Theorem neiss 8999

Description: Any neighborhood of a set S is also a neighborhood of any subset R C_ S. Bourbaki TG I.2. (Contributed by FL, 25-Sep-2006.)

Assertion

Ref	Expression
neiss	|- ((J e. Top /\ N e. ((nei` J)` S) /\ R C_ S) -> N e. ((nei` J)` R))

Proof of Theorem neiss

Step	Hyp	Ref	Expression
1		simp1 876	. . 3 |- ((J e. Top /\ N e. ((nei` J)` S) /\ R C_ S) -> J e. Top)
2		simp3 878	. . . 4 |- ((J e. Top /\ N e. ((nei` J)` S) /\ R C_ S) -> R C_ S)
3		eqid 1884	. . . . . 6 |- U.J = U.J
4	3	neiss2 8992	. . . . 5 |- ((J e. Top /\ N e. ((nei` J)` S)) -> S C_ U.J)
5	4	3adant3 896	. . . 4 |- ((J e. Top /\ N e. ((nei` J)` S) /\ R C_ S) -> S C_ U.J)
6	2, 5	sstrd 2627	. . 3 |- ((J e. Top /\ N e. ((nei` J)` S) /\ R C_ S) -> R C_ U.J)
7	3	isnei 8994	. . 3 |- ((J e. Top /\ R C_ U.J) -> (N e. ((nei` J)` R) <-> (N C_ U.J /\ E.g e. J (R C_ g /\ g C_ N))))
8	1, 6, 7	syl11anc 524	. 2 |- ((J e. Top /\ N e. ((nei` J)` S) /\ R C_ S) -> (N e. ((nei` J)` R) <-> (N C_ U.J /\ E.g e. J (R C_ g /\ g C_ N))))
9	3	neii1 8997	. . 3 |- ((J e. Top /\ N e. ((nei` J)` S)) -> N C_ U.J)
10	9	3adant3 896	. 2 |- ((J e. Top /\ N e. ((nei` J)` S) /\ R C_ S) -> N C_ U.J)
11		neii2 8998	. . . 4 |- ((J e. Top /\ N e. ((nei` J)` S)) -> E.g e. J (S C_ g /\ g C_ N))
12	11	3adant3 896	. . 3 |- ((J e. Top /\ N e. ((nei` J)` S) /\ R C_ S) -> E.g e. J (S C_ g /\ g C_ N))
13		sstr2 2623	. . . . . 6 |- (R C_ S -> (S C_ g -> R C_ g))
14	13	anim1d 619	. . . . 5 |- (R C_ S -> ((S C_ g /\ g C_ N) -> (R C_ g /\ g C_ N)))
15	14	reximdv 2202	. . . 4 |- (R C_ S -> (E.g e. J (S C_ g /\ g C_ N) -> E.g e. J (R C_ g /\ g C_ N)))
16	15	3ad2ant3 899	. . 3 |- ((J e. Top /\ N e. ((nei` J)` S) /\ R C_ S) -> (E.g e. J (S C_ g /\ g C_ N) -> E.g e. J (R C_ g /\ g C_ N)))
17	12, 16	mpd 29	. 2 |- ((J e. Top /\ N e. ((nei` J)` S) /\ R C_ S) -> E.g e. J (R C_ g /\ g C_ N))
18	8, 10, 17	mpbir2and 802	1 |- ((J e. Top /\ N e. ((nei` J)` S) /\ R C_ S) -> N e. ((nei` J)` R))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   /\ w3a 858   e. wcel 1300  E.wrex 2106   C_ wss 2593  U.cuni 3177  ` cfv 3998  Topctop 8857  neicnei 8988

This theorem is referenced by:  neips 9003  neissex 9014

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-uni 3178  df-br 3339  df-opab 3396  df-id 3586  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-fv 4014  df-nei 8989

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