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Theorem fullfunfnv 24484
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 24481 . . . . 5  |-  Fun Funpart F
2 funfn 5283 . . . . 5  |-  ( Fun Funpart F 
<-> Funpart F  Fn  dom Funpart F )
31, 2mpbi 199 . . . 4  |- Funpart F  Fn  dom Funpart F
4 0ex 4150 . . . . . 6  |-  (/)  e.  _V
54fconst 5427 . . . . 5  |-  ( ( _V  \  dom Funpart F )  X.  { (/) } ) : ( _V  \  dom Funpart F ) --> { (/) }
6 ffn 5389 . . . . 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 3526 . . 3  |-  ( dom Funpart F  i^i  ( _V  \  dom Funpart F ) )  =  (/)
10 fnun 5350 . . 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 24416 . . . 4  |- FullFun F  =  (Funpart F  u.  (
( _V  \  dom Funpart F )  X.  { (/) } ) )
1312fneq1i 5338 . . 3  |-  (FullFun F  Fn  _V  <->  (Funpart F  u.  (
( _V  \  dom Funpart F )  X.  { (/) } ) )  Fn  _V )
14 undifv 3528 . . . . 5  |-  ( dom Funpart F  u.  ( _V  \  dom Funpart F ) )  =  _V
1514eqcomi 2287 . . . 4  |-  _V  =  ( dom Funpart F  u.  ( _V  \  dom Funpart F ) )
1615fneq2i 5339 . . 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 1623   _Vcvv 2788    \ cdif 3149    u. cun 3150    i^i cin 3151   (/)c0 3455   {csn 3640    X. cxp 4687   dom cdm 4689   Fun wfun 5249    Fn wfn 5250   -->wf 5251  Funpartcfunpart 24392  FullFuncfullfn 24393
This theorem is referenced by:  brfullfun  24486
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-eprel 4305  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-fo 5261  df-fv 5263  df-1st 6122  df-2nd 6123  df-symdif 24362  df-txp 24395  df-singleton 24403  df-singles 24404  df-image 24405  df-funpart 24415  df-fullfun 24416
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