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Theorem fullfunfv 24557
Description: The function value of the full function of  F agrees with  F. (Contributed by Scott Fenton, 17-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
Assertion
Ref Expression
fullfunfv  |-  (FullFun F `  A )  =  ( F `  A )

Proof of Theorem fullfunfv
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 fveq2 5541 . . . 4  |-  ( x  =  A  ->  (FullFun F `
 x )  =  (FullFun F `  A
) )
2 fveq2 5541 . . . 4  |-  ( x  =  A  ->  ( F `  x )  =  ( F `  A ) )
31, 2eqeq12d 2310 . . 3  |-  ( x  =  A  ->  (
(FullFun F `  x )  =  ( F `  x )  <->  (FullFun F `  A )  =  ( F `  A ) ) )
4 df-fullfun 24487 . . . . 5  |- FullFun F  =  (Funpart F  u.  (
( _V  \  dom Funpart F )  X.  { (/) } ) )
54fveq1i 5542 . . . 4  |-  (FullFun F `  x )  =  ( (Funpart F  u.  (
( _V  \  dom Funpart F )  X.  { (/) } ) ) `  x
)
6 disjdif 3539 . . . . . 6  |-  ( dom Funpart F  i^i  ( _V  \  dom Funpart F ) )  =  (/)
7 funpartfun 24553 . . . . . . . 8  |-  Fun Funpart F
8 funfn 5299 . . . . . . . 8  |-  ( Fun Funpart F 
<-> Funpart F  Fn  dom Funpart F )
97, 8mpbi 199 . . . . . . 7  |- Funpart F  Fn  dom Funpart F
10 0ex 4166 . . . . . . . . 9  |-  (/)  e.  _V
1110fconst 5443 . . . . . . . 8  |-  ( ( _V  \  dom Funpart F )  X.  { (/) } ) : ( _V  \  dom Funpart F ) --> { (/) }
12 ffn 5405 . . . . . . . 8  |-  ( ( ( _V  \  dom Funpart F )  X.  { (/) } ) : ( _V 
\  dom Funpart F ) --> {
(/) }  ->  ( ( _V  \  dom Funpart F )  X.  { (/) } )  Fn  ( _V  \  dom Funpart F ) )
1311, 12ax-mp 8 . . . . . . 7  |-  ( ( _V  \  dom Funpart F )  X.  { (/) } )  Fn  ( _V  \  dom Funpart F )
14 fvun1 5606 . . . . . . 7  |-  ( (Funpart
F  Fn  dom Funpart F  /\  ( ( _V  \  dom Funpart F )  X.  { (/)
} )  Fn  ( _V  \  dom Funpart F )  /\  ( ( dom Funpart F  i^i  ( _V  \  dom Funpart F ) )  =  (/)  /\  x  e.  dom Funpart F ) )  -> 
( (Funpart F  u.  ( ( _V  \  dom Funpart F )  X.  { (/)
} ) ) `  x )  =  (Funpart
F `  x )
)
159, 13, 14mp3an12 1267 . . . . . 6  |-  ( ( ( dom Funpart F  i^i  ( _V  \  dom Funpart F ) )  =  (/)  /\  x  e.  dom Funpart F )  ->  (
(Funpart F  u.  (
( _V  \  dom Funpart F )  X.  { (/) } ) ) `  x
)  =  (Funpart F `  x ) )
166, 15mpan 651 . . . . 5  |-  ( x  e.  dom Funpart F  ->  (
(Funpart F  u.  (
( _V  \  dom Funpart F )  X.  { (/) } ) ) `  x
)  =  (Funpart F `  x ) )
17 vex 2804 . . . . . . . 8  |-  x  e. 
