Theorem fullfunfnv 32053

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

Step	Hyp	Ref	Expression
1		funpartfun 32050	. . . . 5 |- Fun Funpart F
2		funfn 5918	. . . . 5 |- ( Fun Funpart F <-> Funpart F Fn dom Funpart F )
3	1, 2	mpbi 220	. . . 4 |- Funpart F Fn dom Funpart F
4		0ex 4790	. . . . . 6 |- (/) e. _V
5	4	fconst 6091	. . . . 5 |- ( ( _V \ dom Funpart F ) X. { (/) } ) : ( _V \ dom Funpart F ) --> { (/) }
6		ffn 6045	. . . . 5 |- ( ( ( _V \ dom Funpart F ) X. { (/) } ) : ( _V \ dom Funpart F ) --> { (/) } -> ( ( _V \ dom Funpart F ) X. { (/) } ) Fn ( _V \ dom Funpart F ) )
7	5, 6	ax-mp 5	. . . 4 |- ( ( _V \ dom Funpart F ) X. { (/) } ) Fn ( _V \ dom Funpart F )
8	3, 7	pm3.2i 471	. . 3 |- (Funpart F Fn dom Funpart F /\ ( ( _V \ dom Funpart F ) X. { (/) } ) Fn ( _V \ dom Funpart F ) )
9		disjdif 4040	. . 3 |- ( dom Funpart F i^i ( _V \ dom Funpart F ) ) = (/)
10		fnun 5997	. . 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 ) ) )
11	8, 9, 10	mp2an 708	. 2 |- (Funpart F u. ( ( _V \ dom Funpart F ) X. { (/) } ) ) Fn ( dom Funpart F u. ( _V \ dom Funpart F ) )
12		df-fullfun 31982	. . . 4 |- FullFun F = (Funpart F u. ( ( _V \ dom Funpart F ) X. { (/) } ) )
13	12	fneq1i 5985	. . 3 |- (FullFun F Fn _V <-> (Funpart F u. ( ( _V \ dom Funpart F ) X. { (/) } ) ) Fn _V )
14		unvdif 4042	. . . . 5 |- ( dom Funpart F u. ( _V \ dom Funpart F ) ) = _V
15	14	eqcomi 2631	. . . 4 |- _V = ( dom Funpart F u. ( _V \ dom Funpart F ) )
16	15	fneq2i 5986	. . 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 ) ) )
17	13, 16	bitri 264	. 2 |- (FullFun F Fn _V <-> (Funpart F u. ( ( _V \ dom Funpart F ) X. { (/) } ) ) Fn ( dom Funpart F u. ( _V \ dom Funpart F ) ) )
18	11, 17	mpbir 221	1 |- FullFun F Fn _V

Syntax hints:   /\ wa 384   = wceq 1483   _Vcvv 3200   \ cdif 3571   u. cun 3572   i^i cin 3573   (/)c0 3915   {csn 4177   X. cxp 5112   dom cdm 5114   Fun wfun 5882   Fn wfn 5883   -->wf 5884  Funpartcfunpart 31956  FullFuncfullfn 31957

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-symdif 3844  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-eprel 5029  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fo 5894  df-fv 5896  df-1st 7168  df-2nd 7169  df-txp 31961  df-singleton 31969  df-singles 31970  df-image 31971  df-funpart 31981  df-fullfun 31982

This theorem is referenced by:  brfullfun  32055