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Theorem funpartfv 24555
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 24486 . . 3  |- Funpart F  =  ( F  |`  dom  (
(Image F  o. Singleton )  i^i  ( _V  X.  Singletons ) ) )
21fveq1i 5542 . 2  |-  (Funpart F `  A )  =  ( ( F  |`  dom  (
(Image F  o. Singleton )  i^i  ( _V  X.  Singletons ) ) ) `
 A )
3 fvres 5558 . . 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 5573 . . . 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 24552 . . . . . . . . 9  |-  ( A  e.  dom  ( (Image
F  o. Singleton )  i^i  ( _V  X.  Singletons ) )  <->  E. x
( F " { A } )  =  {
x } )
6 eusn 3716 . . . . . . . . 9  |-  ( E! x  x  e.  ( F " { A } )  <->  E. x
( F " { A } )  =  {
x } )
75, 6bitr4i 243 . . . . . . . 8  |-  ( A  e.  dom  ( (Image
F  o. Singleton )  i^i  ( _V  X.  Singletons ) )  <->  E! x  x  e.  ( F " { A } ) )
8 vex 2804 . . . . . . . . . . 11  |-  x  e. 
_V
9 elimasng 5055 . . . . . . . . . . 11  |-  ( ( A  e.  _V  /\  x  e.  _V )  ->  ( x  e.  ( F " { A } )  <->  <. A ,  x >.  e.  F ) )
108, 9mpan2 652 . . . . . . . . . 10  |-  ( A  e.  _V  ->  (
x  e.  ( F
" { A }
)  <->  <. A ,  x >.  e.  F ) )
11 df-br 4040 . . . . . . . . . 10  |-  ( A F x  <->  <. A ,  x >.  e.  F )
1210, 11syl6bbr 254 . . . . . . . . 9  |-  ( A  e.  _V  ->  (
x  e.  ( F
" { A }
)  <->  A F x ) )
1312eubidv 2164 . . . . . . . 8  |-  ( A  e.  _V  ->  ( E! x  x  e.  ( F " { A } )  <->  E! x  A F x ) )
147, 13syl5bb 248 . . . . . . 7  |-  ( A  e.  _V  ->  ( A  e.  dom  ( (Image
F  o. Singleton )  i^i  ( _V  X.  Singletons ) )  <->  E! x  A F x ) )
1514notbid 285 . . . . . 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 5532 . . . . . 6  |-  ( -.  E! x  A F x  ->  ( F `  A )  =  (/) )
1715, 16syl6bi 219 . . . . 5  |-  ( A  e.  _V  ->  ( -.  A  e.  dom  ( (Image F  o. Singleton )  i^i  ( _V  X.  Singletons ) )  -> 
( F `  A
)  =  (/) ) )
18 fvprc 5535 . . . . . 6  |-  ( -.  A  e.  _V  ->  ( F `  A )  =  (/) )
1918a1d 22 . . . . 5  |-  ( -.  A  e.  _V  ->  ( -.  A  e.  dom  ( (Image F  o. Singleton )  i^i  ( _V  X.  Singletons ) )  -> 
( F `  A
)  =  (/) ) )
2017, 19pm2.61i 156 . . . 4  |-  ( -.  A  e.  dom  (
(Image F  o. Singleton )  i^i  ( _V  X.  Singletons ) )  -> 
( F `  A
)  =  (/) )
214, 20eqtr4d 2331 . . 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 156 . 2  |-  ( ( F  |`  dom  ( (Image
F  o. Singleton )  i^i  ( _V  X.  Singletons ) ) ) `
 A )  =  ( F `  A
)
232, 22eqtri 2316 1  |-  (Funpart F `  A )  =  ( F `  A )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176   E.wex 1531    = wceq 1632    e. wcel 1696   E!weu 2156   _Vcvv 2801    i^i cin 3164   (/)c0 3468   {csn 3653   <.cop 3656   class class class wbr 4039    X. cxp 4703   dom cdm 4705    |` cres 4707   "cima 4708    o. ccom 4709   ` cfv 5271  Singletoncsingle 24452   Singletonscsingles 24453  Imagecimage 24454  Funpartcfunpart 24463
This theorem is referenced by:  fullfunfv  24557
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-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-br 4040  df-opab 4094  df-mpt 4095  df-eprel 4321  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-fo 5277  df-fv 5279  df-1st 6138  df-2nd 6139  df-symdif 24433  df-txp 24466  df-singleton 24474  df-singles 24475  df-image 24476  df-funpart 24486
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