Theorem funpartfv 32052

Description: The function value of the functional part is identical to the original functional value. (Contributed by Scott Fenton, 17-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)

Assertion

Ref	Expression
funpartfv	|- (Funpart F ` A ) = ( F ` A )

Proof of Theorem funpartfv

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-funpart 31981	. . 3 |- Funpart F = ( F |` dom ( (Image F o. Singleton ) i^i ( _V X. Singletons ) ) )
2	1	fveq1i 6192	. 2 |- (Funpart F ` A ) = ( ( F |` dom ( (Image F o. Singleton ) i^i ( _V X. Singletons ) ) ) ` A )
3		fvres 6207	. . 3 |- ( A e. dom ( (Image F o. Singleton ) i^i ( _V X. Singletons ) ) -> ( ( F |` dom ( (Image F o. Singleton ) i^i ( _V X. Singletons ) ) ) ` A ) = ( F ` A ) )
4		nfvres 6224	. . . 4 |- ( -. A e. dom ( (Image F o. Singleton ) i^i ( _V X. Singletons ) ) -> ( ( F |` dom ( (Image F o. Singleton ) i^i ( _V X. Singletons ) ) ) ` A ) = (/) )
5		funpartlem 32049	. . . . . . . . 9 |- ( A e. dom ( (Image F o. Singleton ) i^i ( _V X. Singletons ) ) <-> E. x ( F " { A } ) = { x } )
6		eusn 4265	. . . . . . . . 9 |- ( E! x x e. ( F " { A } ) <-> E. x ( F " { A } ) = { x } )
7	5, 6	bitr4i 267	. . . . . . . 8 |- ( A e. dom ( (Image F o. Singleton ) i^i ( _V X. Singletons ) ) <-> E! x x e. ( F " { A } ) )
8		vex 3203	. . . . . . . . . . 11 |- x e. _V
9		elimasng 5491	. . . . . . . . . . 11 |- ( ( A e. _V /\ x e. _V ) -> ( x e. ( F " { A } ) <-> <. A , x >. e. F ) )
10	8, 9	mpan2 707	. . . . . . . . . 10 |- ( A e. _V -> ( x e. ( F " { A } ) <-> <. A , x >. e. F ) )
11		df-br 4654	. . . . . . . . . 10 |- ( A F x <-> <. A , x >. e. F )
12	10, 11	syl6bbr 278	. . . . . . . . 9 |- ( A e. _V -> ( x e. ( F " { A } ) <-> A F x ) )
13	12	eubidv 2490	. . . . . . . 8 |- ( A e. _V -> ( E! x x e. ( F " { A } ) <-> E! x A F x ) )
14	7, 13	syl5bb 272	. . . . . . 7 |- ( A e. _V -> ( A e. dom ( (Image F o. Singleton ) i^i ( _V X. Singletons ) ) <-> E! x A F x ) )
15	14	notbid 308	. . . . . 6 |- ( A e. _V -> ( -. A e. dom ( (Image F o. Singleton ) i^i ( _V X. Singletons ) ) <-> -. E! x A F x ) )
16		tz6.12-2 6182	. . . . . 6 |- ( -. E! x A F x -> ( F ` A ) = (/) )
17	15, 16	syl6bi 243	. . . . 5 |- ( A e. _V -> ( -. A e. dom ( (Image F o. Singleton ) i^i ( _V X. Singletons ) ) -> ( F ` A ) = (/) ) )
18		fvprc 6185	. . . . . 6 |- ( -. A e. _V -> ( F ` A ) = (/) )
19	18	a1d 25	. . . . 5 |- ( -. A e. _V -> ( -. A e. dom ( (Image F o. Singleton ) i^i ( _V X. Singletons ) ) -> ( F ` A ) = (/) ) )
20	17, 19	pm2.61i 176	. . . 4 |- ( -. A e. dom ( (Image F o. Singleton ) i^i ( _V X. Singletons ) ) -> ( F ` A ) = (/) )
21	4, 20	eqtr4d 2659	. . 3 |- ( -. A e. dom ( (Image F o. Singleton ) i^i ( _V X. Singletons ) ) -> ( ( F |` dom ( (Image F o. Singleton ) i^i ( _V X. Singletons ) ) ) ` A ) = ( F ` A ) )
22	3, 21	pm2.61i 176	. 2 |- ( ( F |` dom ( (Image F o. Singleton ) i^i ( _V X. Singletons ) ) ) ` A ) = ( F ` A )
23	2, 22	eqtri 2644	1 |- (Funpart F ` A ) = ( F ` A )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   = wceq 1483   E.wex 1704   e. wcel 1990   E!weu 2470   _Vcvv 3200   i^i cin 3573   (/)c0 3915   {csn 4177   <.cop 4183   class class class wbr 4653   X. cxp 5112   dom cdm 5114   |` cres 5116   "cima 5117   o. ccom 5118   ` cfv 5888  Singletoncsingle 31945   Singletonscsingles 31946  Imagecimage 31947  Funpartcfunpart 31956

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

This theorem is referenced by:  fullfunfv  32054