Theorem funpartfun 30710

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 5132	. 2 |- Rel ( F |` dom ( (Image F o. Singleton ) i^i ( _V X. Singletons ) ) )
2		vex 3048	. . . . . . 7 |- z e. _V
3	2	brres 5111	. . . . . 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 462	. . . . 5 |- ( x ( F |` dom ( (Image F o. Singleton ) i^i ( _V X. Singletons ) ) ) z -> x F z )
5		vex 3048	. . . . . . . 8 |- y e. _V
6	5	brres 5111	. . . . . . 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 452	. . . . . . . 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 30709	. . . . . . . . 9 |- ( x e. dom ( (Image F o. Singleton ) i^i ( _V X. Singletons ) ) <-> E. w ( F " { x } ) = { w } )
9	8	anbi1i 701	. . . . . . . 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 253	. . . . . . 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 253	. . . . . 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 4403	. . . . . . . . . . 11 |- ( x F y <-> <. x , y >. e. F )
13		df-br 4403	. . . . . . . . . . 11 |- ( x F z <-> <. x , z >. e. F )
14	12, 13	anbi12i 703	. . . . . . . . . 10 |- ( ( x F y /\ x F z ) <-> ( <. x , y >. e. F /\ <. x , z >. e. F ) )
15		vex 3048	. . . . . . . . . . . 12 |- x e. _V
16	15, 5	elimasn 5193	. . . . . . . . . . 11 |- ( y e. ( F " { x } ) <-> <. x , y >. e. F )
17	15, 2	elimasn 5193	. . . . . . . . . . 11 |- ( z e. ( F " { x } ) <-> <. x , z >. e. F )
18	16, 17	anbi12i 703	. . . . . . . . . 10 |- ( ( y e. ( F " { x } ) /\ z e. ( F " { x } ) ) <-> ( <. x , y >. e. F /\ <. x , z >. e. F ) )
19	14, 18	bitr4i 256	. . . . . . . . 9 |- ( ( x F y /\ x F z ) <-> ( y e. ( F " { x } ) /\ z e. ( F " { x } ) ) )
20		eleq2 2518	. . . . . . . . . . 11 |- ( ( F " { x } ) = { w } -> ( y e. ( F " { x } ) <-> y e. { w } ) )
21		eleq2 2518	. . . . . . . . . . 11 |- ( ( F " { x } ) = { w } -> ( z e. ( F " { x } ) <-> z e. { w } ) )
22	20, 21	anbi12d 717	. . . . . . . . . 10 |- ( ( F " { x } ) = { w } -> ( ( y e. ( F " { x } ) /\ z e. ( F " { x } ) ) <-> ( y e. { w } /\ z e. { w } ) ) )
23		elsn 3982	. . . . . . . . . . 11 |- ( y e. { w } <-> y = w )
24		elsn 3982	. . . . . . . . . . 11 |- ( z e. { w } <-> z = w )
25		equtr2 1869	. . . . . . . . . . 11 |- ( ( y = w /\ z = w ) -> y = z )
26	23, 24, 25	syl2anb 482	. . . . . . . . . 10 |- ( ( y e. { w } /\ z e. { w } ) -> y = z )
27	22, 26	syl6bi 232	. . . . . . . . 9 |- ( ( F " { x } ) = { w } -> ( ( y e. ( F " { x } ) /\ z e. ( F " { x } ) ) -> y = z ) )
28	19, 27	syl5bi 221	. . . . . . . 8 |- ( ( F " { x } ) = { w } -> ( ( x F y /\ x F z ) -> y = z ) )
29	28	exlimiv 1776	. . . . . . 7 |- ( E. w ( F " { x } ) = { w } -> ( ( x F y /\ x F z ) -> y = z ) )
30	29	impl 626	. . . . . 6 |- ( ( ( E. w ( F " { x } ) = { w } /\ x F y ) /\ x F z ) -> y = z )
31	11, 30	sylanb 475	. . . . 5 |- ( ( x ( F |` dom ( (Image F o. Singleton ) i^i ( _V X. Singletons ) ) ) y /\ x F z ) -> y = z )
32	4, 31	sylan2 477	. . . 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 1670	. . 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 1669	. 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 30640	. . . 4 |- Funpart F = ( F |` dom ( (Image F o. Singleton ) i^i ( _V X. Singletons ) ) )
36	35	funeqi 5602	. . 3 |- ( Fun Funpart F <-> Fun ( F |` dom ( (Image F o. Singleton ) i^i ( _V X. Singletons ) ) ) )
37		dffun2 5592	. . 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 253	. 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 931	1 |- Fun Funpart F

Syntax hints:   -> wi 4   /\ wa 371   A.wal 1442   = wceq 1444   E.wex 1663   e. wcel 1887   _Vcvv 3045   i^i cin 3403   {csn 3968   <.cop 3974   class class class wbr 4402   X. cxp 4832   dom cdm 4834   |` cres 4836   "cima 4837   o. ccom 4838   Rel wrel 4839   Fun wfun 5576  Singletoncsingle 30604   Singletonscsingles 30605  Imagecimage 30606  Funpartcfunpart 30615

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-ral 2742  df-rex 2743  df-rab 2746  df-v 3047  df-sbc 3268  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-symdif 3663  df-nul 3732  df-if 3882  df-sn 3969  df-pr 3971  df-op 3975  df-uni 4199  df-br 4403  df-opab 4462  df-mpt 4463  df-eprel 4745  df-id 4749  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-fo 5588  df-fv 5590  df-1st 6793  df-2nd 6794  df-txp 30620  df-singleton 30628  df-singles 30629  df-image 30630  df-funpart 30640

This theorem is referenced by:  fullfunfnv  30713  fullfunfv  30714