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Theorem neiuni 17179
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 17178 . . . 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 4036 . . 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 17163 . . . . . 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 2781 . . 3  |-  ( ( J  e.  Top  /\  S  C_  X )  ->  A. x  e.  (
( nei `  J
) `  S )
x  C_  X )
10 unissb 4038 . . 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 3358 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 1652    e. wcel 1725   A.wral 2698    C_ wss 3313   U.cuni 4008   ` cfv 5447   Topctop 16951   neicnei 17154
This theorem is referenced by:  neifil  17905
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4313  ax-sep 4323  ax-nul 4331  ax-pow 4370  ax-pr 4396
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2703  df-rex 2704  df-reu 2705  df-rab 2707  df-v 2951  df-sbc 3155  df-csb 3245  df-dif 3316  df-un 3318  df-in 3320  df-ss 3327  df-nul 3622  df-if 3733  df-pw 3794  df-sn 3813  df-pr 3814  df-op 3816  df-uni 4009  df-iun 4088  df-br 4206  df-opab 4260  df-mpt 4261  df-id 4491  df-xp 4877  df-rel 4878  df-cnv 4879  df-co 4880  df-dm 4881  df-rn 4882  df-res 4883  df-ima 4884  df-iota 5411  df-fun 5449  df-fn 5450  df-f 5451  df-f1 5452  df-fo 5453  df-f1o 5454  df-fv 5455  df-top 16956  df-nei 17155
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