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Theorem neiuni 17109
Description: The union of the neighborhoods of a set equals the topology's underlying set. (Contributed by FL, 18-Sep-2007.) (Revised by Mario Carneiro, 9-Apr-2015.)
Hypothesis
Ref Expression
tpnei.1  |-  X  = 
U. J
Assertion
Ref Expression
neiuni  |-  ( ( J  e.  Top  /\  S  C_  X )  ->  X  =  U. (
( nei `  J
) `  S )
)

Proof of Theorem neiuni
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 tpnei.1 . . . . 5  |-  X  = 
U. J
21tpnei 17108 . . . 4  |-  ( J  e.  Top  ->  ( S  C_  X  <->  X  e.  ( ( nei `  J
) `  S )
) )
32biimpa 471 . . 3  |-  ( ( J  e.  Top  /\  S  C_  X )  ->  X  e.  ( ( nei `  J ) `  S ) )
4 elssuni 3985 . . 3  |-  ( X  e.  ( ( nei `  J ) `  S
)  ->  X  C_  U. (
( nei `  J
) `  S )
)
53, 4syl 16 . 2  |-  ( ( J  e.  Top  /\  S  C_  X )  ->  X  C_  U. ( ( nei `  J ) `
 S ) )
61neii1 17093 . . . . . 6  |-  ( ( J  e.  Top  /\  x  e.  ( ( nei `  J ) `  S ) )  ->  x  C_  X )
76ex 424 . . . . 5  |-  ( J  e.  Top  ->  (
x  e.  ( ( nei `  J ) `
 S )  ->  x  C_  X ) )
87adantr 452 . . . 4  |-  ( ( J  e.  Top  /\  S  C_  X )  -> 
( x  e.  ( ( nei `  J
) `  S )  ->  x  C_  X )
)
98ralrimiv 2731 . . 3  |-  ( ( J  e.  Top  /\  S  C_  X )  ->  A. x  e.  (
( nei `  J
) `  S )
x  C_  X )
10 unissb 3987 . . 3  |-  ( U. ( ( nei `  J
) `  S )  C_  X  <->  A. x  e.  ( ( nei `  J
) `  S )
x  C_  X )
119, 10sylibr 204 . 2  |-  ( ( J  e.  Top  /\  S  C_  X )  ->  U. ( ( nei `  J
) `  S )  C_  X )
125, 11eqssd 3308 1  |-  ( ( J  e.  Top  /\  S  C_  X )  ->  X  =  U. (
( nei `  J
) `  S )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1717   A.wral 2649    C_ wss 3263   U.cuni 3957   ` cfv 5394   Topctop 16881   neicnei 17084
This theorem is referenced by:  neifil  17833
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 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-rep 4261  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344
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 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-ral 2654  df-rex 2655  df-reu 2656  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-op 3766  df-uni 3958  df-iun 4037  df-br 4154  df-opab 4208  df-mpt 4209  df-id 4439  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-top 16886  df-nei 17085
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