Theorem neifg 29781

Description: The neighborhood filter of a nonempty set is generated by its open supersets. See comments for opnfbas 20073. (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 20073	. . 3 |- ( ( J e. Top /\ S C_ X /\ S =/= (/) ) -> { x e. J | S C_ x } e. ( fBas ` X ) )
3		fgval 20101	. . 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 16	. 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 4008	. . . . . . 7 |- ( t = u -> ~P t = ~P u )
6	5	ineq2d 3695	. . . . . 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 2739	. . . . 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 3256	. . . 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 4012	. . . . . . 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 3789	. . . . . . . 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 3682	. . . . . . . . . 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 3521	. . . . . . . . . . . 12 |- ( x = z -> ( S C_ x <-> S C_ z ) )
14	13	elrab 3256	. . . . . . . . . . 11 |- ( z e. { x e. J | S C_ x } <-> ( z e. J /\ S C_ z ) )
15		selpw 4012	. . . . . . . . . . 11 |- ( z e. ~P u <-> z C_ u )
16	14, 15	anbi12i 697	. . . . . . . . . 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 249	. . . . . . . . 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 1639	. . . . . . . 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 249	. . . . . . 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 710	. . . . 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 19365	. . . . . . 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 1011	. . . . . 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 649	. . . . . . . . 9 |- ( ( ( z e. J /\ S C_ z ) /\ z C_ u ) <-> ( z e. J /\ ( S C_ z /\ z C_ u ) ) )
25	24	exbii 1639	. . . . . . . 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 2815	. . . . . . . 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 252	. . . . . . 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 694	. . . . . 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 264	. . . . 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 253	. . . 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 257	. . 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 2459	. 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 2503	1 |- ( ( J e. Top /\ S C_ X /\ S =/= (/) ) -> ( X filGen { x e. J | S C_ x } ) = ( ( nei ` J ) ` S ) )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 369   /\ w3a 968   = wceq 1374   E.wex 1591   e. wcel 1762   =/= wne 2657   E.wrex 2810   {crab 2813   i^i cin 3470   C_ wss 3471   (/)c0 3780   ~Pcpw 4005   U.cuni 4240   ` cfv 5581  (class class class)co 6277   fBascfbas 18172   filGencfg 18173   Topctop 19156   neicnei 19359

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-rep 4553  ax-sep 4563  ax-nul 4571  ax-pow 4620  ax-pr 4681

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-nel 2660  df-ral 2814  df-rex 2815  df-reu 2816  df-rab 2818  df-v 3110  df-sbc 3327  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3781  df-if 3935  df-pw 4007  df-sn 4023  df-pr 4025  df-op 4029  df-uni 4241  df-iun 4322  df-br 4443  df-opab 4501  df-mpt 4502  df-id 4790  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-rn 5005  df-res 5006  df-ima 5007  df-iota 5544  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-fbas 18182  df-fg 18183  df-top 19161  df-nei 19360

This theorem is referenced by: (None)