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Theorem ssnei2 12253
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 506 . 2  |-  ( ( ( J  e.  Top  /\  N  e.  ( ( nei `  J ) `
 S ) )  /\  ( N  C_  M  /\  M  C_  X
) )  ->  M  C_  X )
2 neii2 12245 . . . 4  |-  ( ( J  e.  Top  /\  N  e.  ( ( nei `  J ) `  S ) )  ->  E. g  e.  J  ( S  C_  g  /\  g  C_  N ) )
3 sstr2 3074 . . . . . . 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 2510 . . . 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 470 . 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 12238 . . . 4  |-  ( ( J  e.  Top  /\  N  e.  ( ( nei `  J ) `  S ) )  ->  S  C_  X )
119isnei 12240 . . . 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 913 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 1316    e. wcel 1465   E.wrex 2394    C_ wss 3041   U.cuni 3706   ` cfv 5093   Topctop 12091   neicnei 12234
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 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-coll 4013  ax-sep 4016  ax-pow 4068  ax-pr 4101
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-reu 2400  df-rab 2402  df-v 2662  df-sbc 2883  df-csb 2976  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-iun 3785  df-br 3900  df-opab 3960  df-mpt 3961  df-id 4185  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-res 4521  df-ima 4522  df-iota 5058  df-fun 5095  df-fn 5096  df-f 5097  df-f1 5098  df-fo 5099  df-f1o 5100  df-fv 5101  df-top 12092  df-nei 12235
This theorem is referenced by:  topssnei  12258
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