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Theorem fullfunfnv 30699
Description: The full functional part of  F is a function over  _V. (Contributed by Scott Fenton, 17-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
Assertion
Ref Expression
fullfunfnv  |- FullFun F  Fn  _V

Proof of Theorem fullfunfnv
StepHypRef Expression
1 funpartfun 30696 . . . . 5  |-  Fun Funpart F
2 funfn 5622 . . . . 5  |-  ( Fun Funpart F 
<-> Funpart F  Fn  dom Funpart F )
31, 2mpbi 211 . . . 4  |- Funpart F  Fn  dom Funpart F
4 0ex 4549 . . . . . 6  |-  (/)  e.  _V
54fconst 5778 . . . . 5  |-  ( ( _V  \  dom Funpart F )  X.  { (/) } ) : ( _V  \  dom Funpart F ) --> { (/) }
6 ffn 5738 . . . . 5  |-  ( ( ( _V  \  dom Funpart F )  X.  { (/) } ) : ( _V 
\  dom Funpart F ) --> {
(/) }  ->  ( ( _V  \  dom Funpart F )  X.  { (/) } )  Fn  ( _V  \  dom Funpart F ) )
75, 6ax-mp 5 . . . 4  |-  ( ( _V  \  dom Funpart F )  X.  { (/) } )  Fn  ( _V  \  dom Funpart F )
83, 7pm3.2i 456 . . 3  |-  (Funpart F  Fn  dom Funpart F  /\  (
( _V  \  dom Funpart F )  X.  { (/) } )  Fn  ( _V 
\  dom Funpart F ) )
9 disjdif 3864 . . 3  |-  ( dom Funpart F  i^i  ( _V  \  dom Funpart F ) )  =  (/)
10 fnun 5692 . . 3  |-  ( ( (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 ) )  =  (/) )  ->  (Funpart F  u.  ( ( _V 
\  dom Funpart F )  X. 
{ (/) } ) )  Fn  ( dom Funpart F  u.  ( _V  \  dom Funpart F ) ) )
118, 9, 10mp2an 676 . 2  |-  (Funpart F  u.  ( ( _V  \  dom Funpart F )  X.  { (/)
} ) )  Fn  ( dom Funpart F  u.  ( _V  \  dom Funpart F ) )
12 df-fullfun 30627 . . . 4  |- FullFun F  =  (Funpart F  u.  (
( _V  \  dom Funpart F )  X.  { (/) } ) )
1312fneq1i 5680 . . 3  |-  (FullFun F  Fn  _V  <->  (Funpart F  u.  (
( _V  \  dom Funpart F )  X.  { (/) } ) )  Fn  _V )
14 unvdif 3866 . . . . 5  |-  ( dom Funpart F  u.  ( _V  \  dom Funpart F ) )  =  _V
1514eqcomi 2433 . . . 4  |-  _V  =  ( dom Funpart F  u.  ( _V  \  dom Funpart F ) )
1615fneq2i 5681 . . 3  |-  ( (Funpart
F  u.  ( ( _V  \  dom Funpart F )  X.  { (/) } ) )  Fn  _V  <->  (Funpart F  u.  ( ( _V  \  dom Funpart F )  X.  { (/)
} ) )  Fn  ( dom Funpart F  u.  ( _V  \  dom Funpart F ) ) )
1713, 16bitri 252 . 2  |-  (FullFun F  Fn  _V  <->  (Funpart F  u.  (
( _V  \  dom Funpart F )  X.  { (/) } ) )  Fn  ( dom Funpart F  u.  ( _V 
\  dom Funpart F ) ) )
1811, 17mpbir 212 1  |- FullFun F  Fn  _V
Colors of variables: wff setvar class
Syntax hints:    /\ wa 370    = wceq 1437   _Vcvv 3078    \ cdif 3430    u. cun 3431    i^i cin 3432   (/)c0 3758   {csn 3993    X. cxp 4844   dom cdm 4846   Fun wfun 5587    Fn wfn 5588   -->wf 5589  Funpartcfunpart 30601  FullFuncfullfn 30602
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-8 1869  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-sep 4540  ax-nul 4548  ax-pow 4595  ax-pr 4653  ax-un 6589
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-ral 2778  df-rex 2779  df-rab 2782  df-v 3080  df-sbc 3297  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-symdif 3690  df-nul 3759  df-if 3907  df-sn 3994  df-pr 3996  df-op 4000  df-uni 4214  df-br 4418  df-opab 4477  df-mpt 4478  df-eprel 4757  df-id 4761  df-xp 4852  df-rel 4853  df-cnv 4854  df-co 4855  df-dm 4856  df-rn 4857  df-res 4858  df-ima 4859  df-iota 5557  df-fun 5595  df-fn 5596  df-f 5597  df-fo 5599  df-fv 5601  df-1st 6799  df-2nd 6800  df-txp 30606  df-singleton 30614  df-singles 30615  df-image 30616  df-funpart 30626  df-fullfun 30627
This theorem is referenced by:  brfullfun  30701
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