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Theorem ssnei2 17135
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 734 . 2  |-  ( ( ( J  e.  Top  /\  N  e.  ( ( nei `  J ) `
 S ) )  /\  ( N  C_  M  /\  M  C_  X
) )  ->  M  C_  X )
2 neii2 17127 . . . 4  |-  ( ( J  e.  Top  /\  N  e.  ( ( nei `  J ) `  S ) )  ->  E. g  e.  J  ( S  C_  g  /\  g  C_  N ) )
3 sstr2 3315 . . . . . . 7  |-  ( g 
C_  N  ->  ( N  C_  M  ->  g  C_  M ) )
43com12 29 . . . . . 6  |-  ( N 
C_  M  ->  (
g  C_  N  ->  g 
C_  M ) )
54anim2d 549 . . . . 5  |-  ( N 
C_  M  ->  (
( S  C_  g  /\  g  C_  N )  ->  ( S  C_  g  /\  g  C_  M
) ) )
65reximdv 2777 . . . 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 456 . . 3  |-  ( ( ( J  e.  Top  /\  N  e.  ( ( nei `  J ) `
 S ) )  /\  N  C_  M
)  ->  E. g  e.  J  ( S  C_  g  /\  g  C_  M ) )
87adantrr 698 . 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 17120 . . . 4  |-  ( ( J  e.  Top  /\  N  e.  ( ( nei `  J ) `  S ) )  ->  S  C_  X )
119isnei 17122 . . . 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 457 . . 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 452 . 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 889 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 177    /\ wa 359    = wceq 1649    e. wcel 1721   E.wrex 2667    C_ wss 3280   U.cuni 3975   ` cfv 5413   Topctop 16913   neicnei 17116
This theorem is referenced by:  topssnei  17143  nllyrest  17502  nllyidm  17505  hausllycmp  17510  cldllycmp  17511  txnlly  17622  neifil  17865  utop2nei  18233  cnllycmp  18934
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-top 16918  df-nei 17117
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