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Theorem fullfunfnv 25822
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 25819 . . . . 5  |-  Fun Funpart F
2 funfn 5511 . . . . 5  |-  ( Fun Funpart F 
<-> Funpart F  Fn  dom Funpart F )
31, 2mpbi 201 . . . 4  |- Funpart F  Fn  dom Funpart F
4 0ex 4364 . . . . . 6  |-  (/)  e.  _V
54fconst 5658 . . . . 5  |-  ( ( _V  \  dom Funpart F )  X.  { (/) } ) : ( _V  \  dom Funpart F ) --> { (/) }
6 ffn 5620 . . . . 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 443 . . 3  |-  (Funpart F  Fn  dom Funpart F  /\  (
( _V  \  dom Funpart F )  X.  { (/) } )  Fn  ( _V 
\  dom Funpart F ) )
9 disjdif 3724 . . 3  |-  ( dom Funpart F  i^i  ( _V  \  dom Funpart F ) )  =  (/)
10 fnun 5580 . . 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 655 . 2  |-  (Funpart F  u.  ( ( _V  \  dom Funpart F )  X.  { (/)
} ) )  Fn  ( dom Funpart F  u.  ( _V  \  dom Funpart F ) )
12 df-fullfun 25750 . . . 4  |- FullFun F  =  (Funpart F  u.  (
( _V  \  dom Funpart F )  X.  { (/) } ) )
1312fneq1i 5568 . . 3  |-  (FullFun F  Fn  _V  <->  (Funpart F  u.  (
( _V  \  dom Funpart F )  X.  { (/) } ) )  Fn  _V )
14 unvdif 3726 . . . . 5  |-  ( dom Funpart F  u.  ( _V  \  dom Funpart F ) )  =  _V
1514eqcomi 2446 . . . 4  |-  _V  =  ( dom Funpart F  u.  ( _V  \  dom Funpart F ) )
1615fneq2i 5569 . . 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 242 . 2  |-  (FullFun F  Fn  _V  <->  (Funpart F  u.  (
( _V  \  dom Funpart F )  X.  { (/) } ) )  Fn  ( dom Funpart F  u.  ( _V 
\  dom Funpart F ) ) )
1811, 17mpbir 202 1  |- FullFun F  Fn  _V
Colors of variables: wff set class
Syntax hints:    /\ wa 360    = wceq 1653   _Vcvv 2962    \ cdif 3303    u. cun 3304    i^i cin 3305   (/)c0 3613   {csn 3838    X. cxp 4905   dom cdm 4907   Fun wfun 5477    Fn wfn 5478   -->wf 5479  Funpartcfunpart 25724  FullFuncfullfn 25725
This theorem is referenced by:  brfullfun  25824
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-13 1729  ax-14 1731  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423  ax-sep 4355  ax-nul 4363  ax-pow 4406  ax-pr 4432  ax-un 4730
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2291  df-mo 2292  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-ral 2716  df-rex 2717  df-rab 2720  df-v 2964  df-sbc 3168  df-dif 3309  df-un 3311  df-in 3313  df-ss 3320  df-nul 3614  df-if 3764  df-sn 3844  df-pr 3845  df-op 3847  df-uni 4040  df-br 4238  df-opab 4292  df-mpt 4293  df-eprel 4523  df-id 4527  df-xp 4913  df-rel 4914  df-cnv 4915  df-co 4916  df-dm 4917  df-rn 4918  df-res 4919  df-ima 4920  df-iota 5447  df-fun 5485  df-fn 5486  df-f 5487  df-fo 5489  df-fv 5491  df-1st 6378  df-2nd 6379  df-symdif 25694  df-txp 25729  df-singleton 25737  df-singles 25738  df-image 25739  df-funpart 25749  df-fullfun 25750
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