Theorem neiss 20913

Description: Any neighborhood of a set S is also a neighborhood of any subset R C_ S. Theorem 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 2622	. . . 4 |- U. J = U. J
2	1	neii1 20910	. . 3 |- ( ( J e. Top /\ N e. ( ( nei ` J ) ` S ) ) -> N C_ U. J )
3	2	3adant3 1081	. 2 |- ( ( J e. Top /\ N e. ( ( nei ` J ) ` S ) /\ R C_ S ) -> N C_ U. J )
4		neii2 20912	. . . 4 |- ( ( J e. Top /\ N e. ( ( nei ` J ) ` S ) ) -> E. g e. J ( S C_ g /\ g C_ N ) )
5	4	3adant3 1081	. . 3 |- ( ( J e. Top /\ N e. ( ( nei ` J ) ` S ) /\ R C_ S ) -> E. g e. J ( S C_ g /\ g C_ N ) )
6		sstr2 3610	. . . . . 6 |- ( R C_ S -> ( S C_ g -> R C_ g ) )
7	6	anim1d 588	. . . . 5 |- ( R C_ S -> ( ( S C_ g /\ g C_ N ) -> ( R C_ g /\ g C_ N ) ) )
8	7	reximdv 3016	. . . 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 1084	. . 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 15	. 2 |- ( ( J e. Top /\ N e. ( ( nei ` J ) ` S ) /\ R C_ S ) -> E. g e. J ( R C_ g /\ g C_ N ) )
11		simp1 1061	. . 3 |- ( ( J e. Top /\ N e. ( ( nei ` J ) ` S ) /\ R C_ S ) -> J e. Top )
12		simp3 1063	. . . 4 |- ( ( J e. Top /\ N e. ( ( nei ` J ) ` S ) /\ R C_ S ) -> R C_ S )
13	1	neiss2 20905	. . . . 5 |- ( ( J e. Top /\ N e. ( ( nei ` J ) ` S ) ) -> S C_ U. J )
14	13	3adant3 1081	. . . 4 |- ( ( J e. Top /\ N e. ( ( nei ` J ) ` S ) /\ R C_ S ) -> S C_ U. J )
15	12, 14	sstrd 3613	. . 3 |- ( ( J e. Top /\ N e. ( ( nei ` J ) ` S ) /\ R C_ S ) -> R C_ U. J )
16	1	isnei 20907	. . 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 693	. 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 957	1 |- ( ( J e. Top /\ N e. ( ( nei ` J ) ` S ) /\ R C_ S ) -> N e. ( ( nei ` J ) ` R ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   e. wcel 1990   E.wrex 2913   C_ wss 3574   U.cuni 4436   ` cfv 5888   Topctop 20698   neicnei 20901

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-top 20699  df-nei 20902

This theorem is referenced by:  neips  20917  neissex  20931