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Theorem fullfunfv 29160
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 5857 . . . 4  |-  ( x  =  A  ->  (FullFun F `
 x )  =  (FullFun F `  A
) )
2 fveq2 5857 . . . 4  |-  ( x  =  A  ->  ( F `  x )  =  ( F `  A ) )
31, 2eqeq12d 2482 . . 3  |-  ( x  =  A  ->  (
(FullFun F `  x )  =  ( F `  x )  <->  (FullFun F `  A )  =  ( F `  A ) ) )
4 df-fullfun 29087 . . . . 5  |- FullFun F  =  (Funpart F  u.  (
( _V  \  dom Funpart F )  X.  { (/) } ) )
54fveq1i 5858 . . . 4  |-  (FullFun F `  x )  =  ( (Funpart F  u.  (
( _V  \  dom Funpart F )  X.  { (/) } ) ) `  x
)
6 disjdif 3892 . . . . . 6  |-  ( dom Funpart F  i^i  ( _V  \  dom Funpart F ) )  =  (/)
7 funpartfun 29156 . . . . . . . 8  |-  Fun Funpart F
8 funfn 5608 . . . . . . . 8  |-  ( Fun Funpart F 
<-> Funpart F  Fn  dom Funpart F )
97, 8mpbi 208 . . . . . . 7  |- Funpart F  Fn  dom Funpart F
10 0ex 4570 . . . . . . . . 9  |-  (/)  e.  _V
1110fconst 5762 . . . . . . . 8  |-  ( ( _V  \  dom Funpart F )  X.  { (/) } ) : ( _V  \  dom Funpart F ) --> { (/) }
12 ffn 5722 . . . . . . . 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 5929 . . . . . . 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 1309 . . . . . 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 670 . . . . 5  |-  ( x  e.  dom Funpart F  ->  (
(Funpart F  u.  (
( _V  \  dom Funpart F )  X.  { (/) } ) ) `  x
)  =  (Funpart F `  x ) )
17 vex 3109 . . . . . . . 8  |-  x  e. 
_V
18 eldif 3479 . . . . . . . 8  |-  ( x  e.  ( _V  \  dom Funpart F )  <->  ( x  e.  _V  /\  -.  x  e.  dom Funpart F ) )
1917, 18mpbiran 911 . . . . . . 7  |-  ( x  e.  ( _V  \  dom Funpart F )  <->  -.  x  e.  dom Funpart F )
20 fvun2 5930 . . . . . . . . . 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 1309 . . . . . . . . 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 670 . . . . . . . 8  |-  ( x  e.  ( _V  \  dom Funpart F )  ->  (
(Funpart F  u.  (
( _V  \  dom Funpart F )  X.  { (/) } ) ) `  x
)  =  ( ( ( _V  \  dom Funpart F )  X.  { (/) } ) `  x ) )
2310fvconst2 6107 . . . . . . . 8  |-  ( x  e.  ( _V  \  dom Funpart F )  ->  (
( ( _V  \  dom Funpart F )  X.  { (/)
} ) `  x
)  =  (/) )
2422, 23eqtrd 2501 . . . . . . 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 5881 . . . . . 6  |-  ( -.  x  e.  dom Funpart F  -> 
(Funpart F `  x )  =  (/) )
2725, 26eqtr4d 2504 . . . . 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 29158 . . . 4  |-  (Funpart F `  x )  =  ( F `  x )
305, 28, 293eqtri 2493 . . 3  |-  (FullFun F `  x )  =  ( F `  x )
313, 30vtoclg 3164 . 2  |-  ( A  e.  _V  ->  (FullFun F `
 A )  =  ( F `  A
) )
32 fvprc 5851 . . 3  |-  ( -.  A  e.  _V  ->  (FullFun
F `  A )  =  (/) )
33 fvprc 5851 . . 3  |-  ( -.  A  e.  _V  ->  ( F `  A )  =  (/) )
3432, 33eqtr4d 2504 . 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 369    = wceq 1374    e. wcel 1762   _Vcvv 3106    \ cdif 3466    u. cun 3467    i^i cin 3468   (/)c0 3778   {csn 4020    X. cxp 4990   dom cdm 4992   Fun wfun 5573    Fn wfn 5574   -->wf 5575   ` cfv 5579  Funpartcfunpart 29061  FullFuncfullfn 29062
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3108  df-sbc 3325  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-br 4441  df-opab 4499  df-mpt 4500  df-eprel 4784  df-id 4788  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-fo 5585  df-fv 5587  df-1st 6774  df-2nd 6775  df-symdif 29031  df-txp 29066  df-singleton 29074  df-singles 29075  df-image 29076  df-funpart 29086  df-fullfun 29087
This theorem is referenced by:  brfullfun  29161
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