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Theorem regsep 17118
Description: In a regular space, every neighborhood of a point contains a closed subneighborhood. (Contributed by Mario Carneiro, 25-Aug-2015.)
Assertion
Ref Expression
regsep  |-  ( ( J  e.  Reg  /\  U  e.  J  /\  A  e.  U )  ->  E. x  e.  J  ( A  e.  x  /\  ( ( cls `  J
) `  x )  C_  U ) )
Distinct variable groups:    x, A    x, J    x, U

Proof of Theorem regsep
Dummy variables  y 
z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 isreg 17116 . . . . 5  |-  ( J  e.  Reg  <->  ( J  e.  Top  /\  A. y  e.  J  A. z  e.  y  E. x  e.  J  ( z  e.  x  /\  (
( cls `  J
) `  x )  C_  y ) ) )
21simprbi 450 . . . 4  |-  ( J  e.  Reg  ->  A. y  e.  J  A. z  e.  y  E. x  e.  J  ( z  e.  x  /\  (
( cls `  J
) `  x )  C_  y ) )
3 sseq2 3234 . . . . . . . 8  |-  ( y  =  U  ->  (
( ( cls `  J
) `  x )  C_  y  <->  ( ( cls `  J ) `  x
)  C_  U )
)
43anbi2d 684 . . . . . . 7  |-  ( y  =  U  ->  (
( z  e.  x  /\  ( ( cls `  J
) `  x )  C_  y )  <->  ( z  e.  x  /\  (
( cls `  J
) `  x )  C_  U ) ) )
54rexbidv 2598 . . . . . 6  |-  ( y  =  U  ->  ( E. x  e.  J  ( z  e.  x  /\  ( ( cls `  J
) `  x )  C_  y )  <->  E. x  e.  J  ( z  e.  x  /\  (
( cls `  J
) `  x )  C_  U ) ) )
65raleqbi1dv 2778 . . . . 5  |-  ( y  =  U  ->  ( A. z  e.  y  E. x  e.  J  ( z  e.  x  /\  ( ( cls `  J
) `  x )  C_  y )  <->  A. z  e.  U  E. x  e.  J  ( z  e.  x  /\  (
( cls `  J
) `  x )  C_  U ) ) )
76rspccv 2915 . . . 4  |-  ( A. y  e.  J  A. z  e.  y  E. x  e.  J  (
z  e.  x  /\  ( ( cls `  J
) `  x )  C_  y )  ->  ( U  e.  J  ->  A. z  e.  U  E. x  e.  J  (
z  e.  x  /\  ( ( cls `  J
) `  x )  C_  U ) ) )
82, 7syl 15 . . 3  |-  ( J  e.  Reg  ->  ( U  e.  J  ->  A. z  e.  U  E. x  e.  J  (
z  e.  x  /\  ( ( cls `  J
) `  x )  C_  U ) ) )
9 eleq1 2376 . . . . . 6  |-  ( z  =  A  ->  (
z  e.  x  <->  A  e.  x ) )
109anbi1d 685 . . . . 5  |-  ( z  =  A  ->  (
( z  e.  x  /\  ( ( cls `  J
) `  x )  C_  U )  <->  ( A  e.  x  /\  (
( cls `  J
) `  x )  C_  U ) ) )
1110rexbidv 2598 . . . 4  |-  ( z  =  A  ->  ( E. x  e.  J  ( z  e.  x  /\  ( ( cls `  J
) `  x )  C_  U )  <->  E. x  e.  J  ( A  e.  x  /\  (
( cls `  J
) `  x )  C_  U ) ) )
1211rspccv 2915 . . 3  |-  ( A. z  e.  U  E. x  e.  J  (
z  e.  x  /\  ( ( cls `  J
) `  x )  C_  U )  ->  ( A  e.  U  ->  E. x  e.  J  ( A  e.  x  /\  ( ( cls `  J
) `  x )  C_  U ) ) )
138, 12syl6 29 . 2  |-  ( J  e.  Reg  ->  ( U  e.  J  ->  ( A  e.  U  ->  E. x  e.  J  ( A  e.  x  /\  ( ( cls `  J
) `  x )  C_  U ) ) ) )
14133imp 1145 1  |-  ( ( J  e.  Reg  /\  U  e.  J  /\  A  e.  U )  ->  E. x  e.  J  ( A  e.  x  /\  ( ( cls `  J
) `  x )  C_  U ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1633    e. wcel 1701   A.wral 2577   E.wrex 2578    C_ wss 3186   ` cfv 5292   Topctop 16687   clsccl 16811   Regcreg 17093
This theorem is referenced by:  regsep2  17160  regr1lem  17486  kqreglem1  17488  kqreglem2  17489  reghmph  17540
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ral 2582  df-rex 2583  df-rab 2586  df-v 2824  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-nul 3490  df-if 3600  df-sn 3680  df-pr 3681  df-op 3683  df-uni 3865  df-br 4061  df-iota 5256  df-fv 5300  df-reg 17100
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