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Theorem fullfunfnv 25043
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 25040 . . . . 5  |-  Fun Funpart F
2 funfn 5362 . . . . 5  |-  ( Fun Funpart F 
<-> Funpart F  Fn  dom Funpart F )
31, 2mpbi 199 . . . 4  |- Funpart F  Fn  dom Funpart F
4 0ex 4229 . . . . . 6  |-  (/)  e.  _V
54fconst 5507 . . . . 5  |-  ( ( _V  \  dom Funpart F )  X.  { (/) } ) : ( _V  \  dom Funpart F ) --> { (/) }
6 ffn 5469 . . . . 5  |-  ( ( ( _V  \  dom Funpart F )  X.  { (/) } ) : ( _V 
\  dom Funpart F ) --> {
(/) }  ->  ( ( _V  \  dom Funpart F )  X.  { (/) } )  Fn  ( _V  \  dom Funpart F ) )
75, 6ax-mp 8 . . . 4  |-  ( ( _V  \  dom Funpart F )  X.  { (/) } )  Fn  ( _V  \  dom Funpart F )
83, 7pm3.2i 441 . . 3  |-  (Funpart F  Fn  dom Funpart F  /\  (
( _V  \  dom Funpart F )  X.  { (/) } )  Fn  ( _V 
\  dom Funpart F ) )
9 disjdif 3602 . . 3  |-  ( dom Funpart F  i^i  ( _V  \  dom Funpart F ) )  =  (/)
10 fnun 5429 . . 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 653 . 2  |-  (Funpart F  u.  ( ( _V  \  dom Funpart F )  X.  { (/)
} ) )  Fn  ( dom Funpart F  u.  ( _V  \  dom Funpart F ) )
12 df-fullfun 24974 . . . 4  |- FullFun F  =  (Funpart F  u.  (
( _V  \  dom Funpart F )  X.  { (/) } ) )
1312fneq1i 5417 . . 3  |-  (FullFun F  Fn  _V  <->  (Funpart F  u.  (
( _V  \  dom Funpart F )  X.  { (/) } ) )  Fn  _V )
14 undifv 3604 . . . . 5  |-  ( dom Funpart F  u.  ( _V  \  dom Funpart F ) )  =  _V
1514eqcomi 2362 . . . 4  |-  _V  =  ( dom Funpart F  u.  ( _V  \  dom Funpart F ) )
1615fneq2i 5418 . . 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 240 . 2  |-  (FullFun F  Fn  _V  <->  (Funpart F  u.  (
( _V  \  dom Funpart F )  X.  { (/) } ) )  Fn  ( dom Funpart F  u.  ( _V 
\  dom Funpart F ) ) )
1811, 17mpbir 200 1  |- FullFun F  Fn  _V
Colors of variables: wff set class
Syntax hints:    /\ wa 358    = wceq 1642   _Vcvv 2864    \ cdif 3225    u. cun 3226    i^i cin 3227   (/)c0 3531   {csn 3716    X. cxp 4766   dom cdm 4768   Fun wfun 5328    Fn wfn 5329   -->wf 5330  Funpartcfunpart 24950  FullFuncfullfn 24951
This theorem is referenced by:  brfullfun  25045
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4220  ax-nul 4228  ax-pow 4267  ax-pr 4293  ax-un 4591
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-rab 2628  df-v 2866  df-sbc 3068  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3907  df-br 4103  df-opab 4157  df-mpt 4158  df-eprel 4384  df-id 4388  df-xp 4774  df-rel 4775  df-cnv 4776  df-co 4777  df-dm 4778  df-rn 4779  df-res 4780  df-ima 4781  df-iota 5298  df-fun 5336  df-fn 5337  df-f 5338  df-fo 5340  df-fv 5342  df-1st 6206  df-2nd 6207  df-symdif 24920  df-txp 24953  df-singleton 24961  df-singles 24962  df-image 24963  df-funpart 24973  df-fullfun 24974
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