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Theorem neiss 16846
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 2283 . . . 4  |-  U. J  =  U. J
21neii1 16843 . . 3  |-  ( ( J  e.  Top  /\  N  e.  ( ( nei `  J ) `  S ) )  ->  N  C_  U. J )
323adant3 975 . 2  |-  ( ( J  e.  Top  /\  N  e.  ( ( nei `  J ) `  S )  /\  R  C_  S )  ->  N  C_ 
U. J )
4 neii2 16845 . . . 4  |-  ( ( J  e.  Top  /\  N  e.  ( ( nei `  J ) `  S ) )  ->  E. g  e.  J  ( S  C_  g  /\  g  C_  N ) )
543adant3 975 . . 3  |-  ( ( J  e.  Top  /\  N  e.  ( ( nei `  J ) `  S )  /\  R  C_  S )  ->  E. g  e.  J  ( S  C_  g  /\  g  C_  N ) )
6 sstr2 3186 . . . . . 6  |-  ( R 
C_  S  ->  ( S  C_  g  ->  R  C_  g ) )
76anim1d 547 . . . . 5  |-  ( R 
C_  S  ->  (
( S  C_  g  /\  g  C_  N )  ->  ( R  C_  g  /\  g  C_  N
) ) )
87reximdv 2654 . . . 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 978 . . 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 14 . 2  |-  ( ( J  e.  Top  /\  N  e.  ( ( nei `  J ) `  S )  /\  R  C_  S )  ->  E. g  e.  J  ( R  C_  g  /\  g  C_  N ) )
11 simp1 955 . . 3  |-  ( ( J  e.  Top  /\  N  e.  ( ( nei `  J ) `  S )  /\  R  C_  S )  ->  J  e.  Top )
12 simp3 957 . . . 4  |-  ( ( J  e.  Top  /\  N  e.  ( ( nei `  J ) `  S )  /\  R  C_  S )  ->  R  C_  S )
131neiss2 16838 . . . . 5  |-  ( ( J  e.  Top  /\  N  e.  ( ( nei `  J ) `  S ) )  ->  S  C_  U. J )
14133adant3 975 . . . 4  |-  ( ( J  e.  Top  /\  N  e.  ( ( nei `  J ) `  S )  /\  R  C_  S )  ->  S  C_ 
U. J )
1512, 14sstrd 3189 . . 3  |-  ( ( J  e.  Top  /\  N  e.  ( ( nei `  J ) `  S )  /\  R  C_  S )  ->  R  C_ 
U. J )
161isnei 16840 . . 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 642 . 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 888 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 4    <-> wb 176    /\ wa 358    /\ w3a 934    e. wcel 1684   E.wrex 2544    C_ wss 3152   U.cuni 3827   ` cfv 5255   Topctop 16631   neicnei 16834
This theorem is referenced by:  neips  16850  neissex  16864
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-top 16636  df-nei 16835
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