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Theorem tpnei 16958
Description: The underlying set of a topology is a neighborhood of any of its subsets. Special case of opnneiss 16955. (Contributed by FL, 2-Oct-2006.)
Hypothesis
Ref Expression
tpnei.1  |-  X  = 
U. J
Assertion
Ref Expression
tpnei  |-  ( J  e.  Top  ->  ( S  C_  X  <->  X  e.  ( ( nei `  J
) `  S )
) )

Proof of Theorem tpnei
StepHypRef Expression
1 tpnei.1 . . . 4  |-  X  = 
U. J
21topopn 16752 . . 3  |-  ( J  e.  Top  ->  X  e.  J )
3 opnneiss 16955 . . . 4  |-  ( ( J  e.  Top  /\  X  e.  J  /\  S  C_  X )  ->  X  e.  ( ( nei `  J ) `  S ) )
433exp 1150 . . 3  |-  ( J  e.  Top  ->  ( X  e.  J  ->  ( S  C_  X  ->  X  e.  ( ( nei `  J ) `  S
) ) ) )
52, 4mpd 14 . 2  |-  ( J  e.  Top  ->  ( S  C_  X  ->  X  e.  ( ( nei `  J
) `  S )
) )
6 ssnei 16947 . . 3  |-  ( ( J  e.  Top  /\  X  e.  ( ( nei `  J ) `  S ) )  ->  S  C_  X )
76ex 423 . 2  |-  ( J  e.  Top  ->  ( X  e.  ( ( nei `  J ) `  S )  ->  S  C_  X ) )
85, 7impbid 183 1  |-  ( J  e.  Top  ->  ( S  C_  X  <->  X  e.  ( ( nei `  J
) `  S )
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    = wceq 1642    e. wcel 1710    C_ wss 3228   U.cuni 3906   ` cfv 5334   Topctop 16731   neicnei 16934
This theorem is referenced by:  neiuni  16959  neifil  17671
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-rep 4210  ax-sep 4220  ax-nul 4228  ax-pow 4267  ax-pr 4293  ax-un 4591
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-reu 2626  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3907  df-iun 3986  df-br 4103  df-opab 4157  df-mpt 4158  df-id 4388  df-xp 4774  df-rel 4775  df-cnv 4776  df-co 4777  df-dm 4778  df-rn 4779  df-res 4780  df-ima 4781  df-iota 5298  df-fun 5336  df-fn 5337  df-f 5338  df-f1 5339  df-fo 5340  df-f1o 5341  df-fv 5342  df-top 16736  df-nei 16935
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