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Theorem funpartfv 29823
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 29751 . . 3  |- Funpart F  =  ( F  |`  dom  (
(Image F  o. Singleton )  i^i  ( _V  X.  Singletons ) ) )
21fveq1i 5849 . 2  |-  (Funpart F `  A )  =  ( ( F  |`  dom  (
(Image F  o. Singleton )  i^i  ( _V  X.  Singletons ) ) ) `
 A )
3 fvres 5862 . . 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 5878 . . . 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 29820 . . . . . . . . 9  |-  ( A  e.  dom  ( (Image
F  o. Singleton )  i^i  ( _V  X.  Singletons ) )  <->  E. x
( F " { A } )  =  {
x } )
6 eusn 4092 . . . . . . . . 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 3109 . . . . . . . . . . 11  |-  x  e. 
_V
9 elimasng 5351 . . . . . . . . . . 11  |-  ( ( A  e.  _V  /\  x  e.  _V )  ->  ( x  e.  ( F " { A } )  <->  <. A ,  x >.  e.  F ) )
108, 9mpan2 669 . . . . . . . . . 10  |-  ( A  e.  _V  ->  (
x  e.  ( F
" { A }
)  <->  <. A ,  x >.  e.  F ) )
11 df-br 4440 . . . . . . . . . 10  |-  ( A F x  <->  <. A ,  x >.  e.  F )
1210, 11syl6bbr 263 . . . . . . . . 9  |-  ( A  e.  _V  ->  (
x  e.  ( F
" { A }
)  <->  A F x ) )
1312eubidv 2306 . . . . . . . 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 292 . . . . . 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 5839 . . . . . 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 5842 . . . . . 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 2498 . . 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 2483 1  |-  (Funpart F `  A )  =  ( F `  A )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    = wceq 1398   E.wex 1617    e. wcel 1823   E!weu 2284   _Vcvv 3106    i^i cin 3460   (/)c0 3783   {csn 4016   <.cop 4022   class class class wbr 4439    X. cxp 4986   dom cdm 4988    |` cres 4990   "cima 4991    o. ccom 4992   ` cfv 5570  Singletoncsingle 29715   Singletonscsingles 29716  Imagecimage 29717  Funpartcfunpart 29726
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-symdif 3715  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-mpt 4499  df-eprel 4780  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-fo 5576  df-fv 5578  df-1st 6773  df-2nd 6774  df-txp 29731  df-singleton 29739  df-singles 29740  df-image 29741  df-funpart 29751
This theorem is referenced by:  fullfunfv  29825
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