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Theorem funpartfv 28115
Description: The function value of the functional part is identical to the original functional value. (Contributed by Scott Fenton, 17-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
Assertion
Ref Expression
funpartfv  |-  (Funpart F `  A )  =  ( F `  A )

Proof of Theorem funpartfv
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 df-funpart 28043 . . 3  |- Funpart F  =  ( F  |`  dom  (
(Image F  o. Singleton )  i^i  ( _V  X.  Singletons ) ) )
21fveq1i 5795 . 2  |-  (Funpart F `  A )  =  ( ( F  |`  dom  (
(Image F  o. Singleton )  i^i  ( _V  X.  Singletons ) ) ) `
 A )
3 fvres 5808 . . 3  |-  ( A  e.  dom  ( (Image
F  o. Singleton )  i^i  ( _V  X.  Singletons ) )  -> 
( ( F  |`  dom  ( (Image F  o. Singleton )  i^i  ( _V  X.  Singletons ) ) ) `  A
)  =  ( F `
 A ) )
4 nfvres 5824 . . . 4  |-  ( -.  A  e.  dom  (
(Image F  o. Singleton )  i^i  ( _V  X.  Singletons ) )  -> 
( ( F  |`  dom  ( (Image F  o. Singleton )  i^i  ( _V  X.  Singletons ) ) ) `  A
)  =  (/) )
5 funpartlem 28112 . . . . . . . . 9  |-  ( A  e.  dom  ( (Image
F  o. Singleton )  i^i  ( _V  X.  Singletons ) )  <->  E. x
( F " { A } )  =  {
x } )
6 eusn 4054 . . . . . . . . 9  |-  ( E! x  x  e.  ( F " { A } )  <->  E. x
( F " { A } )  =  {
x } )
75, 6bitr4i 252 . . . . . . . 8  |-  ( A  e.  dom  ( (Image
F  o. Singleton )  i^i  ( _V  X.  Singletons ) )  <->  E! x  x  e.  ( F " { A } ) )
8 vex 3075 . . . . . . . . . . 11  |-  x  e. 
_V
9 elimasng 5298 . . . . . . . . . . 11  |-  ( ( A  e.  _V  /\  x  e.  _V )  ->  ( x  e.  ( F " { A } )  <->  <. A ,  x >.  e.  F ) )
108, 9mpan2 671 . . . . . . . . . 10  |-  ( A  e.  _V  ->  (
x  e.  ( F
" { A }
)  <->  <. A ,  x >.  e.  F ) )
11 df-br 4396 . . . . . . . . . 10  |-  ( A F x  <->  <. A ,  x >.  e.  F )
1210, 11syl6bbr 263 . . . . . . . . 9  |-  ( A  e.  _V  ->  (
x  e.  ( F
" { A }
)  <->  A F x ) )
1312eubidv 2284 . . . . . . . 8  |-  ( A  e.  _V  ->  ( E! x  x  e.  ( F " { A } )  <->  E! x  A F x ) )
147, 13syl5bb 257 . . . . . . 7  |-  ( A  e.  _V  ->  ( A  e.  dom  ( (Image
F  o. Singleton )  i^i  ( _V  X.  Singletons ) )  <->  E! x  A F x ) )
1514notbid 294 . . . . . 6  |-  ( A  e.  _V  ->  ( -.  A  e.  dom  ( (Image F  o. Singleton )  i^i  ( _V  X.  Singletons ) )  <->  -.  E! x  A F x ) )
16 tz6.12-2 5785 . . . . . 6  |-  ( -.  E! x  A F x  ->  ( F `  A )  =  (/) )
1715, 16syl6bi 228 . . . . 5  |-  ( A  e.  _V  ->  ( -.  A  e.  dom  ( (Image F  o. Singleton )  i^i  ( _V  X.  Singletons ) )  -> 
( F `  A
)  =  (/) ) )
18 fvprc 5788 . . . . . 6  |-  ( -.  A  e.  _V  ->  ( F `  A )  =  (/) )
1918a1d 25 . . . . 5  |-  ( -.  A  e.  _V  ->  ( -.  A  e.  dom  ( (Image F  o. Singleton )  i^i  ( _V  X.  Singletons ) )  -> 
( F `  A
)  =  (/) ) )
2017, 19pm2.61i 164 . . . 4  |-  ( -.  A  e.  dom  (
(Image F  o. Singleton )  i^i  ( _V  X.  Singletons ) )  -> 
( F `  A
)  =  (/) )
214, 20eqtr4d 2496 . . 3  |-  ( -.  A  e.  dom  (
(Image F  o. Singleton )  i^i  ( _V  X.  Singletons ) )  -> 
( ( F  |`  dom  ( (Image F  o. Singleton )  i^i  ( _V  X.  Singletons ) ) ) `  A
)  =  ( F `
 A ) )
223, 21pm2.61i 164 . 2  |-  ( ( F  |`  dom  ( (Image
F  o. Singleton )  i^i  ( _V  X.  Singletons ) ) ) `
 A )  =  ( F `  A
)
232, 22eqtri 2481 1  |-  (Funpart F `  A )  =  ( F `  A )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    = wceq 1370   E.wex 1587    e. wcel 1758   E!weu 2261   _Vcvv 3072    i^i cin 3430   (/)c0 3740   {csn 3980   <.cop 3986   class class class wbr 4395    X. cxp 4941   dom cdm 4943    |` cres 4945   "cima 4946    o. ccom 4947   ` cfv 5521  Singletoncsingle 28007   Singletonscsingles 28008  Imagecimage 28009  Funpartcfunpart 28018
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1954  ax-ext 2431  ax-sep 4516  ax-nul 4524  ax-pow 4573  ax-pr 4634  ax-un 6477
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2265  df-mo 2266  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ne 2647  df-ral 2801  df-rex 2802  df-rab 2805  df-v 3074  df-sbc 3289  df-dif 3434  df-un 3436  df-in 3438  df-ss 3445  df-nul 3741  df-if 3895  df-sn 3981  df-pr 3983  df-op 3987  df-uni 4195  df-br 4396  df-opab 4454  df-mpt 4455  df-eprel 4735  df-id 4739  df-xp 4949  df-rel 4950  df-cnv 4951  df-co 4952  df-dm 4953  df-rn 4954  df-res 4955  df-ima 4956  df-iota 5484  df-fun 5523  df-fn 5524  df-f 5525  df-fo 5527  df-fv 5529  df-1st 6682  df-2nd 6683  df-symdif 27988  df-txp 28023  df-singleton 28031  df-singles 28032  df-image 28033  df-funpart 28043
This theorem is referenced by:  fullfunfv  28117
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