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Theorem neiuni 18851
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 18850 . . . 4  |-  ( J  e.  Top  ->  ( S  C_  X  <->  X  e.  ( ( nei `  J
) `  S )
) )
32biimpa 484 . . 3  |-  ( ( J  e.  Top  /\  S  C_  X )  ->  X  e.  ( ( nei `  J ) `  S ) )
4 elssuni 4222 . . 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 18835 . . . . . 6  |-  ( ( J  e.  Top  /\  x  e.  ( ( nei `  J ) `  S ) )  ->  x  C_  X )
76ex 434 . . . . 5  |-  ( J  e.  Top  ->  (
x  e.  ( ( nei `  J ) `
 S )  ->  x  C_  X ) )
87adantr 465 . . . 4  |-  ( ( J  e.  Top  /\  S  C_  X )  -> 
( x  e.  ( ( nei `  J
) `  S )  ->  x  C_  X )
)
98ralrimiv 2823 . . 3  |-  ( ( J  e.  Top  /\  S  C_  X )  ->  A. x  e.  (
( nei `  J
) `  S )
x  C_  X )
10 unissb 4224 . . 3  |-  ( U. ( ( nei `  J
) `  S )  C_  X  <->  A. x  e.  ( ( nei `  J
) `  S )
x  C_  X )
119, 10sylibr 212 . 2  |-  ( ( J  e.  Top  /\  S  C_  X )  ->  U. ( ( nei `  J
) `  S )  C_  X )
125, 11eqssd 3474 1  |-  ( ( J  e.  Top  /\  S  C_  X )  ->  X  =  U. (
( nei `  J
) `  S )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1370    e. wcel 1758   A.wral 2795    C_ wss 3429   U.cuni 4192   ` cfv 5519   Topctop 18623   neicnei 18826
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4504  ax-sep 4514  ax-nul 4522  ax-pow 4571  ax-pr 4632
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-reu 2802  df-rab 2804  df-v 3073  df-sbc 3288  df-csb 3390  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-nul 3739  df-if 3893  df-pw 3963  df-sn 3979  df-pr 3981  df-op 3985  df-uni 4193  df-iun 4274  df-br 4394  df-opab 4452  df-mpt 4453  df-id 4737  df-xp 4947  df-rel 4948  df-cnv 4949  df-co 4950  df-dm 4951  df-rn 4952  df-res 4953  df-ima 4954  df-iota 5482  df-fun 5521  df-fn 5522  df-f 5523  df-f1 5524  df-fo 5525  df-f1o 5526  df-fv 5527  df-top 18628  df-nei 18827
This theorem is referenced by:  neifil  19578
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