Theorem funpartfun 29568

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 5291	. 2 |- Rel ( F |` dom ( (Image F o. Singleton ) i^i ( _V X. Singletons ) ) )
2		vex 3098	. . . . . . 7 |- z e. _V
3	2	brres 5270	. . . . . 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 460	. . . . 5 |- ( x ( F |` dom ( (Image F o. Singleton ) i^i ( _V X. Singletons ) ) ) z -> x F z )
5		vex 3098	. . . . . . . 8 |- y e. _V
6	5	brres 5270	. . . . . . 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 450	. . . . . . . 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 29567	. . . . . . . . 9 |- ( x e. dom ( (Image F o. Singleton ) i^i ( _V X. Singletons ) ) <-> E. w ( F " { x } ) = { w } )
9	8	anbi1i 695	. . . . . . . 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 249	. . . . . . 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 249	. . . . . 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 4438	. . . . . . . . . . 11 |- ( x F y <-> <. x , y >. e. F )
13		df-br 4438	. . . . . . . . . . 11 |- ( x F z <-> <. x , z >. e. F )
14	12, 13	anbi12i 697	. . . . . . . . . 10 |- ( ( x F y /\ x F z ) <-> ( <. x , y >. e. F /\ <. x , z >. e. F ) )
15		vex 3098	. . . . . . . . . . . 12 |- x e. _V
16	15, 5	elimasn 5352	. . . . . . . . . . 11 |- ( y e. ( F " { x } ) <-> <. x , y >. e. F )
17	15, 2	elimasn 5352	. . . . . . . . . . 11 |- ( z e. ( F " { x } ) <-> <. x , z >. e. F )
18	16, 17	anbi12i 697	. . . . . . . . . 10 |- ( ( y e. ( F " { x } ) /\ z e. ( F " { x } ) ) <-> ( <. x , y >. e. F /\ <. x , z >. e. F ) )
19	14, 18	bitr4i 252	. . . . . . . . 9 |- ( ( x F y /\ x F z ) <-> ( y e. ( F " { x } ) /\ z e. ( F " { x } ) ) )
20		eleq2 2516	. . . . . . . . . . 11 |- ( ( F " { x } ) = { w } -> ( y e. ( F " { x } ) <-> y e. { w } ) )
21		eleq2 2516	. . . . . . . . . . 11 |- ( ( F " { x } ) = { w } -> ( z e. ( F " { x } ) <-> z e. { w } ) )
22	20, 21	anbi12d 710	. . . . . . . . . 10 |- ( ( F " { x } ) = { w } -> ( ( y e. ( F " { x } ) /\ z e. ( F " { x } ) ) <-> ( y e. { w } /\ z e. { w } ) ) )
23		elsn 4028	. . . . . . . . . . 11 |- ( y e. { w } <-> y = w )
24		elsn 4028	. . . . . . . . . . 11 |- ( z e. { w } <-> z = w )
25		equtr2 1788	. . . . . . . . . . 11 |- ( ( y = w /\ z = w ) -> y = z )
26	23, 24, 25	syl2anb 479	. . . . . . . . . 10 |- ( ( y e. { w } /\ z e. { w } ) -> y = z )
27	22, 26	syl6bi 228	. . . . . . . . 9 |- ( ( F " { x } ) = { w } -> ( ( y e. ( F " { x } ) /\ z e. ( F " { x } ) ) -> y = z ) )
28	19, 27	syl5bi 217	. . . . . . . 8 |- ( ( F " { x } ) = { w } -> ( ( x F y /\ x F z ) -> y = z ) )
29	28	exlimiv 1709	. . . . . . 7 |- ( E. w ( F " { x } ) = { w } -> ( ( x F y /\ x F z ) -> y = z ) )
30	29	impl 620	. . . . . 6 |- ( ( ( E. w ( F " { x } ) = { w } /\ x F y ) /\ x F z ) -> y = z )
31	11, 30	sylanb 472	. . . . 5 |- ( ( x ( F |` dom ( (Image F o. Singleton ) i^i ( _V X. Singletons ) ) ) y /\ x F z ) -> y = z )
32	4, 31	sylan2 474	. . . 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 1606	. . 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 1605	. 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 29498	. . . 4 |- Funpart F = ( F |` dom ( (Image F o. Singleton ) i^i ( _V X. Singletons ) ) )
36	35	funeqi 5598	. . 3 |- ( Fun Funpart F <-> Fun ( F |` dom ( (Image F o. Singleton ) i^i ( _V X. Singletons ) ) ) )
37		dffun2 5588	. . 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 249	. 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 920	1 |- Fun Funpart F

Syntax hints:   -> wi 4   /\ wa 369   A.wal 1381   = wceq 1383   E.wex 1599   e. wcel 1804   _Vcvv 3095   i^i cin 3460   {csn 4014   <.cop 4020   class class class wbr 4437   X. cxp 4987   dom cdm 4989   |` cres 4991   "cima 4992   o. ccom 4993   Rel wrel 4994   Fun wfun 5572  Singletoncsingle 29462   Singletonscsingles 29463  Imagecimage 29464  Funpartcfunpart 29473

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rex 2799  df-rab 2802  df-v 3097  df-sbc 3314  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3771  df-if 3927  df-sn 4015  df-pr 4017  df-op 4021  df-uni 4235  df-br 4438  df-opab 4496  df-mpt 4497  df-eprel 4781  df-id 4785  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-fo 5584  df-fv 5586  df-1st 6785  df-2nd 6786  df-symdif 29443  df-txp 29478  df-singleton 29486  df-singles 29487  df-image 29488  df-funpart 29498

This theorem is referenced by:  fullfunfnv  29571  fullfunfv  29572