Theorem neiss 12944

Description: Any neighborhood of a set S is also a neighborhood of any subset R C_ S. Similar to Proposition 1 of [BourbakiTop1] p. 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

Dummy variable g is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2170	. . . 4 |- U. J = U. J
2	1	neii1 12941	. . 3 |- ( ( J e. Top /\ N e. ( ( nei ` J ) ` S ) ) -> N C_ U. J )
3	2	3adant3 1012	. 2 |- ( ( J e. Top /\ N e. ( ( nei ` J ) ` S ) /\ R C_ S ) -> N C_ U. J )
4		neii2 12943	. . . 4 |- ( ( J e. Top /\ N e. ( ( nei ` J ) ` S ) ) -> E. g e. J ( S C_ g /\ g C_ N ) )
5	4	3adant3 1012	. . 3 |- ( ( J e. Top /\ N e. ( ( nei ` J ) ` S ) /\ R C_ S ) -> E. g e. J ( S C_ g /\ g C_ N ) )
6		sstr2 3154	. . . . . 6 |- ( R C_ S -> ( S C_ g -> R C_ g ) )
7	6	anim1d 334	. . . . 5 |- ( R C_ S -> ( ( S C_ g /\ g C_ N ) -> ( R C_ g /\ g C_ N ) ) )
8	7	reximdv 2571	. . . 4 |- ( R C_ S -> ( E. g e. J ( S C_ g /\ g C_ N ) -> E. g e. J ( R C_ g /\ g C_ N ) ) )
9	8	3ad2ant3 1015	. . 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 ) ) )
10	5, 9	mpd 13	. 2 |- ( ( J e. Top /\ N e. ( ( nei ` J ) ` S ) /\ R C_ S ) -> E. g e. J ( R C_ g /\ g C_ N ) )
11		simp1 992	. . 3 |- ( ( J e. Top /\ N e. ( ( nei ` J ) ` S ) /\ R C_ S ) -> J e. Top )
12		simp3 994	. . . 4 |- ( ( J e. Top /\ N e. ( ( nei ` J ) ` S ) /\ R C_ S ) -> R C_ S )
13	1	neiss2 12936	. . . . 5 |- ( ( J e. Top /\ N e. ( ( nei ` J ) ` S ) ) -> S C_ U. J )
14	13	3adant3 1012	. . . 4 |- ( ( J e. Top /\ N e. ( ( nei ` J ) ` S ) /\ R C_ S ) -> S C_ U. J )
15	12, 14	sstrd 3157	. . 3 |- ( ( J e. Top /\ N e. ( ( nei ` J ) ` S ) /\ R C_ S ) -> R C_ U. J )
16	1	isnei 12938	. . 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 ) ) ) )
17	11, 15, 16	syl2anc 409	. 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 ) ) ) )
18	3, 10, 17	mpbir2and 939	1 |- ( ( J e. Top /\ N e. ( ( nei ` J ) ` S ) /\ R C_ S ) -> N e. ( ( nei ` J ) ` R ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   /\ w3a 973   e. wcel 2141   E.wrex 2449   C_ wss 3121   U.cuni 3796   ` cfv 5198   Topctop 12789   neicnei 12932

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-pow 4160  ax-pr 4194

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-top 12790  df-nei 12933

This theorem is referenced by:  neipsm  12948  neissex  12959