Theorem opnssneib 14815

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 528	. . . . . 6 |- ( ( ( S e. J /\ N C_ X ) /\ S C_ N ) -> N C_ X )
2		sseq2 3248	. . . . . . . . . 10 |- ( g = S -> ( S C_ g <-> S C_ S ) )
3		sseq1 3247	. . . . . . . . . 10 |- ( g = S -> ( g C_ N <-> S C_ N ) )
4	2, 3	anbi12d 473	. . . . . . . . 9 |- ( g = S -> ( ( S C_ g /\ g C_ N ) <-> ( S C_ S /\ S C_ N ) ) )
5		ssid 3244	. . . . . . . . . 10 |- S C_ S
6	5	biantrur 303	. . . . . . . . 9 |- ( S C_ N <-> ( S C_ S /\ S C_ N ) )
7	4, 6	bitr4di 198	. . . . . . . 8 |- ( g = S -> ( ( S C_ g /\ g C_ N ) <-> S C_ N ) )
8	7	rspcev 2907	. . . . . . 7 |- ( ( S e. J /\ S C_ N ) -> E. g e. J ( S C_ g /\ g C_ N ) )
9	8	adantlr 477	. . . . . 6 |- ( ( ( S e. J /\ N C_ X ) /\ S C_ N ) -> E. g e. J ( S C_ g /\ g C_ N ) )
10	1, 9	jca 306	. . . . 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 115	. . . 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 1039	. . 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 14668	. . . . 5 |- ( ( J e. Top /\ S e. J ) -> S C_ X )
15	13	isnei 14803	. . . . 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 282	. . . 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 1041	. . 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 169	. 2 |- ( ( J e. Top /\ S e. J /\ N C_ X ) -> ( S C_ N -> N e. ( ( nei ` J ) ` S ) ) )
19		ssnei 14810	. . . 4 |- ( ( J e. Top /\ N e. ( ( nei ` J ) ` S ) ) -> S C_ N )
20	19	ex 115	. . 3 |- ( J e. Top -> ( N e. ( ( nei ` J ) ` S ) -> S C_ N ) )
21	20	3ad2ant1 1042	. 2 |- ( ( J e. Top /\ S e. J /\ N C_ X ) -> ( N e. ( ( nei ` J ) ` S ) -> S C_ N ) )
22	18, 21	impbid 129	1 |- ( ( J e. Top /\ S e. J /\ N C_ X ) -> ( S C_ N <-> N e. ( ( nei ` J ) ` S ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1002   = wceq 1395   e. wcel 2200   E.wrex 2509   C_ wss 3197   U.cuni 3887   ` cfv 5314   Topctop 14656   neicnei 14797

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-coll 4198  ax-sep 4201  ax-pow 4257  ax-pr 4292

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-iun 3966  df-br 4083  df-opab 4145  df-mpt 4146  df-id 4381  df-xp 4722  df-rel 4723  df-cnv 4724  df-co 4725  df-dm 4726  df-rn 4727  df-res 4728  df-ima 4729  df-iota 5274  df-fun 5316  df-fn 5317  df-f 5318  df-f1 5319  df-fo 5320  df-f1o 5321  df-fv 5322  df-top 14657  df-nei 14798

This theorem is referenced by:  neissex  14824