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Theorem fullfunfnv 25707
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 25704 . . . . 5  |-  Fun Funpart F
2 funfn 5449 . . . . 5  |-  ( Fun Funpart F 
<-> Funpart F  Fn  dom Funpart F )
31, 2mpbi 200 . . . 4  |- Funpart F  Fn  dom Funpart F
4 0ex 4307 . . . . . 6  |-  (/)  e.  _V
54fconst 5596 . . . . 5  |-  ( ( _V  \  dom Funpart F )  X.  { (/) } ) : ( _V  \  dom Funpart F ) --> { (/) }
6 ffn 5558 . . . . 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 442 . . 3  |-  (Funpart F  Fn  dom Funpart F  /\  (
( _V  \  dom Funpart F )  X.  { (/) } )  Fn  ( _V 
\  dom Funpart F ) )
9 disjdif 3668 . . 3  |-  ( dom Funpart F  i^i  ( _V  \  dom Funpart F ) )  =  (/)
10 fnun 5518 . . 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 654 . 2  |-  (Funpart F  u.  ( ( _V  \  dom Funpart F )  X.  { (/)
} ) )  Fn  ( dom Funpart F  u.  ( _V  \  dom Funpart F ) )
12 df-fullfun 25638 . . . 4  |- FullFun F  =  (Funpart F  u.  (
( _V  \  dom Funpart F )  X.  { (/) } ) )
1312fneq1i 5506 . . 3  |-  (FullFun F  Fn  _V  <->  (Funpart F  u.  (
( _V  \  dom Funpart F )  X.  { (/) } ) )  Fn  _V )
14 undifv 3670 . . . . 5  |-  ( dom Funpart F  u.  ( _V  \  dom Funpart F ) )  =  _V
1514eqcomi 2416 . . . 4  |-  _V  =  ( dom Funpart F  u.  ( _V  \  dom Funpart F ) )
1615fneq2i 5507 . . 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 241 . 2  |-  (FullFun F  Fn  _V  <->  (Funpart F  u.  (
( _V  \  dom Funpart F )  X.  { (/) } ) )  Fn  ( dom Funpart F  u.  ( _V 
\  dom Funpart F ) ) )
1811, 17mpbir 201 1  |- FullFun F  Fn  _V
Colors of variables: wff set class
Syntax hints:    /\ wa 359    = wceq 1649   _Vcvv 2924    \ cdif 3285    u. cun 3286    i^i cin 3287   (/)c0 3596   {csn 3782    X. cxp 4843   dom cdm 4845   Fun wfun 5415    Fn wfn 5416   -->wf 5417  Funpartcfunpart 25614  FullFuncfullfn 25615
This theorem is referenced by:  brfullfun  25709
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-ral 2679  df-rex 2680  df-rab 2683  df-v 2926  df-sbc 3130  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-sn 3788  df-pr 3789  df-op 3791  df-uni 3984  df-br 4181  df-opab 4235  df-mpt 4236  df-eprel 4462  df-id 4466  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-fo 5427  df-fv 5429  df-1st 6316  df-2nd 6317  df-symdif 25584  df-txp 25617  df-singleton 25625  df-singles 25626  df-image 25627  df-funpart 25637  df-fullfun 25638
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