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Theorem fullfunfv 25792
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 5728 . . . 4  |-  ( x  =  A  ->  (FullFun F `
 x )  =  (FullFun F `  A
) )
2 fveq2 5728 . . . 4  |-  ( x  =  A  ->  ( F `  x )  =  ( F `  A ) )
31, 2eqeq12d 2450 . . 3  |-  ( x  =  A  ->  (
(FullFun F `  x )  =  ( F `  x )  <->  (FullFun F `  A )  =  ( F `  A ) ) )
4 df-fullfun 25719 . . . . 5  |- FullFun F  =  (Funpart F  u.  (
( _V  \  dom Funpart F )  X.  { (/) } ) )
54fveq1i 5729 . . . 4  |-  (FullFun F `  x )  =  ( (Funpart F  u.  (
( _V  \  dom Funpart F )  X.  { (/) } ) ) `  x
)
6 disjdif 3700 . . . . . 6  |-  ( dom Funpart F  i^i  ( _V  \  dom Funpart F ) )  =  (/)
7 funpartfun 25788 . . . . . . . 8  |-  Fun Funpart F
8 funfn 5482 . . . . . . . 8  |-  ( Fun Funpart F 
<-> Funpart F  Fn  dom Funpart F )
97, 8mpbi 200 . . . . . . 7  |- Funpart F  Fn  dom Funpart F
10 0ex 4339 . . . . . . . . 9  |-  (/)  e.  _V
1110fconst 5629 . . . . . . . 8  |-  ( ( _V  \  dom Funpart F )  X.  { (/) } ) : ( _V  \  dom Funpart F ) --> { (/) }
12 ffn 5591 . . . . . . . 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 5794 . . . . . . 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 1269 . . . . . 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 652 . . . . 5  |-  ( x  e.  dom Funpart F  ->  (
(Funpart F  u.  (
( _V  \  dom Funpart F )  X.  { (/) } ) ) `  x
)  =  (Funpart F `  x ) )
17 vex 2959 . . . . . . . 8  |-  x  e. 
_V
18 eldif 3330 . . . . . . . 8  |-  ( x  e.  ( _V  \  dom Funpart F )  <->  ( x  e.  _V  /\  -.  x  e.  dom Funpart F ) )
1917, 18mpbiran 885 . . . . . . 7  |-  ( x  e.  ( _V  \  dom Funpart F )  <->  -.  x  e.  dom Funpart F )
20 fvun2 5795 . . . . . . . . . 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 1269 . . . . . . . . 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 652 . . . . . . . 8  |-  ( x  e.  ( _V  \  dom Funpart F )  ->  (
(Funpart F  u.  (
( _V  \  dom Funpart F )  X.  { (/) } ) ) `  x
)  =  ( ( ( _V  \  dom Funpart F )  X.  { (/) } ) `  x ) )
2310fvconst2 5947 . . . . . . . 8  |-  ( x  e.  ( _V  \  dom Funpart F )  ->  (
( ( _V  \  dom Funpart F )  X.  { (/)
} ) `  x
)  =  (/) )
2422, 23eqtrd 2468 . . . . . . 7  |-  ( x  e.  ( _V  \  dom Funpart F )  ->  (
(Funpart F  u.  (
( _V  \  dom Funpart F )  X.  { (/) } ) ) `  x
)  =  (/) )
2519, 24sylbir 205 . . . . . 6  |-  ( -.  x  e.  dom Funpart F  -> 
( (Funpart F  u.  ( ( _V  \  dom Funpart F )  X.  { (/)
} ) ) `  x )  =  (/) )
26 ndmfv 5755 . . . . . 6  |-  ( -.  x  e.  dom Funpart F  -> 
(Funpart F `  x )  =  (/) )
2725, 26eqtr4d 2471 . . . . 5  |-  ( -.  x  e.  dom Funpart F  -> 
( (Funpart F  u.  ( ( _V  \  dom Funpart F )  X.  { (/)
} ) ) `  x )  =  (Funpart
F `  x )
)
2816, 27pm2.61i 158 . . . 4  |-  ( (Funpart
F  u.  ( ( _V  \  dom Funpart F )  X.  { (/) } ) ) `  x )  =  (Funpart F `  x )
29 funpartfv 25790 . . . 4  |-  (Funpart F `  x )  =  ( F `  x )
305, 28, 293eqtri 2460 . . 3  |-  (FullFun F `  x )  =  ( F `  x )
313, 30vtoclg 3011 . 2  |-  ( A  e.  _V  ->  (FullFun F `
 A )  =  ( F `  A
) )
32 fvprc 5722 . . 3  |-  ( -.  A  e.  _V  ->  (FullFun
F `  A )  =  (/) )
33 fvprc 5722 . . 3  |-  ( -.  A  e.  _V  ->  ( F `  A )  =  (/) )
3432, 33eqtr4d 2471 . 2  |-  ( -.  A  e.  _V  ->  (FullFun
F `  A )  =  ( F `  A ) )
3531, 34pm2.61i 158 1  |-  (FullFun F `  A )  =  ( F `  A )
Colors of variables: wff set class
Syntax hints:   -. wn 3    /\ wa 359    = wceq 1652    e. wcel 1725   _Vcvv 2956    \ cdif 3317    u. cun 3318    i^i cin 3319   (/)c0 3628   {csn 3814    X. cxp 4876   dom cdm 4878   Fun wfun 5448    Fn wfn 5449   -->wf 5450   ` cfv 5454  Funpartcfunpart 25693  FullFuncfullfn 25694
This theorem is referenced by:  brfullfun  25793
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-mpt 4268  df-eprel 4494  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-fo 5460  df-fv 5462  df-1st 6349  df-2nd 6350  df-symdif 25663  df-txp 25698  df-singleton 25706  df-singles 25707  df-image 25708  df-funpart 25718  df-fullfun 25719
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