Theorem funpartfun 32050

Description: The functional part of F is a function. (Contributed by Scott Fenton, 16-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)

Assertion

Ref	Expression
funpartfun	|- Fun Funpart F

Proof of Theorem funpartfun

Dummy variables x y z w are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		relres 5426	. 2 |- Rel ( F |` dom ( (Image F o. Singleton ) i^i ( _V X. Singletons ) ) )
2		vex 3203	. . . . . . 7 |- z e. _V
3	2	brres 5402	. . . . . 6 |- ( x ( F |` dom ( (Image F o. Singleton ) i^i ( _V X. Singletons ) ) ) z <-> ( x F z /\ x e. dom ( (Image F o. Singleton ) i^i ( _V X. Singletons ) ) ) )
4	3	simplbi 476	. . . . 5 |- ( x ( F |` dom ( (Image F o. Singleton ) i^i ( _V X. Singletons ) ) ) z -> x F z )
5		vex 3203	. . . . . . . 8 |- y e. _V
6	5	brres 5402	. . . . . . 7 |- ( x ( F |` dom ( (Image F o. Singleton ) i^i ( _V X. Singletons ) ) ) y <-> ( x F y /\ x e. dom ( (Image F o. Singleton ) i^i ( _V X. Singletons ) ) ) )
7		ancom 466	. . . . . . . 8 |- ( ( x F y /\ x e. dom ( (Image F o. Singleton ) i^i ( _V X. Singletons ) ) ) <-> ( x e. dom ( (Image F o. Singleton ) i^i ( _V X. Singletons ) ) /\ x F y ) )
8		funpartlem 32049	. . . . . . . . 9 |- ( x e. dom ( (Image F o. Singleton ) i^i ( _V X. Singletons ) ) <-> E. w ( F " { x } ) = { w } )
9	8	anbi1i 731	. . . . . . . 8 |- ( ( x e. dom ( (Image F o. Singleton ) i^i ( _V X. Singletons ) ) /\ x F y ) <-> ( E. w ( F " { x } ) = { w } /\ x F y ) )
10	7, 9	bitri 264	. . . . . . 7 |- ( ( x F y /\ x e. dom ( (Image F o. Singleton ) i^i ( _V X. Singletons ) ) ) <-> ( E. w ( F " { x } ) = { w } /\ x F y ) )
11	6, 10	bitri 264	. . . . . 6 |- ( x ( F |` dom ( (Image F o. Singleton ) i^i ( _V X. Singletons ) ) ) y <-> ( E. w ( F " { x } ) = { w } /\ x F y ) )
12		df-br 4654	. . . . . . . . . . 11 |- ( x F y <-> <. x , y >. e. F )
13		df-br 4654	. . . . . . . . . . 11 |- ( x F z <-> <. x , z >. e. F )
14	12, 13	anbi12i 733	. . . . . . . . . 10 |- ( ( x F y /\ x F z ) <-> ( <. x , y >. e. F /\ <. x , z >. e. F ) )
15		vex 3203	. . . . . . . . . . . 12 |- x e. _V
16	15, 5	elimasn 5490	. . . . . . . . . . 11 |- ( y e. ( F " { x } ) <-> <. x , y >. e. F )
17	15, 2	elimasn 5490	. . . . . . . . . . 11 |- ( z e. ( F " { x } ) <-> <. x , z >. e. F )
18	16, 17	anbi12i 733	. . . . . . . . . 10 |- ( ( y e. ( F " { x } ) /\ z e. ( F " { x } ) ) <-> ( <. x , y >. e. F /\ <. x , z >. e. F ) )
19	14, 18	bitr4i 267	. . . . . . . . 9 |- ( ( x F y /\ x F z ) <-> ( y e. ( F " { x } ) /\ z e. ( F " { x } ) ) )
20		eleq2 2690	. . . . . . . . . . 11 |- ( ( F " { x } ) = { w } -> ( y e. ( F " { x } ) <-> y e. { w } ) )
21		eleq2 2690	. . . . . . . . . . 11 |- ( ( F " { x } ) = { w } -> ( z e. ( F " { x } ) <-> z e. { w } ) )
22	20, 21	anbi12d 747	. . . . . . . . . 10 |- ( ( F " { x } ) = { w } -> ( ( y e. ( F " { x } ) /\ z e. ( F " { x } ) ) <-> ( y e. { w } /\ z e. { w } ) ) )
