Theorem opnneissb 7728

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 7603	. . . . . . 7 |- ((J e. Top /\ N e. J) -> N (_ X)
3	2	adantr 389	. . . . . 6 |- (((J e. Top /\ N e. J) /\ (S (_ X /\ S (_ N)) -> N (_ X)
4		ssid 2080	. . . . . . . 8 |- N (_ N
5		sseq2 2083	. . . . . . . . . 10 |- (g = N -> (S (_ g <-> S (_ N))
6		sseq1 2082	. . . . . . . . . 10 |- (g = N -> (g (_ N <-> N (_ N))
7	5, 6	anbi12d 628	. . . . . . . . 9 |- (g = N -> ((S (_ g /\ g (_ N) <-> (S (_ N /\ N (_ N)))
8	7	rcla4ev 1877	. . . . . . . 8 |- ((N e. J /\ (S (_ N /\ N (_ N)) -> E.g e. J (S (_ g /\ g (_ N))
9	4, 8	mpanr2 710	. . . . . . 7 |- ((N e. J /\ S (_ N) -> E.g e. J (S (_ g /\ g (_ N))
10	9	ad2ant2l 408	. . . . . 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 7718	. . . . . 6 |- ((J e. Top /\ S (_ X) -> (N e. ((nei` J)` S) <-> (N (_ X /\ E.g e. J (S (_ g /\ g (_ N))))
13	12	ad2ant2r 409	. . . . 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 384	. . 3 |- (J e. Top -> (N e. J -> (S (_ X -> (S (_ N -> N e. ((nei` J)` S)))))
16	15	3imp 827	. 2 |- ((J e. Top /\ N e. J /\ S (_ X) -> (S (_ N -> N e. ((nei` J)` S)))
17		ssnei 7724	. . . 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 800	. 2 |- ((J e. Top /\ N e. J /\ S (_ X) -> (N e. ((nei` J)` S) -> S (_ N))
20	16, 19	impbid 516	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 775   = wceq 956   e. wcel 958  E.wrex 1646   (_ wss 2047  U.cuni 2503  ` cfv 3182  Topctop 7588  neicnei 7712

This theorem is referenced by:  opnneiss 7732

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-9 965  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-rep 2693  ax-sep 2703  ax-pow 2742  ax-pr 2779  ax-un 2866

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 777  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-ral 1649  df-rex 1650  df-rab 1652  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-uni 2504  df-br 2620  df-opab 2667  df-id 2835  df-xp 3184  df-rel 3185  df-cnv 3186  df-co 3187  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fun 3192  df-fn 3193  df-fv 3198  df-nei 7713

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