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Theorem neiss 18831
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.
StepHypRef Expression
1 eqid 2451 . . . 4  |-  U. J  =  U. J
21neii1 18828 . . 3  |-  ( ( J  e.  Top  /\  N  e.  ( ( nei `  J ) `  S ) )  ->  N  C_  U. J )
323adant3 1008 . 2  |-  ( ( J  e.  Top  /\  N  e.  ( ( nei `  J ) `  S )  /\  R  C_  S )  ->  N  C_ 
U. J )
4 neii2 18830 . . . 4  |-  ( ( J  e.  Top  /\  N  e.  ( ( nei `  J ) `  S ) )  ->  E. g  e.  J  ( S  C_  g  /\  g  C_  N ) )
543adant3 1008 . . 3  |-  ( ( J  e.  Top  /\  N  e.  ( ( nei `  J ) `  S )  /\  R  C_  S )  ->  E. g  e.  J  ( S  C_  g  /\  g  C_  N ) )
6 sstr2 3463 . . . . . 6  |-  ( R 
C_  S  ->  ( S  C_  g  ->  R  C_  g ) )
76anim1d 564 . . . . 5  |-  ( R 
C_  S  ->  (
( S  C_  g  /\  g  C_  N )  ->  ( R  C_  g  /\  g  C_  N
) ) )
87reximdv 2925 . . . 4  |-  ( R 
C_  S  ->  ( E. g  e.  J  ( S  C_  g  /\  g  C_  N )  ->  E. g  e.  J  ( R  C_  g  /\  g  C_  N ) ) )
983ad2ant3 1011 . . 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 ) ) )
105, 9mpd 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 988 . . 3  |-  ( ( J  e.  Top  /\  N  e.  ( ( nei `  J ) `  S )  /\  R  C_  S )  ->  J  e.  Top )
12 simp3 990 . . . 4  |-  ( ( J  e.  Top  /\  N  e.  ( ( nei `  J ) `  S )  /\  R  C_  S )  ->  R  C_  S )
131neiss2 18823 . . . . 5  |-  ( ( J  e.  Top  /\  N  e.  ( ( nei `  J ) `  S ) )  ->  S  C_  U. J )
14133adant3 1008 . . . 4  |-  ( ( J  e.  Top  /\  N  e.  ( ( nei `  J ) `  S )  /\  R  C_  S )  ->  S  C_ 
U. J )
1512, 14sstrd 3466 . . 3  |-  ( ( J  e.  Top  /\  N  e.  ( ( nei `  J ) `  S )  /\  R  C_  S )  ->  R  C_ 
U. J )
161isnei 18825 . . 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 ) ) ) )
1711, 15, 16syl2anc 661 . 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 ) ) ) )
183, 10, 17mpbir2and 913 1  |-  ( ( J  e.  Top  /\  N  e.  ( ( nei `  J ) `  S )  /\  R  C_  S )  ->  N  e.  ( ( nei `  J
) `  R )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    e. wcel 1758   E.wrex 2796    C_ wss 3428   U.cuni 4191   ` cfv 5518   Topctop 18616   neicnei 18819
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4503  ax-sep 4513  ax-nul 4521  ax-pow 4570  ax-pr 4631
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-reu 2802  df-rab 2804  df-v 3072  df-sbc 3287  df-csb 3389  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-nul 3738  df-if 3892  df-pw 3962  df-sn 3978  df-pr 3980  df-op 3984  df-uni 4192  df-iun 4273  df-br 4393  df-opab 4451  df-mpt 4452  df-id 4736  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5481  df-fun 5520  df-fn 5521  df-f 5522  df-f1 5523  df-fo 5524  df-f1o 5525  df-fv 5526  df-top 18621  df-nei 18820
This theorem is referenced by:  neips  18835  neissex  18849
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