23		velsn 4193	. . . . . . . . . . 11 |- ( y e. { w } <-> y = w )
24		velsn 4193	. . . . . . . . . . 11 |- ( z e. { w } <-> z = w )
25		equtr2 1954	. . . . . . . . . . 11 |- ( ( y = w /\ z = w ) -> y = z )
26	23, 24, 25	syl2anb 496	. . . . . . . . . 10 |- ( ( y e. { w } /\ z e. { w } ) -> y = z )
27	22, 26	syl6bi 243	. . . . . . . . 9 |- ( ( F " { x } ) = { w } -> ( ( y e. ( F " { x } ) /\ z e. ( F " { x } ) ) -> y = z ) )
28	19, 27	syl5bi 232	. . . . . . . 8 |- ( ( F " { x } ) = { w } -> ( ( x F y /\ x F z ) -> y = z ) )
29	28	exlimiv 1858	. . . . . . 7 |- ( E. w ( F " { x } ) = { w } -> ( ( x F y /\ x F z ) -> y = z ) )
30	29	impl 650	. . . . . 6 |- ( ( ( E. w ( F " { x } ) = { w } /\ x F y ) /\ x F z ) -> y = z )
31	11, 30	sylanb 489	. . . . 5 |- ( ( x ( F |` dom ( (Image F o. Singleton ) i^i ( _V X. Singletons ) ) ) y /\ x F z ) -> y = z )
32	4, 31	sylan2 491	. . . 4 |- ( ( x ( F |` dom ( (Image F o. Singleton ) i^i ( _V X. Singletons ) ) ) y /\ x ( F |` dom ( (Image F o. Singleton ) i^i ( _V X. Singletons ) ) ) z ) -> y = z )
33	32	gen2 1723	. . 3 |- A. y A. z ( ( x ( F |` dom ( (Image F o. Singleton ) i^i ( _V X. Singletons ) ) ) y /\ x ( F |` dom ( (Image F o. Singleton ) i^i ( _V X. Singletons ) ) ) z ) -> y = z )
34	33	ax-gen 1722	. 2 |- A. x A. y A. z ( ( x ( F |` dom ( (Image F o. Singleton ) i^i ( _V X. Singletons ) ) ) y /\ x ( F |` dom ( (Image F o. Singleton ) i^i ( _V X. Singletons ) ) ) z ) -> y = z )
35		df-funpart 31981	. . . 4 |- Funpart F = ( F |` dom ( (Image F o. Singleton ) i^i ( _V X. Singletons ) ) )
36	35	funeqi 5909	. . 3 |- ( Fun Funpart F <-> Fun ( F |` dom ( (Image F o. Singleton ) i^i ( _V X. Singletons ) ) ) )
37		dffun2 5898	. . 3 |- ( Fun ( F |` dom ( (Image F o. Singleton ) i^i ( _V X. Singletons ) ) ) <-> ( Rel ( F |` dom ( (Image F o. Singleton ) i^i ( _V X. Singletons ) ) ) /\ A. x A. y A. z ( ( x ( F |` dom ( (Image F o. Singleton ) i^i ( _V X. Singletons ) ) ) y /\ x ( F |` dom ( (Image F o. Singleton ) i^i ( _V X. Singletons ) ) ) z ) -> y = z ) ) )
38	36, 37	bitri 264	. 2 |- ( Fun Funpart F <-> ( Rel ( F |` dom ( (Image F o. Singleton ) i^i ( _V X. Singletons ) ) ) /\ A. x A. y A. z ( ( x ( F |` dom ( (Image F o. Singleton ) i^i ( _V X. Singletons ) ) ) y /\ x ( F |` dom ( (Image F o. Singleton ) i^i ( _V X. Singletons ) ) ) z ) -> y = z ) ) )
39	1, 34, 38	mpbir2an 955	1 |- Fun Funpart F

Syntax hints:   -> wi 4   /\ wa 384   A.wal 1481   = wceq 1483   E.wex 1704   e. wcel 1990   _Vcvv 3200   i^i cin 3573   {csn 4177   <.cop 4183   class class class wbr 4653   X. cxp 5112   dom cdm 5114   |` cres 5116   "cima 5117   o. ccom 5118   Rel wrel 5119   Fun wfun 5882  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:  fullfunfnv  32053  fullfunfv  32054