Theorem opnssneib 12325

Description: Any superset of an open set is a neighborhood of it. (Contributed by NM, 14-Feb-2007.)

Hypothesis

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

Assertion

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

Proof of Theorem opnssneib

Dummy variable g is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simplr 519	. . . . . 6 |- ( ( ( S e. J /\ N C_ X ) /\ S C_ N ) -> N C_ X )
2		sseq2 3121	. . . . . . . . . 10 |- ( g = S -> ( S C_ g <-> S C_ S ) )
3		sseq1 3120	. . . . . . . . . 10 |- ( g = S -> ( g C_ N <-> S C_ N ) )
4	2, 3	anbi12d 464	. . . . . . . . 9 |- ( g = S -> ( ( S C_ g /\ g C_ N ) <-> ( S C_ S /\ S C_ N ) ) )
5		ssid 3117	. . . . . . . . . 10 |- S C_ S
6	5	biantrur 301	. . . . . . . . 9 |- ( S C_ N <-> ( S C_ S /\ S C_ N ) )
7	4, 6	syl6bbr 197	. . . . . . . 8 |- ( g = S -> ( ( S C_ g /\ g C_ N ) <-> S C_ N ) )
8	7	rspcev 2789	. . . . . . 7 |- ( ( S e. J /\ S C_ N ) -> E. g e. J ( S C_ g /\ g C_ N ) )
9	8	adantlr 468	. . . . . 6 |- ( ( ( S e. J /\ N C_ X ) /\ S C_ N ) -> E. g e. J ( S C_ g /\ g C_ N ) )
10	1, 9	jca 304	. . . . 5 |- ( ( ( S e. J /\ N C_ X ) /\ S C_ N ) -> ( N C_ X /\ E. g e. J ( S C_ g /\ g C_ N ) ) )
11	10	ex 114	. . . 4 |- ( ( S e. J /\ N C_ X ) -> ( S C_ N -> ( N C_ X /\ E. g e. J ( S C_ g /\ g C_ N ) ) ) )
12	11	3adant1 999	. . 3 |- ( ( J e. Top /\ S e. J /\ N C_ X ) -> ( S C_ N -> ( N C_ X /\ E. g e. J ( S C_ g /\ g C_ N ) ) ) )
13		neips.1	. . . . . 6 |- X = U. J
14	13	eltopss 12176	. . . . 5 |- ( ( J e. Top /\ S e. J ) -> S C_ X )
15	13	isnei 12313	. . . . 5 |- ( ( J e. Top /\ S C_ X ) -> ( N e. ( ( nei ` J ) ` S ) <-> ( N C_ X /\ E. g e. J ( S C_ g /\ g C_ N ) ) ) )
16	14, 15	syldan 280	. . . 4 |- ( ( J e. Top /\ S e. J ) -> ( N e. ( ( nei ` J ) ` S ) <-> ( N C_ X /\ E. g e. J ( S C_ g /\ g C_ N ) ) ) )
17	16	3adant3 1001	. . 3 |- ( ( J e. Top /\ S e. J /\ N C_ X ) -> ( N e. ( ( nei ` J ) ` S ) <-> ( N C_ X /\ E. g e. J ( S C_ g /\ g C_ N ) ) ) )
18	12, 17	sylibrd 168	. 2 |- ( ( J e. Top /\ S e. J /\ N C_ X ) -> ( S C_ N -> N e. ( ( nei ` J ) ` S ) ) )
19		ssnei 12320	. . . 4 |- ( ( J e. Top /\ N e. ( ( nei ` J ) ` S ) ) -> S C_ N )
20	19	ex 114	. . 3 |- ( J e. Top -> ( N e. ( ( nei ` J ) ` S ) -> S C_ N ) )
21	20	3ad2ant1 1002	. 2 |- ( ( J e. Top /\ S e. J /\ N C_ X ) -> ( N e. ( ( nei ` J ) ` S ) -> S C_ N ) )
22	18, 21	impbid 128	1 |- ( ( J e. Top /\ S e. J /\ N 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 2417   C_ wss 3071   U.cuni 3736   ` cfv 5123   Topctop 12164   neicnei 12307

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 2121  ax-coll 4043  ax-sep 4046  ax-pow 4098  ax-pr 4131

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-top 12165  df-nei 12308

This theorem is referenced by:  neissex  12334