Theorem neiss 14622

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 2205	. . . 4 |- U. J = U. J
2	1	neii1 14619	. . 3 |- ( ( J e. Top /\ N e. ( ( nei ` J ) ` S ) ) -> N C_ U. J )
3	2	3adant3 1020	. 2 |- ( ( J e. Top /\ N e. ( ( nei ` J ) ` S ) /\ R C_ S ) -> N C_ U. J )
4		neii2 14621	. . . 4 |- ( ( J e. Top /\ N e. ( ( nei ` J ) ` S ) ) -> E. g e. J ( S C_ g /\ g C_ N ) )
5	4	3adant3 1020	. . 3 |- ( ( J e. Top /\ N e. ( ( nei ` J ) ` S ) /\ R C_ S ) -> E. g e. J ( S C_ g /\ g C_ N ) )
6		sstr2 3200	. . . . . 6 |- ( R C_ S -> ( S C_ g -> R C_ g ) )
7	6	anim1d 336	. . . . 5 |- ( R C_ S -> ( ( S C_ g /\ g C_ N ) -> ( R C_ g /\ g C_ N ) ) )
8	7	reximdv 2607	. . . 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 1023	. . 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 1000	. . 3 |- ( ( J e. Top /\ N e. ( ( nei ` J ) ` S ) /\ R C_ S ) -> J e. Top )
12		simp3 1002	. . . 4 |- ( ( J e. Top /\ N e. ( ( nei ` J ) ` S ) /\ R C_ S ) -> R C_ S )
13	1	neiss2 14614	. . . . 5 |- ( ( J e. Top /\ N e. ( ( nei ` J ) ` S ) ) -> S C_ U. J )
14	13	3adant3 1020	. . . 4 |- ( ( J e. Top /\ N e. ( ( nei ` J ) ` S ) /\ R C_ S ) -> S C_ U. J )
15	12, 14	sstrd 3203	. . 3 |- ( ( J e. Top /\ N e. ( ( nei ` J ) ` S ) /\ R C_ S ) -> R C_ U. J )
16	1	isnei 14616	. . 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 411	. 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 947	1 |- ( ( J e. Top /\ N e. ( ( nei ` J ) ` S ) /\ R C_ S ) -> N e. ( ( nei ` J ) ` R ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 981   e. wcel 2176   E.wrex 2485   C_ wss 3166   U.cuni 3850   ` cfv 5271   Topctop 14469   neicnei 14610

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-14 2179  ax-ext 2187  ax-coll 4159  ax-sep 4162  ax-pow 4218  ax-pr 4253

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-rex 2490  df-reu 2491  df-rab 2493  df-v 2774  df-sbc 2999  df-csb 3094  df-un 3170  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-iun 3929  df-br 4045  df-opab 4106  df-mpt 4107  df-id 4340  df-xp 4681  df-rel 4682  df-cnv 4683  df-co 4684  df-dm 4685  df-rn 4686  df-res 4687  df-ima 4688  df-iota 5232  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-top 14470  df-nei 14611

This theorem is referenced by:  neipsm  14626  neissex  14637