Theorem neifg 31098

Description: The neighborhood filter of a nonempty set is generated by its open supersets. See comments for opnfbas 20935. (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 20935	. . 3 |- ( ( J e. Top /\ S C_ X /\ S =/= (/) ) -> { x e. J | S C_ x } e. ( fBas ` X ) )
3		fgval 20963	. . 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 3945	. . . . . . 7 |- ( t = u -> ~P t = ~P u )
6	5	ineq2d 3625	. . . . . 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 2702	. . . . 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 3184	. . . 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 3949	. . . . . . 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 3732	. . . . . . . 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 3608	. . . . . . . . . 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 3440	. . . . . . . . . . . 12 |- ( x = z -> ( S C_ x <-> S C_ z ) )
14	13	elrab 3184	. . . . . . . . . . 11 |- ( z e. { x e. J | S C_ x } <-> ( z e. J /\ S C_ z ) )
15		selpw 3949	. . . . . . . . . . 11 |- ( z e. ~P u <-> z C_ u )
16	14, 15	anbi12i 711	. . . . . . . . . 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 257	. . . . . . . . 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 1726	. . . . . . . 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 257	. . . . . . 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 725	. . . . 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 20196	. . . . . . 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 1050	. . . . . 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 661	. . . . . . . . 9 |- ( ( ( z e. J /\ S C_ z ) /\ z C_ u ) <-> ( z e. J /\ ( S C_ z /\ z C_ u ) ) )
25	24	exbii 1726	. . . . . . . 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 2762	. . . . . . . 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 260	. . . . . . 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 708	. . . . . 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 272	. . . . 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 261	. . . 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 265	. . 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 2469	. 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 2505	1 |- ( ( J e. Top /\ S C_ X /\ S =/= (/) ) -> ( X filGen { x e. J | S C_ x } ) = ( ( nei ` J ) ` S ) )

Syntax hints:   -> wi 4   <-> wb 189   /\ wa 376   /\ w3a 1007   = wceq 1452   E.wex 1671   e. wcel 1904   =/= wne 2641   E.wrex 2757   {crab 2760   i^i cin 3389   C_ wss 3390   (/)c0 3722   ~Pcpw 3942   U.cuni 4190   ` cfv 5589  (class class class)co 6308   fBascfbas 19035   filGencfg 19036   Topctop 19994   neicnei 20190

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-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639

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-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  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-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-fbas 19044  df-fg 19045  df-top 19998  df-nei 20191

This theorem is referenced by: (None)