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Theorem ssnei2 16869
Description: Any subset of  X containing a neigborhood of a set is a neighborhood of this set. Proposition Vi of [BourbakiTop1] p. I.3. (Contributed by FL, 2-Oct-2006.)
Hypothesis
Ref Expression
neips.1  |-  X  = 
U. J
Assertion
Ref Expression
ssnei2  |-  ( ( ( J  e.  Top  /\  N  e.  ( ( nei `  J ) `
 S ) )  /\  ( N  C_  M  /\  M  C_  X
) )  ->  M  e.  ( ( nei `  J
) `  S )
)

Proof of Theorem ssnei2
Dummy variable  g is distinct from all other variables.
StepHypRef Expression
1 simprr 733 . 2  |-  ( ( ( J  e.  Top  /\  N  e.  ( ( nei `  J ) `
 S ) )  /\  ( N  C_  M  /\  M  C_  X
) )  ->  M  C_  X )
2 neii2 16861 . . . 4  |-  ( ( J  e.  Top  /\  N  e.  ( ( nei `  J ) `  S ) )  ->  E. g  e.  J  ( S  C_  g  /\  g  C_  N ) )
3 sstr2 3199 . . . . . . 7  |-  ( g 
C_  N  ->  ( N  C_  M  ->  g  C_  M ) )
43com12 27 . . . . . 6  |-  ( N 
C_  M  ->  (
g  C_  N  ->  g 
C_  M ) )
54anim2d 548 . . . . 5  |-  ( N 
C_  M  ->  (
( S  C_  g  /\  g  C_  N )  ->  ( S  C_  g  /\  g  C_  M
) ) )
65reximdv 2667 . . . 4  |-  ( N 
C_  M  ->  ( E. g  e.  J  ( S  C_  g  /\  g  C_  N )  ->  E. g  e.  J  ( S  C_  g  /\  g  C_  M ) ) )
72, 6mpan9 455 . . 3  |-  ( ( ( J  e.  Top  /\  N  e.  ( ( nei `  J ) `
 S ) )  /\  N  C_  M
)  ->  E. g  e.  J  ( S  C_  g  /\  g  C_  M ) )
87adantrr 697 . 2  |-  ( ( ( J  e.  Top  /\  N  e.  ( ( nei `  J ) `
 S ) )  /\  ( N  C_  M  /\  M  C_  X
) )  ->  E. g  e.  J  ( S  C_  g  /\  g  C_  M ) )
9 neips.1 . . . . 5  |-  X  = 
U. J
109neiss2 16854 . . . 4  |-  ( ( J  e.  Top  /\  N  e.  ( ( nei `  J ) `  S ) )  ->  S  C_  X )
119isnei 16856 . . . 4  |-  ( ( J  e.  Top  /\  S  C_  X )  -> 
( M  e.  ( ( nei `  J
) `  S )  <->  ( M  C_  X  /\  E. g  e.  J  ( S  C_  g  /\  g  C_  M ) ) ) )
1210, 11syldan 456 . . 3  |-  ( ( J  e.  Top  /\  N  e.  ( ( nei `  J ) `  S ) )  -> 
( M  e.  ( ( nei `  J
) `  S )  <->  ( M  C_  X  /\  E. g  e.  J  ( S  C_  g  /\  g  C_  M ) ) ) )
1312adantr 451 . 2  |-  ( ( ( J  e.  Top  /\  N  e.  ( ( nei `  J ) `
 S ) )  /\  ( N  C_  M  /\  M  C_  X
) )  ->  ( M  e.  ( ( nei `  J ) `  S )  <->  ( M  C_  X  /\  E. g  e.  J  ( S  C_  g  /\  g  C_  M ) ) ) )
141, 8, 13mpbir2and 888 1  |-  ( ( ( J  e.  Top  /\  N  e.  ( ( nei `  J ) `
 S ) )  /\  ( N  C_  M  /\  M  C_  X
) )  ->  M  e.  ( ( nei `  J
) `  S )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696   E.wrex 2557    C_ wss 3165   U.cuni 3843   ` cfv 5271   Topctop 16647   neicnei 16850
This theorem is referenced by:  topssnei  16877  nllyrest  17228  nllyidm  17231  hausllycmp  17236  cldllycmp  17237  txnlly  17347  neifil  17591  cnllycmp  18470  islimrs4  25685
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-top 16652  df-nei 16851
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