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Theorem fullfunfv 30258
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 5805 . . . 4  |-  ( x  =  A  ->  (FullFun F `
 x )  =  (FullFun F `  A
) )
2 fveq2 5805 . . . 4  |-  ( x  =  A  ->  ( F `  x )  =  ( F `  A ) )
31, 2eqeq12d 2424 . . 3  |-  ( x  =  A  ->  (
(FullFun F `  x )  =  ( F `  x )  <->  (FullFun F `  A )  =  ( F `  A ) ) )
4 df-fullfun 30185 . . . . 5  |- FullFun F  =  (Funpart F  u.  (
( _V  \  dom Funpart F )  X.  { (/) } ) )
54fveq1i 5806 . . . 4  |-  (FullFun F `  x )  =  ( (Funpart F  u.  (
( _V  \  dom Funpart F )  X.  { (/) } ) ) `  x
)
6 disjdif 3843 . . . . . 6  |-  ( dom Funpart F  i^i  ( _V  \  dom Funpart F ) )  =  (/)
7 funpartfun 30254 . . . . . . . 8  |-  Fun Funpart F
8 funfn 5554 . . . . . . . 8  |-  ( Fun Funpart F 
<-> Funpart F  Fn  dom Funpart F )
97, 8mpbi 208 . . . . . . 7  |- Funpart F  Fn  dom Funpart F
10 0ex 4525 . . . . . . . . 9  |-  (/)  e.  _V
1110fconst 5710 . . . . . . . 8  |-  ( ( _V  \  dom Funpart F )  X.  { (/) } ) : ( _V  \  dom Funpart F ) --> { (/) }
12 ffn 5670 . . . . . . . 8  |-  ( ( ( _V  \  dom Funpart F )  X.  { (/) } ) : ( _V 
\  dom Funpart F ) --> {
(/) }  ->  ( ( _V  \  dom Funpart F )  X.  { (/) } )  Fn  ( _V  \  dom Funpart F ) )
1311, 12ax-mp 5 . . . . . . 7  |-  ( ( _V  \  dom Funpart F )  X.  { (/) } )  Fn  ( _V  \  dom Funpart F )
14 fvun1 5876 . . . . . . 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 1316 . . . . . 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 668 . . . . 5  |-  ( x  e.  dom Funpart F  ->  (
(Funpart F  u.  (
( _V  \  dom Funpart F )  X.  { (/) } ) ) `  x
)  =  (Funpart F `  x ) )
17 vex 3061 . . . . . . . 8  |-  x  e. 
_V
18 eldif 3423 . . . . . . . 8  |-  ( x  e.  ( _V  \  dom Funpart F )  <->  ( x  e.  _V  /\  -.  x  e.  dom Funpart F ) )
1917, 18mpbiran 919 . . . . . . 7  |-  ( x  e.  ( _V  \  dom Funpart F )  <->  -.  x  e.  dom Funpart F )
20 fvun2 5877 . . . . . . . . . 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 1316 . . . . . . . . 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 668 . . . . . . . 8  |-  ( x  e.  ( _V  \  dom Funpart F )  ->  (
(Funpart F  u.  (
( _V  \  dom Funpart F )  X.  { (/) } ) ) `  x
)  =  ( ( ( _V  \  dom Funpart F )  X.  { (/) } ) `  x ) )
2310fvconst2 6063 . . . . . . . 8  |-  ( x  e.  ( _V  \  dom Funpart F )  ->  (
( ( _V  \  dom Funpart F )  X.  { (/)
} ) `  x
)  =  (/) )
2422, 23eqtrd 2443 . . . . . . 7  |-  ( x  e.  ( _V  \  dom Funpart F )  ->  (
(Funpart F  u.  (
( _V  \  dom Funpart F )  X.  { (/) } ) ) `  x
)  =  (/) )
2519, 24sylbir 213 . . . . . 6  |-  ( -.  x  e.  dom Funpart F  -> 
( (Funpart F  u.  ( ( _V  \  dom Funpart F )  X.  { (/)
} ) ) `  x )  =  (/) )
26 ndmfv 5829 . . . . . 6  |-  ( -.  x  e.  dom Funpart F  -> 
(Funpart F `  x )  =  (/) )
2725, 26eqtr4d 2446 . . . . 5  |-  ( -.  x  e.  dom Funpart F  -> 
( (Funpart F  u.  ( ( _V  \  dom Funpart F )  X.  { (/)
} ) ) `  x )  =  (Funpart
F `  x )
)
2816, 27pm2.61i 164 . . . 4  |-  ( (Funpart
F  u.  ( ( _V  \  dom Funpart F )  X.  { (/) } ) ) `  x )  =  (Funpart F `  x )
29 funpartfv 30256 . . . 4  |-  (Funpart F `  x )  =  ( F `  x )
305, 28, 293eqtri 2435 . . 3  |-  (FullFun F `  x )  =  ( F `  x )
313, 30vtoclg 3116 . 2  |-  ( A  e.  _V  ->  (FullFun F `
 A )  =  ( F `  A
) )
32 fvprc 5799 . . 3  |-  ( -.  A  e.  _V  ->  (FullFun
F `  A )  =  (/) )
33 fvprc 5799 . . 3  |-  ( -.  A  e.  _V  ->  ( F `  A )  =  (/) )
3432, 33eqtr4d 2446 . 2  |-  ( -.  A  e.  _V  ->  (FullFun
F `  A )  =  ( F `  A ) )
3531, 34pm2.61i 164 1  |-  (FullFun F `  A )  =  ( F `  A )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    /\ wa 367    = wceq 1405    e. wcel 1842   _Vcvv 3058    \ cdif 3410    u. cun 3411    i^i cin 3412   (/)c0 3737   {csn 3971    X. cxp 4940   dom cdm 4942   Fun wfun 5519    Fn wfn 5520   -->wf 5521   ` cfv 5525  Funpartcfunpart 30159  FullFuncfullfn 30160
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6530
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-rab 2762  df-v 3060  df-sbc 3277  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-symdif 3669  df-nul 3738  df-if 3885  df-sn 3972  df-pr 3974  df-op 3978  df-uni 4191  df-br 4395  df-opab 4453  df-mpt 4454  df-eprel 4733  df-id 4737  df-xp 4948  df-rel 4949  df-cnv 4950  df-co 4951  df-dm 4952  df-rn 4953  df-res 4954  df-ima 4955  df-iota 5489  df-fun 5527  df-fn 5528  df-f 5529  df-fo 5531  df-fv 5533  df-1st 6738  df-2nd 6739  df-txp 30164  df-singleton 30172  df-singles 30173  df-image 30174  df-funpart 30184  df-fullfun 30185
This theorem is referenced by:  brfullfun  30259
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