Theorem opnneissb 15037

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 14891	. . . . . 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 3260	. . . . . . 7 |- N C_ N
5		sseq2 3264	. . . . . . . . 9 |- ( g = N -> ( S C_ g <-> S C_ N ) )
6		sseq1 3263	. . . . . . . . 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 2923	. . . . . . 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 15026	. . . . . 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 953	. . . 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 1220	. 2 |- ( ( J e. Top /\ N e. J /\ S C_ X ) -> ( S C_ N -> N e. ( ( nei ` J ) ` S ) ) )
16		ssnei 15033	. . . 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 1045	. 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 1005   = wceq 1398   e. wcel 2205   E.wrex 2523   C_ wss 3213   U.cuni 3916   ` cfv 5354   Topctop 14879   neicnei 15020

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 2208  ax-ext 2216  ax-coll 4227  ax-sep 4230  ax-pow 4289  ax-pr 4324

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-reu 2529  df-rab 2531  df-v 2817  df-sbc 3045  df-csb 3141  df-un 3217  df-in 3219  df-ss 3226  df-pw 3673  df-sn 3697  df-pr 3698  df-op 3700  df-uni 3917  df-iun 3995  df-br 4112  df-opab 4174  df-mpt 4175  df-id 4416  df-xp 4757  df-rel 4758  df-cnv 4759  df-co 4760  df-dm 4761  df-rn 4762  df-res 4763  df-ima 4764  df-iota 5314  df-fun 5356  df-fn 5357  df-f 5358  df-f1 5359  df-fo 5360  df-f1o 5361  df-fv 5362  df-top 14880  df-nei 15021

This theorem is referenced by:  opnneiss  15040