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Theorem ssnei2 16853
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 16845 . . . 4  |-  ( ( J  e.  Top  /\  N  e.  ( ( nei `  J ) `  S ) )  ->  E. g  e.  J  ( S  C_  g  /\  g  C_  N ) )
3 sstr2 3186 . . . . . . 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 2654 . . . 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 16838 . . . 4  |-  ( ( J  e.  Top  /\  N  e.  ( ( nei `  J ) `  S ) )  ->  S  C_  X )
119isnei 16840 . . . 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 1623    e. wcel 1684   E.wrex 2544    C_ wss 3152   U.cuni 3827   ` cfv 5255   Topctop 16631   neicnei 16834
This theorem is referenced by:  topssnei  16861  nllyrest  17212  nllyidm  17215  hausllycmp  17220  cldllycmp  17221  txnlly  17331  neifil  17575  cnllycmp  18454  islimrs4  25582
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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