Theorem opnneissb 14660

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 14514	. . . . . 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 3213	. . . . . . 7 |- N C_ N
5		sseq2 3217	. . . . . . . . 9 |- ( g = N -> ( S C_ g <-> S C_ N ) )
6		sseq1 3216	. . . . . . . . 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 2877	. . . . . . 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 14649	. . . . . 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 947	. . . 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 1196	. 2 |- ( ( J e. Top /\ N e. J /\ S C_ X ) -> ( S C_ N -> N e. ( ( nei ` J ) ` S ) ) )
16		ssnei 14656	. . . 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 1021	. 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 981   = wceq 1373   e. wcel 2176   E.wrex 2485   C_ wss 3166   U.cuni 3850   ` cfv 5272   Topctop 14502   neicnei 14643

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-14 2179  ax-ext 2187  ax-coll 4160  ax-sep 4163  ax-pow 4219  ax-pr 4254

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-rex 2490  df-reu 2491  df-rab 2493  df-v 2774  df-sbc 2999  df-csb 3094  df-un 3170  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-iun 3929  df-br 4046  df-opab 4107  df-mpt 4108  df-id 4341  df-xp 4682  df-rel 4683  df-cnv 4684  df-co 4685  df-dm 4686  df-rn 4687  df-res 4688  df-ima 4689  df-iota 5233  df-fun 5274  df-fn 5275  df-f 5276  df-f1 5277  df-fo 5278  df-f1o 5279  df-fv 5280  df-top 14503  df-nei 14644

This theorem is referenced by:  opnneiss  14663