Theorem opnneissb 14107

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 13961	. . . . . 6 |- ( ( J e. Top /\ N e. J ) -> N C_ X )
3	2	adantr 276	. . . . 5 |- ( ( ( J e. Top /\ N e. J ) /\ ( S C_ X /\ S C_ N ) ) -> N C_ X )
4		ssid 3190	. . . . . . 7 |- N C_ N
5		sseq2 3194	. . . . . . . . 9 |- ( g = N -> ( S C_ g <-> S C_ N ) )
6		sseq1 3193	. . . . . . . . 9 |- ( g = N -> ( g C_ N <-> N C_ N ) )
7	5, 6	anbi12d 473	. . . . . . . 8 |- ( g = N -> ( ( S C_ g /\ g C_ N ) <-> ( S C_ N /\ N C_ N ) ) )
8	7	rspcev 2856	. . . . . . 7 |- ( ( N e. J /\ ( S C_ N /\ N C_ N ) ) -> E. g e. J ( S C_ g /\ g C_ N ) )
9	4, 8	mpanr2 438	. . . . . 6 |- ( ( N e. J /\ S C_ N ) -> E. g e. J ( S C_ g /\ g C_ N ) )
10	9	ad2ant2l 508	. . . . 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 14096	. . . . . 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 509	. . . . 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 946	. . . 4 |- ( ( ( J e. Top /\ N e. J ) /\ ( S C_ X /\ S C_ N ) ) -> N e. ( ( nei ` J ) ` S ) )
14	13	exp43 372	. . 3 |- ( J e. Top -> ( N e. J -> ( S C_ X -> ( S C_ N -> N e. ( ( nei ` J ) ` S ) ) ) ) )
15	14	3imp 1195	. 2 |- ( ( J e. Top /\ N e. J /\ S C_ X ) -> ( S C_ N -> N e. ( ( nei ` J ) ` S ) ) )
16		ssnei 14103	. . . 4 |- ( ( J e. Top /\ N e. ( ( nei ` J ) ` S ) ) -> S C_ N )
17	16	ex 115	. . 3 |- ( J e. Top -> ( N e. ( ( nei ` J ) ` S ) -> S C_ N ) )
18	17	3ad2ant1 1020	. 2 |- ( ( J e. Top /\ N e. J /\ S C_ X ) -> ( N e. ( ( nei ` J ) ` S ) -> S C_ N ) )
19	15, 18	impbid 129	1 |- ( ( J e. Top /\ N e. J /\ S C_ X ) -> ( S C_ N <-> N e. ( ( nei ` J ) ` S ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 980   = wceq 1364   e. wcel 2160   E.wrex 2469   C_ wss 3144   U.cuni 3824   ` cfv 5235   Topctop 13949   neicnei 14090

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2163  ax-ext 2171  ax-coll 4133  ax-sep 4136  ax-pow 4192  ax-pr 4227

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-reu 2475  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-id 4311  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-iota 5196  df-fun 5237  df-fn 5238  df-f 5239  df-f1 5240  df-fo 5241  df-f1o 5242  df-fv 5243  df-top 13950  df-nei 14091

This theorem is referenced by:  opnneiss  14110