Theorem neiss 12319

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 2139	. . . 4 |- U. J = U. J
2	1	neii1 12316	. . 3 |- ( ( J e. Top /\ N e. ( ( nei ` J ) ` S ) ) -> N C_ U. J )
3	2	3adant3 1001	. 2 |- ( ( J e. Top /\ N e. ( ( nei ` J ) ` S ) /\ R C_ S ) -> N C_ U. J )
4		neii2 12318	. . . 4 |- ( ( J e. Top /\ N e. ( ( nei ` J ) ` S ) ) -> E. g e. J ( S C_ g /\ g C_ N ) )
5	4	3adant3 1001	. . 3 |- ( ( J e. Top /\ N e. ( ( nei ` J ) ` S ) /\ R C_ S ) -> E. g e. J ( S C_ g /\ g C_ N ) )
6		sstr2 3104	. . . . . 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 2533	. . . 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 1004	. . 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 981	. . 3 |- ( ( J e. Top /\ N e. ( ( nei ` J ) ` S ) /\ R C_ S ) -> J e. Top )
12		simp3 983	. . . 4 |- ( ( J e. Top /\ N e. ( ( nei ` J ) ` S ) /\ R C_ S ) -> R C_ S )
13	1	neiss2 12311	. . . . 5 |- ( ( J e. Top /\ N e. ( ( nei ` J ) ` S ) ) -> S C_ U. J )
14	13	3adant3 1001	. . . 4 |- ( ( J e. Top /\ N e. ( ( nei ` J ) ` S ) /\ R C_ S ) -> S C_ U. J )
15	12, 14	sstrd 3107	. . 3 |- ( ( J e. Top /\ N e. ( ( nei ` J ) ` S ) /\ R C_ S ) -> R C_ U. J )
16	1	isnei 12313	. . 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 408	. 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 928	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 962   e. wcel 1480   E.wrex 2417   C_ wss 3071   U.cuni 3736   ` cfv 5123   Topctop 12164   neicnei 12307

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-pow 4098  ax-pr 4131

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-top 12165  df-nei 12308

This theorem is referenced by:  neipsm  12323  neissex  12334