Theorem opnneissb 7725

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 (_ X) -> (S (_ N <-> N e. ((nei` J)` S)))

Proof of Theorem opnneissb

Step	Hyp	Ref	Expression
1		neips.1	. . . . . . . 8 |- X = U.J
2	1	eltopss 7604	. . . . . . 7 |- ((J e. Top /\ N e. J) -> N (_ X)
3	2	adantr 391	. . . . . 6 |- (((J e. Top /\ N e. J) /\ (S (_ X /\ S (_ N)) -> N (_ X)
4		ssid 2083	. . . . . . . 8 |- N (_ N
5		sseq2 2086	. . . . . . . . . 10 |- (g = N -> (S (_ g <-> S (_ N))
6		sseq1 2085	. . . . . . . . . 10 |- (g = N -> (g (_ N <-> N (_ N))
7	5, 6	anbi12d 630	. . . . . . . . 9 |- (g = N -> ((S (_ g /\ g (_ N) <-> (S (_ N /\ N (_ N)))
8	7	rcla4ev 1880	. . . . . . . 8 |- ((N e. J /\ (S (_ N /\ N (_ N)) -> E.g e. J (S (_ g /\ g (_ N))
9	4, 8	mpanr2 712	. . . . . . 7 |- ((N e. J /\ S (_ N) -> E.g e. J (S (_ g /\ g (_ N))
10	9	ad2ant2l 410	. . . . . 6 |- (((J e. Top /\ N e. J) /\ (S (_ X /\ S (_ N)) -> E.g e. J (S (_ g /\ g (_ N))
11	3, 10	jca 288	. . . . 5 |- (((J e. Top /\ N e. J) /\ (S (_ X /\ S (_ N)) -> (N (_ X /\ E.g e. J (S (_ g /\ g (_ N)))
12	1	isnei 7715	. . . . . 6 |- ((J e. Top /\ S (_ X) -> (N e. ((nei` J)` S) <-> (N (_ X /\ E.g e. J (S (_ g /\ g (_ N))))
13	12	ad2ant2r 411	. . . . 5 |- (((J e. Top /\ N e. J) /\ (S (_ X /\ S (_ N)) -> (N e. ((nei` J)` S) <-> (N (_ X /\ E.g e. J (S (_ g /\ g (_ N))))
14	11, 13	mpbird 196	. . . 4 |- (((J e. Top /\ N e. J) /\ (S (_ X /\ S (_ N)) -> N e. ((nei` J)` S))
15	14	exp43 386	. . 3 |- (J e. Top -> (N e. J -> (S (_ X -> (S (_ N -> N e. ((nei` J)` S)))))
16	15	3imp 829	. 2 |- ((J e. Top /\ N e. J /\ S (_ X) -> (S (_ N -> N e. ((nei` J)` S)))
17		ssnei 7721	. . . 4 |- ((J e. Top /\ N e. ((nei` J)` S)) -> S (_ N)
18	17	ex 373	. . 3 |- (J e. Top -> (N e. ((nei` J)` S) -> S (_ N))
19	18	3ad2ant1 802	. 2 |- ((J e. Top /\ N e. J /\ S (_ X) -> (N e. ((nei` J)` S) -> S (_ N))
20	16, 19	impbid 518	1 |- ((J e. Top /\ N e. J /\ S (_ X) -> (S (_ N <-> N e. ((nei` J)` S)))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   /\ w3a 777   = wceq 958   e. wcel 960  E.wrex 1649   (_ wss 2050  U.cuni 2507  ` cfv 3188  Topctop 7590  neicnei 7709

This theorem is referenced by:  opnneiss 7729

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-9 967  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-rep 2698  ax-sep 2708  ax-pow 2748  ax-pr 2785  ax-un 2872

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 779  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-ral 1652  df-rex 1653  df-rab 1655  df-v 1815  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-pw 2406  df-sn 2416  df-pr 2417  df-op 2420  df-uni 2508  df-br 2625  df-opab 2672  df-id 2841  df-xp 3190  df-rel 3191  df-cnv 3192  df-co 3193  df-dm 3194  df-rn 3195  df-res 3196  df-ima 3197  df-fun 3198  df-fn 3199  df-fv 3204  df-nei 7710

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