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Theorem ssnei2 12363
Description: Any subset  M of  X containing a neighborhood  N of a set  S is a neighborhood of this set. Generalization to subsets of Property 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 522 . 2  |-  ( ( ( J  e.  Top  /\  N  e.  ( ( nei `  J ) `
 S ) )  /\  ( N  C_  M  /\  M  C_  X
) )  ->  M  C_  X )
2 neii2 12355 . . . 4  |-  ( ( J  e.  Top  /\  N  e.  ( ( nei `  J ) `  S ) )  ->  E. g  e.  J  ( S  C_  g  /\  g  C_  N ) )
3 sstr2 3108 . . . . . . 7  |-  ( g 
C_  N  ->  ( N  C_  M  ->  g  C_  M ) )
43com12 30 . . . . . 6  |-  ( N 
C_  M  ->  (
g  C_  N  ->  g 
C_  M ) )
54anim2d 335 . . . . 5  |-  ( N 
C_  M  ->  (
( S  C_  g  /\  g  C_  N )  ->  ( S  C_  g  /\  g  C_  M
) ) )
65reximdv 2536 . . . 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 279 . . 3  |-  ( ( ( J  e.  Top  /\  N  e.  ( ( nei `  J ) `
 S ) )  /\  N  C_  M
)  ->  E. g  e.  J  ( S  C_  g  /\  g  C_  M ) )
87adantrr 471 . 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 12348 . . . 4  |-  ( ( J  e.  Top  /\  N  e.  ( ( nei `  J ) `  S ) )  ->  S  C_  X )
119isnei 12350 . . . 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 280 . . 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 274 . 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 929 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    /\ wa 103    <-> wb 104    = wceq 1332    e. wcel 1481   E.wrex 2418    C_ wss 3075   U.cuni 3743   ` cfv 5130   Topctop 12201   neicnei 12344
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-coll 4050  ax-sep 4053  ax-pow 4105  ax-pr 4138
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-reu 2424  df-rab 2426  df-v 2691  df-sbc 2913  df-csb 3007  df-un 3079  df-in 3081  df-ss 3088  df-pw 3516  df-sn 3537  df-pr 3538  df-op 3540  df-uni 3744  df-iun 3822  df-br 3937  df-opab 3997  df-mpt 3998  df-id 4222  df-xp 4552  df-rel 4553  df-cnv 4554  df-co 4555  df-dm 4556  df-rn 4557  df-res 4558  df-ima 4559  df-iota 5095  df-fun 5132  df-fn 5133  df-f 5134  df-f1 5135  df-fo 5136  df-f1o 5137  df-fv 5138  df-top 12202  df-nei 12345
This theorem is referenced by:  topssnei  12368
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