  A  |->  ( x F C ) ) `  D ) )
4 eqid 2420 . . . . . . . . . . 11  |-  ( x  e.  A  |->  ( x F C ) )  =  ( x  e.  A  |->  ( x F C ) )
54dmmptss 5342 . . . . . . . . . 10  |-  dom  (
x  e.  A  |->  ( x F C ) )  C_  A
65sseli 3457 . . . . . . . . 9  |-  ( D  e.  dom  ( x  e.  A  |->  ( x F C ) )  ->  D  e.  A
)
76con3i 140 . . . . . . . 8  |-  ( -.  D  e.  A  ->  -.  D  e.  dom  ( x  e.  A  |->  ( x F C ) ) )
8 ndmfv 5896 . . . . . . . 8  |-  ( -.  D  e.  dom  (
x  e.  A  |->  ( x F C ) )  ->  ( (
x  e.  A  |->  ( x F C ) ) `  D )  =  (/) )
97, 8syl 17 . . . . . . 7  |-  ( -.  D  e.  A  -> 
( ( x  e.  A  |->  ( x F C ) ) `  D )  =  (/) )
109adantl 467 . . . . . 6  |-  ( ( F  Fn  ( A  X.  B )  /\  -.  D  e.  A
)  ->  ( (
x  e.  A  |->  ( x F C ) ) `  D )  =  (/) )
11 fndm 5684 . . . . . . 7  |-  ( F  Fn  ( A  X.  B )  ->  dom  F  =  ( A  X.  B ) )
12 simpl 458 . . . . . . . 8  |-  ( ( D  e.  A  /\  C  e.  B )  ->  D  e.  A )
1312con3i 140 . . . . . . 7  |-  ( -.  D  e.  A  ->  -.  ( D  e.  A  /\  C  e.  B
) )
14 ndmovg 6457 . . . . . . 7  |-  ( ( dom  F  =  ( A  X.  B )  /\  -.  ( D  e.  A  /\  C  e.  B ) )  -> 
( D F C )  =  (/) )
1511, 13, 14syl2an 479 . . . . . 6  |-  ( ( F  Fn  ( A  X.  B )  /\  -.  D  e.  A
)  ->  ( D F C )  =  (/) )
1610, 15eqtr4d 2464 . . . . 5  |-  ( ( F  Fn  ( A  X.  B )  /\  -.  D  e.  A
)  ->  ( (
x  e.  A  |->  ( x F C ) ) `  D )  =  ( D F C ) )
1716ex 435 . . . 4  |-  ( F  Fn  ( A  X.  B )  ->  ( -.  D  e.  A  ->  ( ( x  e.  A  |->  ( x F C ) ) `  D )  =  ( D F C ) ) )
1817adantr 466 . . 3  |-  ( ( F  Fn  ( A  X.  B )  /\  C  e.  B )  ->  ( -.  D  e.  A  ->  ( (
x  e.  A  |->  ( x F C ) ) `  D )  =  ( D F C ) ) )
19 oveq1 6303 . . . 4  |-  ( x  =  D  ->  (
x F C )  =  ( D F C ) )
20 ovex 6324 . . . 4  |-  ( D F C )  e. 
_V
2119, 4, 20fvmpt 5955 . . 3  |-  ( D  e.  A  ->  (
( x  e.  A  |->  ( x F C ) ) `  D
)  =  ( D F C ) )
2218, 21pm2.61d2 163 . 2  |-  ( ( F  Fn  ( A  X.  B )  /\  C  e.  B )  ->  ( ( x  e.  A  |->  ( x F C ) ) `  D )  =  ( D F C ) )
233, 22eqtrd 2461 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 370    = wceq 1437    e. wcel 1867   _Vcvv 3078   (/)c0 3758   {csn 3993    |-> cmpt 4475    X. cxp 4843   `'ccnv 4844   dom cdm 4845    |` cres 4847    o. ccom 4849    Fn wfn 5587   ` cfv 5592  (class class class)co 6296   1stc1st 6796
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-8 1869  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-sep 4539  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6588
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-ral 2778  df-rex 2779  df-rab 2782  df-v 3080  df-sbc 3297  df-csb 3393  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-nul 3759  df-if 3907  df-sn 3994  df-pr 3996  df-op 4000  df-uni 4214  df-iun 4295  df-br 4418  df-opab 4476  df-mpt 4477  df-id 4760  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5556  df-fun 5594  df-fn 5595  df-f 5596  df-f1 5597  df-fo 5598  df-f1o 5599  df-fv 5600  df-ov 6299  df-1st 6798  df-2nd 6799
This theorem is referenced by:  curry2ima  28126
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