_V
18 eldif 3175 . . . . . . . 8  |-  ( x  e.  ( _V  \  dom Funpart F )  <->  ( x  e.  _V  /\  -.  x  e.  dom Funpart F ) )
1917, 18mpbiran 884 . . . . . . 7  |-  ( x  e.  ( _V  \  dom Funpart F )  <->  -.  x  e.  dom Funpart F )
20 fvun2 5607 . . . . . . . . . 10  |-  ( (Funpart
F  Fn  dom Funpart F  /\  ( ( _V  \  dom Funpart F )  X.  { (/)
} )  Fn  ( _V  \  dom Funpart F )  /\  ( ( dom Funpart F  i^i  ( _V  \  dom Funpart F ) )  =  (/)  /\  x  e.  ( _V  \  dom Funpart F ) ) )  -> 
( (Funpart F  u.  ( ( _V  \  dom Funpart F )  X.  { (/)
} ) ) `  x )  =  ( ( ( _V  \  dom Funpart F )  X.  { (/)
} ) `  x
) )
219, 13, 20mp3an12 1267 . . . . . . . . 9  |-  ( ( ( dom Funpart F  i^i  ( _V  \  dom Funpart F ) )  =  (/)  /\  x  e.  ( _V  \  dom Funpart F ) )  ->  (
(Funpart F  u.  (
( _V  \  dom Funpart F )  X.  { (/) } ) ) `  x
)  =  ( ( ( _V  \  dom Funpart F )  X.  { (/) } ) `  x ) )
226, 21mpan 651 . . . . . . . 8  |-  ( x  e.  ( _V  \  dom Funpart F )  ->  (
(Funpart F  u.  (
( _V  \  dom Funpart F )  X.  { (/) } ) ) `  x
)  =  ( ( ( _V  \  dom Funpart F )  X.  { (/) } ) `  x ) )
2310fvconst2 5745 . . . . . . . 8  |-  ( x  e.  ( _V  \  dom Funpart F )  ->  (
( ( _V  \  dom Funpart F )  X.  { (/)
} ) `  x
)  =  (/) )
2422, 23eqtrd 2328 . . . . . . 7  |-  ( x  e.  ( _V  \  dom Funpart F )  ->  (
(Funpart F  u.  (
( _V  \  dom Funpart F )  X.  { (/) } ) ) `  x
)  =  (/) )
2519, 24sylbir 204 . . . . . 6  |-  ( -.  x  e.  dom Funpart F  -> 
( (Funpart F  u.  ( ( _V  \  dom Funpart F )  X.  { (/)
} ) ) `  x )  =  (/) )
26 ndmfv 5568 . . . . . 6  |-  ( -.  x  e.  dom Funpart F  -> 
(Funpart F `  x )  =  (/) )
2725, 26eqtr4d 2331 . . . . 5  |-  ( -.  x  e.  dom Funpart F  -> 
( (Funpart F  u.  ( ( _V  \  dom Funpart F )  X.  { (/)
} ) ) `  x )  =  (Funpart
F `  x )
)
2816, 27pm2.61i 156 . . . 4  |-  ( (Funpart
F  u.  ( ( _V  \  dom Funpart F )  X.  { (/) } ) ) `  x )  =  (Funpart F `  x )
29 funpartfv 24555 . . . 4  |-  (Funpart F `  x )  =  ( F `  x )
305, 28, 293eqtri 2320 . . 3  |-  (FullFun F `  x )  =  ( F `  x )
313, 30vtoclg 2856 . 2  |-  ( A  e.  _V  ->  (FullFun F `
 A )  =  ( F `  A
) )
32 fvprc 5535 . . 3  |-  ( -.  A  e.  _V  ->  (FullFun
F `  A )  =  (/) )
33 fvprc 5535 . . 3  |-  ( -.  A  e.  _V  ->  ( F `  A )  =  (/) )
3432, 33eqtr4d 2331 . 2  |-  ( -.  A  e.  _V  ->  (FullFun
F `  A )  =  ( F `  A ) )
3531, 34pm2.61i 156 1  |-  (FullFun F `  A )  =  ( F `  A )
Colors of variables: wff set class
Syntax hints:   -. wn 3    /\ wa 358    = wceq 1632    e. wcel 1696   _Vcvv 2801    \ cdif 3162    u. cun 3163    i^i cin 3164   (/)c0 3468   {csn 3653    X. cxp 4703   dom cdm 4705   Fun wfun 5265    Fn wfn 5266   -->wf 5267   ` cfv 5271  Funpartcfunpart 24463  FullFuncfullfn 24464
This theorem is referenced by:  brfullfun  24558
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  df-fullfun 24487
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