Theorem neifg 32366

Description: The neighborhood filter of a nonempty set is generated by its open supersets. See comments for opnfbas 21646. (Contributed by Jeff Hankins, 3-Sep-2009.)

Hypothesis

Ref	Expression
neifg.1	|- X = U. J

Assertion

Ref	Expression
neifg	|- ( ( J e. Top /\ S C_ X /\ S =/= (/) ) -> ( X filGen { x e. J | S C_ x } ) = ( ( nei ` J ) ` S ) )

Distinct variable groups:   x, J   x, S   x, X

Proof of Theorem neifg

Dummy variables u t z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		neifg.1	. . . 4 |- X = U. J
2	1	opnfbas 21646	. . 3 |- ( ( J e. Top /\ S C_ X /\ S =/= (/) ) -> { x e. J | S C_ x } e. ( fBas ` X ) )
3		fgval 21674	. . 3 |- ( { x e. J | S C_ x } e. ( fBas ` X ) -> ( X filGen { x e. J | S C_ x } ) = { t e. ~P X | ( { x e. J | S C_ x } i^i ~P t ) =/= (/) } )
4	2, 3	syl 17	. 2 |- ( ( J e. Top /\ S C_ X /\ S =/= (/) ) -> ( X filGen { x e. J | S C_ x } ) = { t e. ~P X | ( { x e. J | S C_ x } i^i ~P t ) =/= (/) } )
5		pweq 4161	. . . . . . 7 |- ( t = u -> ~P t = ~P u )
6	5	ineq2d 3814	. . . . . 6 |- ( t = u -> ( { x e. J | S C_ x } i^i ~P t ) = ( { x e. J | S C_ x } i^i ~P u ) )
7	6	neeq1d 2853	. . . . 5 |- ( t = u -> ( ( { x e. J | S C_ x } i^i ~P t ) =/= (/) <-> ( { x e. J | S C_ x } i^i ~P u ) =/= (/) ) )
8	7	elrab 3363	. . . 4 |- ( u e. { t e. ~P X | ( { x e. J | S C_ x } i^i ~P t ) =/= (/) } <-> ( u e. ~P X /\ ( { x e. J | S C_ x } i^i ~P u ) =/= (/) ) )
9		selpw 4165	. . . . . . 7 |- ( u e. ~P X <-> u C_ X )
10	9	a1i 11	. . . . . 6 |- ( ( J e. Top /\ S C_ X /\ S =/= (/) ) -> ( u e. ~P X <-> u C_ X ) )
11		n0 3931	. . . . . . . 8 |- ( ( { x e. J | S C_ x } i^i ~P u ) =/= (/) <-> E. z z e. ( { x e. J | S C_ x } i^i ~P u ) )
12		elin 3796	. . . . . . . . . 10 |- ( z e. ( { x e. J | S C_ x } i^i ~P u ) <-> ( z e. { x e. J | S C_ x } /\ z e. ~P u ) )
13		sseq2 3627	. . . . . . . . . . . 12 |- ( x = z -> ( S C_ x <-> S C_ z ) )
14	13	elrab 3363	. . . . . . . . . . 11 |- ( z e. { x e. J | S C_ x } <-> ( z e. J /\ S C_ z ) )
15		selpw 4165	. . . . . . . . . . 11 |- ( z e. ~P u <-> z C_ u )
16	14, 15	anbi12i 733	. . . . . . . . . 10 |- ( ( z e. { x e. J | S C_ x } /\ z e. ~P u ) <-> ( ( z e. J /\ S C_ z ) /\ z C_ u ) )
17	12, 16	bitri 264	. . . . . . . . 9 |- ( z e. ( { x e. J | S C_ x } i^i ~P u ) <-> ( ( z e. J /\ S C_ z ) /\ z C_ u ) )
18	17	exbii 1774	. . . . . . . 8 |- ( E. z z e. ( { x e. J | S C_ x } i^i ~P u ) <-> E. z ( ( z e. J /\ S C_ z ) /\ z C_ u ) )
19	11, 18	bitri 264	. . . . . . 7 |- ( ( { x e. J | S C_ x } i^i ~P u ) =/= (/) <-> E. z ( ( z e. J /\ S C_ z ) /\ z C_ u ) )
20	19	a1i 11	. . . . . 6 |- ( ( J e. Top /\ S C_ X /\ S =/= (/) ) -> ( ( { x e. J | S C_ x } i^i ~P u ) =/= (/) <-> E. z ( ( z e. J /\ S C_ z ) /\ z C_ u ) ) )
