Theorem opnneissb 12313

Description: An open set is a neighborhood of any of its subsets. (Contributed by FL, 2-Oct-2006.)

Hypothesis

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

Assertion

Ref	Expression
opnneissb	|- ( ( J e. Top /\ N e. J /\ S C_ X ) -> ( S C_ N <-> N e. ( ( nei ` J ) ` S ) ) )

Proof of Theorem opnneissb

Dummy variable g is distinct from all other variables.

Step	Hyp	Ref	Expression
1		neips.1	. . . . . . 7 |- X = U. J
2	1	eltopss 12165	. . . . . 6 |- ( ( J e. Top /\ N e. J ) -> N C_ X )
3	2	adantr 274	. . . . 5 |- ( ( ( J e. Top /\ N e. J ) /\ ( S C_ X /\ S C_ N ) ) -> N C_ X )
4		ssid 3112	. . . . . . 7 |- N C_ N
5		sseq2 3116	. . . . . . . . 9 |- ( g = N -> ( S C_ g <-> S C_ N ) )
6		sseq1 3115	. . . . . . . . 9 |- ( g = N -> ( g C_ N <-> N C_ N ) )
7	5, 6	anbi12d 464	. . . . . . . 8 |- ( g = N -> ( ( S C_ g /\ g C_ N ) <-> ( S C_ N /\ N C_ N ) ) )
8	7	rspcev 2784	. . . . . . 7 |- ( ( N e. J /\ ( S C_ N /\ N C_ N ) ) -> E. g e. J ( S C_ g /\ g C_ N ) )
9	4, 8	mpanr2 434	. . . . . 6 |- ( ( N e. J /\ S C_ N ) -> E. g e. J ( S C_ g /\ g C_ N ) )
10	9	ad2ant2l 499	. . . . 5 |- ( ( ( J e. Top /\ N e. J ) /\ ( S C_ X /\ S C_ N ) ) -> E. g e. J ( S C_ g /\ g C_ N ) )
11	1	isnei 12302	. . . . . 6 |- ( ( J e. Top /\ S C_ X ) -> ( N e. ( ( nei ` J ) ` S ) <-> ( N C_ X /\ E. g e. J ( S C_ g /\ g C_ N ) ) ) )
12	11	ad2ant2r 500	. . . . 5 |- ( ( ( J e. Top /\ N e. J ) /\ ( S C_ X /\ S C_ N ) ) -> ( N e. ( ( nei ` J ) ` S ) <-> ( N C_ X /\ E. g e. J ( S C_ g /\ g C_ N ) ) ) )
13	3, 10, 12	mpbir2and 928	. . . 4 |- ( ( ( J e. Top /\ N e. J ) /\ ( S C_ X /\ S C_ N ) ) -> N e. ( ( nei ` J ) ` S ) )
14	13	exp43 369	. . 3 |- ( J e. Top -> ( N e. J -> ( S C_ X -> ( S C_ N -> N e. ( ( nei ` J ) ` S ) ) ) ) )
15	14	3imp 1175	. 2 |- ( ( J e. Top /\ N e. J /\ S C_ X ) -> ( S C_ N -> N e. ( ( nei ` J ) ` S ) ) )
16		ssnei 12309	. . . 4 |- ( ( J e. Top /\ N e. ( ( nei ` J ) ` S ) ) -> S C_ N )
17	16	ex 114	. . 3 |- ( J e. Top -> ( N e. ( ( nei ` J ) ` S ) -> S C_ N ) )
18	17	3ad2ant1 1002	. 2 |- ( ( J e. Top /\ N e. J /\ S C_ X ) -> ( N e. ( ( nei ` J ) ` S ) -> S C_ N ) )
19	15, 18	impbid 128	1 |- ( ( J e. Top /\ N e. J /\ S C_ X ) -> ( S C_ N <-> N e. ( ( nei ` J ) ` S ) ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   /\ w3a 962   = wceq 1331   e. wcel 1480   E.wrex 2415   C_ wss 3066   U.cuni 3731   ` cfv 5118   Topctop 12153   neicnei 12296

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-coll 4038  ax-sep 4041  ax-pow 4093  ax-pr 4126

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-reu 2421  df-rab 2423  df-v 2683  df-sbc 2905  df-csb 2999  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-iun 3810  df-br 3925  df-opab 3985  df-mpt 3986  df-id 4210  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-res 4546  df-ima 4547  df-iota 5083  df-fun 5120  df-fn 5121  df-f 5122  df-f1 5123  df-fo 5124  df-f1o 5125  df-fv 5126  df-top 12154  df-nei 12297

This theorem is referenced by:  opnneiss  12316