MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  curry1val Unicode version

Theorem curry1val 6406
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 6405 . . 3  |-  ( ( F  Fn  ( A  X.  B )  /\  C  e.  A )  ->  G  =  ( x  e.  B  |->  ( C F x ) ) )
32fveq1d 5697 . 2  |-  ( ( F  Fn  ( A  X.  B )  /\  C  e.  A )  ->  ( G `  D
)  =  ( ( x  e.  B  |->  ( C F x ) ) `  D ) )
4 eqid 2412 . . . . . . . . . 10  |-  ( x  e.  B  |->  ( C F x ) )  =  ( x  e.  B  |->  ( C F x ) )
54dmmptss 5333 . . . . . . . . 9  |-  dom  (
x  e.  B  |->  ( C F x ) )  C_  B
65sseli 3312 . . . . . . . 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 5722 . . . . . . 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 5511 . . . . . . 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 6197 . . . . . 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 2447 . . . 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 6056 . . . 4  |-  ( x  =  D  ->  ( C F x )  =  ( C F D ) )
20 ovex 6073 . . . 4  |-  ( C F D )  e. 
_V
2119, 4, 20fvmpt 5773 . . 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 2444 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 1721   _Vcvv 2924   (/)c0 3596   {csn 3782    e. cmpt 4234    X. cxp 4843   `'ccnv 4844   dom cdm 4845    |` cres 4847    o. ccom 4849    Fn wfn 5416   ` cfv 5421  (class class class)co 6048   2ndc2nd 6315
This theorem is referenced by:  nvinvfval  22082  hhssabloi  22723
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 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668
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 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-ral 2679  df-rex 2680  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-sn 3788  df-pr 3789  df-op 3791  df-uni 3984  df-iun 4063  df-br 4181  df-opab 4235  df-mpt 4236  df-id 4466  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 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-ov 6051  df-1st 6316  df-2nd 6317
  Copyright terms: Public domain W3C validator