Theorem opnssneib 15013

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 529	. . . . . 6 |- ( ( ( S e. J /\ N C_ X ) /\ S C_ N ) -> N C_ X )
2		sseq2 3261	. . . . . . . . . 10 |- ( g = S -> ( S C_ g <-> S C_ S ) )
3		sseq1 3260	. . . . . . . . . 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 3257	. . . . . . . . . 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 2920	. . . . . . 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 1042	. . 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 14866	. . . . 5 |- ( ( J e. Top /\ S e. J ) -> S C_ X )
15	13	isnei 15001	. . . . 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 1044	. . 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 15008	. . . 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 1045	. 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 1005   = wceq 1398   e. wcel 2203   E.wrex 2521   C_ wss 3210   U.cuni 3913   ` cfv 5351   Topctop 14854   neicnei 14995

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2206  ax-ext 2214  ax-coll 4224  ax-sep 4227  ax-pow 4286  ax-pr 4321

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-reu 2527  df-rab 2529  df-v 2814  df-sbc 3042  df-csb 3138  df-un 3214  df-in 3216  df-ss 3223  df-pw 3670  df-sn 3694  df-pr 3695  df-op 3697  df-uni 3914  df-iun 3992  df-br 4109  df-opab 4171  df-mpt 4172  df-id 4413  df-xp 4754  df-rel 4755  df-cnv 4756  df-co 4757  df-dm 4758  df-rn 4759  df-res 4760  df-ima 4761  df-iota 5311  df-fun 5353  df-fn 5354  df-f 5355  df-f1 5356  df-fo 5357  df-f1o 5358  df-fv 5359  df-top 14855  df-nei 14996

This theorem is referenced by:  neissex  15022