Theorem neifg 30373

Description: The neighborhood filter of a nonempty set is generated by its open supersets. See comments for opnfbas 20469. (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 20469	. . 3 |- ( ( J e. Top /\ S C_ X /\ S =/= (/) ) -> { x e. J | S C_ x } e. ( fBas ` X ) )
3		fgval 20497	. . 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 4018	. . . . . . 7 |- ( t = u -> ~P t = ~P u )
6	5	ineq2d 3696	. . . . . 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 2734	. . . . 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 3257	. . . 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 4022	. . . . . . 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 3803	. . . . . . . 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 3683	. . . . . . . . . 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 3257	. . . . . . . . . . 11 |- ( z e. { x e. J | S C_ x } <-> ( z e. J /\ S C_ z ) )
15		selpw 4022	. . . . . . . . . . 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 1668	. . . . . . . 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 19731	. . . . . . 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 1016	. . . . . 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 1668	. . . . . . . 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 2813	. . . . . . . 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 2454	. 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 2498	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 973   = wceq 1395   E.wex 1613   e. wcel 1819   =/= wne 2652   E.wrex 2808   {crab 2811   i^i cin 3470   C_ wss 3471   (/)c0 3793   ~Pcpw 4015   U.cuni 4251   ` cfv 5594  (class class class)co 6296   fBascfbas 18533   filGencfg 18534   Topctop 19521   neicnei 19725

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-fbas 18543  df-fg 18544  df-top 19526  df-nei 19726

This theorem is referenced by: (None)