Theorem fullfunfv 32054

Description: The function value of the full function of F agrees with F. (Contributed by Scott Fenton, 17-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)

Assertion

Ref	Expression
fullfunfv	|- (FullFun F ` A ) = ( F ` A )

Proof of Theorem fullfunfv

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 6191	. . . 4 |- ( x = A -> (FullFun F ` x ) = (FullFun F ` A ) )
2		fveq2 6191	. . . 4 |- ( x = A -> ( F ` x ) = ( F ` A ) )
3	1, 2	eqeq12d 2637	. . 3 |- ( x = A -> ( (FullFun F ` x ) = ( F ` x ) <-> (FullFun F ` A ) = ( F ` A ) ) )
4		df-fullfun 31982	. . . . 5 |- FullFun F = (Funpart F u. ( ( _V \ dom Funpart F ) X. { (/) } ) )
5	4	fveq1i 6192	. . . 4 |- (FullFun F ` x ) = ( (Funpart F u. ( ( _V \ dom Funpart F ) X. { (/) } ) ) ` x )
6		disjdif 4040	. . . . . 6 |- ( dom Funpart F i^i ( _V \ dom Funpart F ) ) = (/)
7		funpartfun 32050	. . . . . . . 8 |- Fun Funpart F
8		funfn 5918	. . . . . . . 8 |- ( Fun Funpart F <-> Funpart F Fn dom Funpart F )
9	7, 8	mpbi 220	. . . . . . 7 |- Funpart F Fn dom Funpart F
10		0ex 4790	. . . . . . . . 9 |- (/) e. _V
11	10	fconst 6091	. . . . . . . 8 |- ( ( _V \ dom Funpart F ) X. { (/) } ) : ( _V \ dom Funpart F ) --> { (/) }
12		ffn 6045	. . . . . . . 8 |- ( ( ( _V \ dom Funpart F ) X. { (/) } ) : ( _V \ dom Funpart F ) --> { (/) } -> ( ( _V \ dom Funpart F ) X. { (/) } ) Fn ( _V \ dom Funpart F ) )
13	11, 12	ax-mp 5	. . . . . . 7 |- ( ( _V \ dom Funpart F ) X. { (/) } ) Fn ( _V \ dom Funpart F )
14		fvun1 6269	. . . . . . 7 |- ( (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 ) ) = (/) /\ x e. dom Funpart F ) ) -> ( (Funpart F u. ( ( _V \ dom Funpart F ) X. { (/) } ) ) ` x ) = (Funpart F ` x ) )
15	9, 13, 14	mp3an12 1414	. . . . . 6 |- ( ( ( dom Funpart F i^i ( _V \ dom Funpart F ) ) = (/) /\ x e. dom Funpart F ) -> ( (Funpart F u. ( ( _V \ dom Funpart F ) X. { (/) } ) ) ` x ) = (Funpart F ` x ) )
16	6, 15	mpan 706	. . . . 5 |- ( x e. dom Funpart F -> ( (Funpart F u. ( ( _V \ dom Funpart F ) X. { (/) } ) ) ` x ) = (Funpart F ` x ) )
17		vex 3203	. . . . . . . 8 |- x e. _V
18		eldif 3584	. . . . . . . 8 |- ( x e. ( _V \ dom Funpart F ) <-> ( x e. _V /\ -. x e. dom Funpart F ) )
19	17, 18	mpbiran 953	. . . . . . 7 |- ( x e. ( _V \ dom Funpart F ) <-> -. x e. dom Funpart F )
20		fvun2 6270	. . . . . . . . . 10 |- ( (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 ) ) = (/) /\ x e. ( _V \ dom Funpart F ) ) ) -> ( (Funpart F u. ( ( _V \ dom Funpart F ) X. { (/) } ) ) ` x ) = ( ( ( _V \ dom Funpart F ) X. { (/) } ) ` x ) )
21	9, 13, 20	mp3an12 1414	. . . . . . . . 9 |- ( ( ( dom Funpart F i^i ( _V \ dom Funpart F ) ) = (/) /\ x e. ( _V \ dom Funpart F ) ) -> ( (Funpart F u. ( ( _V \ dom Funpart F ) X. { (/) } ) ) ` x ) = ( ( ( _V \ dom Funpart F ) X. { (/) } ) ` x ) )
22	6, 21	mpan 706	. . . . . . . 8 |- ( x e. ( _V \ dom Funpart F ) -> ( (Funpart F u. ( ( _V \ dom Funpart F ) X. { (/) } ) ) ` x ) = ( ( ( _V \ dom Funpart F ) X. { (/) } ) ` x ) )
23	10	fvconst2 6469	. . . . . . . 8 |- ( x e. ( _V \ dom Funpart F ) -> ( ( ( _V \ dom Funpart F ) X. { (/) } ) ` x ) = (/) )
24	22, 23	eqtrd 2656	. . . . . . 7 |- ( x e. ( _V \ dom Funpart F ) -> ( (Funpart F u. ( ( _V \ dom Funpart F ) X. { (/) } ) ) ` x ) = (/) )
25	19, 24	sylbir 225	. . . . . 6 |- ( -. x e. dom Funpart F -> ( (Funpart F u. ( ( _V \ dom Funpart F ) X. { (/) } ) ) ` x ) = (/) )
26		ndmfv 6218	. . . . . 6 |- ( -. x e. dom Funpart F -> (Funpart F ` x ) = (/) )
27	25, 26	eqtr4d 2659	. . . . 5 |- ( -. x e. dom Funpart F -> ( (Funpart F u. ( ( _V \ dom Funpart F ) X. { (/) } ) ) ` x ) = (Funpart F ` x ) )
28	16, 27	pm2.61i 176	. . . 4 |- ( (Funpart F u. ( ( _V \ dom Funpart F ) X. { (/) } ) ) ` x ) = (Funpart F ` x )
29		funpartfv 32052	. . . 4 |- (Funpart F ` x ) = ( F ` x )
30	5, 28, 29	3eqtri 2648	. . 3 |- (FullFun F ` x ) = ( F ` x )
31	3, 30	vtoclg 3266	. 2 |- ( A e. _V -> (FullFun F ` A ) = ( F ` A ) )
32		fvprc 6185	. . 3 |- ( -. A e. _V -> (FullFun F ` A ) = (/) )
33		fvprc 6185	. . 3 |- ( -. A e. _V -> ( F ` A ) = (/) )
34	32, 33	eqtr4d 2659	. 2 |- ( -. A e. _V -> (FullFun F ` A ) = ( F ` A ) )
35	31, 34	pm2.61i 176	1 |- (FullFun F ` A ) = ( F ` A )

Syntax hints:   -. wn 3   /\ wa 384   = wceq 1483   e. wcel 1990   _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   ` cfv 5888  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