21	10, 20	anbi12d 747	. . . . 5 |- ( ( J e. Top /\ S C_ X /\ S =/= (/) ) -> ( ( u e. ~P X /\ ( { x e. J | S C_ x } i^i ~P u ) =/= (/) ) <-> ( u C_ X /\ E. z ( ( z e. J /\ S C_ z ) /\ z C_ u ) ) ) )
22	1	isnei 20907	. . . . . . 7 |- ( ( J e. Top /\ S C_ X ) -> ( u e. ( ( nei ` J ) ` S ) <-> ( u C_ X /\ E. z e. J ( S C_ z /\ z C_ u ) ) ) )
23	22	3adant3 1081	. . . . . 6 |- ( ( J e. Top /\ S C_ X /\ S =/= (/) ) -> ( u e. ( ( nei ` J ) ` S ) <-> ( u C_ X /\ E. z e. J ( S C_ z /\ z C_ u ) ) ) )
24		anass 681	. . . . . . . . 9 |- ( ( ( z e. J /\ S C_ z ) /\ z C_ u ) <-> ( z e. J /\ ( S C_ z /\ z C_ u ) ) )
25	24	exbii 1774	. . . . . . . 8 |- ( E. z ( ( z e. J /\ S C_ z ) /\ z C_ u ) <-> E. z ( z e. J /\ ( S C_ z /\ z C_ u ) ) )
26		df-rex 2918	. . . . . . . 8 |- ( E. z e. J ( S C_ z /\ z C_ u ) <-> E. z ( z e. J /\ ( S C_ z /\ z C_ u ) ) )
27	25, 26	bitr4i 267	. . . . . . 7 |- ( E. z ( ( z e. J /\ S C_ z ) /\ z C_ u ) <-> E. z e. J ( S C_ z /\ z C_ u ) )
28	27	anbi2i 730	. . . . . 6 |- ( ( u C_ X /\ E. z ( ( z e. J /\ S C_ z ) /\ z C_ u ) ) <-> ( u C_ X /\ E. z e. J ( S C_ z /\ z C_ u ) ) )
29	23, 28	syl6rbbr 279	. . . . 5 |- ( ( J e. Top /\ S C_ X /\ S =/= (/) ) -> ( ( u C_ X /\ E. z ( ( z e. J /\ S C_ z ) /\ z C_ u ) ) <-> u e. ( ( nei ` J ) ` S ) ) )
30	21, 29	bitrd 268	. . . 4 |- ( ( J e. Top /\ S C_ X /\ S =/= (/) ) -> ( ( u e. ~P X /\ ( { x e. J | S C_ x } i^i ~P u ) =/= (/) ) <-> u e. ( ( nei ` J ) ` S ) ) )
31	8, 30	syl5bb 272	. . 3 |- ( ( J e. Top /\ S C_ X /\ S =/= (/) ) -> ( u e. { t e. ~P X | ( { x e. J | S C_ x } i^i ~P t ) =/= (/) } <-> u e. ( ( nei ` J ) ` S ) ) )
32	31	eqrdv 2620	. 2 |- ( ( J e. Top /\ S C_ X /\ S =/= (/) ) -> { t e. ~P X | ( { x e. J | S C_ x } i^i ~P t ) =/= (/) } = ( ( nei ` J ) ` S ) )
33	4, 32	eqtrd 2656	1 |- ( ( J e. Top /\ S C_ X /\ S =/= (/) ) -> ( X filGen { x e. J | S C_ x } ) = ( ( nei ` J ) ` S ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   E.wex 1704   e. wcel 1990   =/= wne 2794   E.wrex 2913   {crab 2916   i^i cin 3573   C_ wss 3574   (/)c0 3915   ~Pcpw 4158   U.cuni 4436   ` cfv 5888  (class class class)co 6650   fBascfbas 19734   filGencfg 19735   Topctop 20698   neicnei 20901

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-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906

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-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  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-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-fbas 19743  df-fg 19744  df-top 20699  df-nei 20902

This theorem is referenced by: (None)