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Theorem fullfunfv 24485
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 5525 . . . 4  |-  ( x  =  A  ->  (FullFun F `
 x )  =  (FullFun F `  A
) )
2 fveq2 5525 . . . 4  |-  ( x  =  A  ->  ( F `  x )  =  ( F `  A ) )
31, 2eqeq12d 2297 . . 3  |-  ( x  =  A  ->  (
(FullFun F `  x )  =  ( F `  x )  <->  (FullFun F `  A )  =  ( F `  A ) ) )
4 df-fullfun 24416 . . . . 5  |- FullFun F  =  (Funpart F  u.  (
( _V  \  dom Funpart F )  X.  { (/) } ) )
54fveq1i 5526 . . . 4  |-  (FullFun F `  x )  =  ( (Funpart F  u.  (
( _V  \  dom Funpart F )  X.  { (/) } ) ) `  x
)
6 disjdif 3526 . . . . . 6  |-  ( dom Funpart F  i^i  ( _V  \  dom Funpart F ) )  =  (/)
7 funpartfun 24481 . . . . . . . 8  |-  Fun Funpart F
8 funfn 5283 . . . . . . . 8  |-  ( Fun Funpart F 
<-> Funpart F  Fn  dom Funpart F )
97, 8mpbi 199 . . . . . . 7  |- Funpart F  Fn  dom Funpart F
10 0ex 4150 . . . . . . . . 9  |-  (/)  e.  _V
1110fconst 5427 . . . . . . . 8  |-  ( ( _V  \  dom Funpart F )  X.  { (/) } ) : ( _V  \  dom Funpart F ) --> { (/) }
12 ffn 5389 . . . . . . . 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 5590 . . . . . . 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 2791 . . . . . . . 8  |-  x  e. 
_V
18 eldif 3162 . . . . . . . 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 5591 . . . . . . . . . 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 5729 . . . . . . . 8  |-  ( x  e.  ( _V  \  dom Funpart F )  ->  (
( ( _V  \  dom Funpart F )  X.  { (/)
} ) `  x
)  =  (/) )
2422, 23eqtrd 2315 . . . . . . 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 5552 . . . . . 6  |-  ( -.  x  e.  dom Funpart F  -> 
(Funpart F `  x )  =  (/) )
2725, 26eqtr4d 2318 . . . . 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 24483 . . . 4  |-  (Funpart F `  x )  =  ( F `  x )
305, 28, 293eqtri 2307 . . 3  |-  (FullFun F `  x )  =  ( F `  x )
313, 30vtoclg 2843 . 2  |-  ( A  e.  _V  ->  (FullFun F `
 A )  =  ( F `  A
) )
32 fvprc 5519 . . 3  |-  ( -.  A  e.  _V  ->  (FullFun
F `  A )  =  (/) )
33 fvprc 5519 . . 3  |-  ( -.  A  e.  _V  ->  ( F `  A )  =  (/) )
3432, 33eqtr4d 2318 . 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 1623    e. wcel 1684   _Vcvv 2788    \ cdif 3149    u. cun 3150    i^i cin 3151   (/)c0 3455   {csn 3640    X. cxp 4687   dom cdm 4689   Fun wfun 5249    Fn wfn 5250   -->wf 5251   ` cfv 5255  Funpartcfunpart 24392  FullFuncfullfn 24393
This theorem is referenced by:  brfullfun  24486
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-eprel 4305  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-fo 5261  df-fv 5263  df-1st 6122  df-2nd 6123  df-symdif 24362  df-txp 24395  df-singleton 24403  df-singles 24404  df-image 24405  df-funpart 24415  df-fullfun 24416
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