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Theorem neiss 16808
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
StepHypRef Expression
1 eqid 2258 . . . 4  |-  U. J  =  U. J
21neii1 16805 . . 3  |-  ( ( J  e.  Top  /\  N  e.  ( ( nei `  J ) `  S ) )  ->  N  C_  U. J )
323adant3 980 . 2  |-  ( ( J  e.  Top  /\  N  e.  ( ( nei `  J ) `  S )  /\  R  C_  S )  ->  N  C_ 
U. J )
4 neii2 16807 . . . 4  |-  ( ( J  e.  Top  /\  N  e.  ( ( nei `  J ) `  S ) )  ->  E. g  e.  J  ( S  C_  g  /\  g  C_  N ) )
543adant3 980 . . 3  |-  ( ( J  e.  Top  /\  N  e.  ( ( nei `  J ) `  S )  /\  R  C_  S )  ->  E. g  e.  J  ( S  C_  g  /\  g  C_  N ) )
6 sstr2 3161 . . . . . 6  |-  ( R 
C_  S  ->  ( S  C_  g  ->  R  C_  g ) )
76anim1d 549 . . . . 5  |-  ( R 
C_  S  ->  (
( S  C_  g  /\  g  C_  N )  ->  ( R  C_  g  /\  g  C_  N
) ) )
87reximdv 2629 . . . 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 983 . . 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 16 . 2  |-  ( ( J  e.  Top  /\  N  e.  ( ( nei `  J ) `  S )  /\  R  C_  S )  ->  E. g  e.  J  ( R  C_  g  /\  g  C_  N ) )
11 simp1 960 . . 3  |-  ( ( J  e.  Top  /\  N  e.  ( ( nei `  J ) `  S )  /\  R  C_  S )  ->  J  e.  Top )
12 simp3 962 . . . 4  |-  ( ( J  e.  Top  /\  N  e.  ( ( nei `  J ) `  S )  /\  R  C_  S )  ->  R  C_  S )
131neiss2 16800 . . . . 5  |-  ( ( J  e.  Top  /\  N  e.  ( ( nei `  J ) `  S ) )  ->  S  C_  U. J )
14133adant3 980 . . . 4  |-  ( ( J  e.  Top  /\  N  e.  ( ( nei `  J ) `  S )  /\  R  C_  S )  ->  S  C_ 
U. J )
1512, 14sstrd 3164 . . 3  |-  ( ( J  e.  Top  /\  N  e.  ( ( nei `  J ) `  S )  /\  R  C_  S )  ->  R  C_ 
U. J )
161isnei 16802 . . 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 645 . 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 893 1  |-  ( ( J  e.  Top  /\  N  e.  ( ( nei `  J ) `  S )  /\  R  C_  S )  ->  N  e.  ( ( nei `  J
) `  R )
)
Colors of variables: wff set class
Syntax hints:    -> wi 6    <-> wb 178    /\ wa 360    /\ w3a 939    e. wcel 1621   E.wrex 2519    C_ wss 3127   U.cuni 3801   ` cfv 4673   Topctop 16593   neicnei 16796
This theorem is referenced by:  neips  16812  neissex  16826
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2239  ax-rep 4105  ax-sep 4115  ax-nul 4123  ax-pow 4160  ax-pr 4186  ax-un 4484
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2122  df-mo 2123  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ne 2423  df-ral 2523  df-rex 2524  df-reu 2525  df-rab 2527  df-v 2765  df-sbc 2967  df-csb 3057  df-dif 3130  df-un 3132  df-in 3134  df-ss 3141  df-nul 3431  df-if 3540  df-pw 3601  df-sn 3620  df-pr 3621  df-op 3623  df-uni 3802  df-iun 3881  df-br 3998  df-opab 4052  df-mpt 4053  df-id 4281  df-xp 4675  df-rel 4676  df-cnv 4677  df-co 4678  df-dm 4679  df-rn 4680  df-res 4681  df-ima 4682  df-fun 4683  df-fn 4684  df-f 4685  df-f1 4686  df-fo 4687  df-f1o 4688  df-fv 4689  df-top 16598  df-nei 16797
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