Theorem opnssneib 12950

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 525	. . . . . 6 |- ( ( ( S e. J /\ N C_ X ) /\ S C_ N ) -> N C_ X )
2		sseq2 3171	. . . . . . . . . 10 |- ( g = S -> ( S C_ g <-> S C_ S ) )
3		sseq1 3170	. . . . . . . . . 10 |- ( g = S -> ( g C_ N <-> S C_ N ) )
4	2, 3	anbi12d 470	. . . . . . . . 9 |- ( g = S -> ( ( S C_ g /\ g C_ N ) <-> ( S C_ S /\ S C_ N ) ) )
5		ssid 3167	. . . . . . . . . 10 |- S C_ S
6	5	biantrur 301	. . . . . . . . 9 |- ( S C_ N <-> ( S C_ S /\ S C_ N ) )
7	4, 6	bitr4di 197	. . . . . . . 8 |- ( g = S -> ( ( S C_ g /\ g C_ N ) <-> S C_ N ) )
8	7	rspcev 2834	. . . . . . 7 |- ( ( S e. J /\ S C_ N ) -> E. g e. J ( S C_ g /\ g C_ N ) )
9	8	adantlr 474	. . . . . 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 1010	. . 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 12801	. . . . 5 |- ( ( J e. Top /\ S e. J ) -> S C_ X )
15	13	isnei 12938	. . . . 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 1012	. . 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 12945	. . . 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 1013	. 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 973   = wceq 1348   e. wcel 2141   E.wrex 2449   C_ wss 3121   U.cuni 3796   ` cfv 5198   Topctop 12789   neicnei 12932

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-pow 4160  ax-pr 4194

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-top 12790  df-nei 12933

This theorem is referenced by:  neissex  12959