Theorem funpartfun 30781

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 5138	. 2 |- Rel ( F |` dom ( (Image F o. Singleton ) i^i ( _V X. Singletons ) ) )
2		vex 3034	. . . . . . 7 |- z e. _V
3	2	brres 5117	. . . . . 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 467	. . . . 5 |- ( x ( F |` dom ( (Image F o. Singleton ) i^i ( _V X. Singletons ) ) ) z -> x F z )
5		vex 3034	. . . . . . . 8 |- y e. _V
6	5	brres 5117	. . . . . . 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 457	. . . . . . . 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 30780	. . . . . . . . 9 |- ( x e. dom ( (Image F o. Singleton ) i^i ( _V X. Singletons ) ) <-> E. w ( F " { x } ) = { w } )
9	8	anbi1i 709	. . . . . . . 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 257	. . . . . . 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 257	. . . . . 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 4396	. . . . . . . . . . 11 |- ( x F y <-> <. x , y >. e. F )
13		df-br 4396	. . . . . . . . . . 11 |- ( x F z <-> <. x , z >. e. F )
14	12, 13	anbi12i 711	. . . . . . . . . 10 |- ( ( x F y /\ x F z ) <-> ( <. x , y >. e. F /\ <. x , z >. e. F ) )
15		vex 3034	. . . . . . . . . . . 12 |- x e. _V
16	15, 5	elimasn 5199	. . . . . . . . . . 11 |- ( y e. ( F " { x } ) <-> <. x , y >. e. F )
17	15, 2	elimasn 5199	. . . . . . . . . . 11 |- ( z e. ( F " { x } ) <-> <. x , z >. e. F )
18	16, 17	anbi12i 711	. . . . . . . . . 10 |- ( ( y e. ( F " { x } ) /\ z e. ( F " { x } ) ) <-> ( <. x , y >. e. F /\ <. x , z >. e. F ) )
19	14, 18	bitr4i 260	. . . . . . . . 9 |- ( ( x F y /\ x F z ) <-> ( y e. ( F " { x } ) /\ z e. ( F " { x } ) ) )
20		eleq2 2538	. . . . . . . . . . 11 |- ( ( F " { x } ) = { w } -> ( y e. ( F " { x } ) <-> y e. { w } ) )
21		eleq2 2538	. . . . . . . . . . 11 |- ( ( F " { x } ) = { w } -> ( z e. ( F " { x } ) <-> z e. { w } ) )
22	20, 21	anbi12d 725	. . . . . . . . . 10 |- ( ( F " { x } ) = { w } -> ( ( y e. ( F " { x } ) /\ z e. ( F " { x } ) ) <-> ( y e. { w } /\ z e. { w } ) ) )
23		elsn 3973	. . . . . . . . . . 11 |- ( y e. { w } <-> y = w )
24		elsn 3973	. . . . . . . . . . 11 |- ( z e. { w } <-> z = w )
25		equtr2 1877	. . . . . . . . . . 11 |- ( ( y = w /\ z = w ) -> y = z )
26	23, 24, 25	syl2anb 487	. . . . . . . . . 10 |- ( ( y e. { w } /\ z e. { w } ) -> y = z )
27	22, 26	syl6bi 236	. . . . . . . . 9 |- ( ( F " { x } ) = { w } -> ( ( y e. ( F " { x } ) /\ z e. ( F " { x } ) ) -> y = z ) )
28	19, 27	syl5bi 225	. . . . . . . 8 |- ( ( F " { x } ) = { w } -> ( ( x F y /\ x F z ) -> y = z ) )
29	28	exlimiv 1784	. . . . . . 7 |- ( E. w ( F " { x } ) = { w } -> ( ( x F y /\ x F z ) -> y = z ) )
30	29	impl 632	. . . . . 6 |- ( ( ( E. w ( F " { x } ) = { w } /\ x F y ) /\ x F z ) -> y = z )
31	11, 30	sylanb 480	. . . . 5 |- ( ( x ( F |` dom ( (Image F o. Singleton ) i^i ( _V X. Singletons ) ) ) y /\ x F z ) -> y = z )
32	4, 31	sylan2 482	. . . 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 1678	. . 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 1677	. 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 30711	. . . 4 |- Funpart F = ( F |` dom ( (Image F o. Singleton ) i^i ( _V X. Singletons ) ) )
36	35	funeqi 5609	. . 3 |- ( Fun Funpart F <-> Fun ( F |` dom ( (Image F o. Singleton ) i^i ( _V X. Singletons ) ) ) )
37		dffun2 5599	. . 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 257	. 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 934	1 |- Fun Funpart F

Syntax hints:   -> wi 4   /\ wa 376   A.wal 1450   = wceq 1452   E.wex 1671   e. wcel 1904   _Vcvv 3031   i^i cin 3389   {csn 3959   <.cop 3965   class class class wbr 4395   X. cxp 4837   dom cdm 4839   |` cres 4841   "cima 4842   o. ccom 4843   Rel wrel 4844   Fun wfun 5583  Singletoncsingle 30675   Singletonscsingles 30676  Imagecimage 30677  Funpartcfunpart 30686

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602

This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-sbc 3256  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-symdif 3654  df-nul 3723  df-if 3873  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-br 4396  df-opab 4455  df-mpt 4456  df-eprel 4750  df-id 4754  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-fo 5595  df-fv 5597  df-1st 6812  df-2nd 6813  df-txp 30691  df-singleton 30699  df-singles 30700  df-image 30701  df-funpart 30711

This theorem is referenced by:  fullfunfnv  30784  fullfunfv  30785