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Theorem regsep 19594
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 19592 . . . . 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 464 . . . 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 3519 . . . . . . . 8  |-  ( y  =  U  ->  (
( ( cls `  J
) `  x )  C_  y  <->  ( ( cls `  J ) `  x
)  C_  U )
)
43anbi2d 703 . . . . . . 7  |-  ( y  =  U  ->  (
( z  e.  x  /\  ( ( cls `  J
) `  x )  C_  y )  <->  ( z  e.  x  /\  (
( cls `  J
) `  x )  C_  U ) ) )
54rexbidv 2966 . . . . . 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 3059 . . . . 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 3204 . . . 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 16 . . 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 2532 . . . . . 6  |-  ( z  =  A  ->  (
z  e.  x  <->  A  e.  x ) )
109anbi1d 704 . . . . 5  |-  ( z  =  A  ->  (
( z  e.  x  /\  ( ( cls `  J
) `  x )  C_  U )  <->  ( A  e.  x  /\  (
( cls `  J
) `  x )  C_  U ) ) )
1110rexbidv 2966 . . . 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 3204 . . 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 33 . 2  |-  ( J  e.  Reg  ->  ( U  e.  J  ->  ( A  e.  U  ->  E. x  e.  J  ( A  e.  x  /\  ( ( cls `  J
) `  x )  C_  U ) ) ) )
14133imp 1185 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 setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 968    = wceq 1374    e. wcel 1762   A.wral 2807   E.wrex 2808    C_ wss 3469   ` cfv 5579   Topctop 19154   clsccl 19278   Regcreg 19569
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3108  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-br 4441  df-iota 5542  df-fv 5587  df-reg 19576
This theorem is referenced by:  regsep2  19636  regr1lem  19968  kqreglem1  19970  kqreglem2  19971  reghmph  20022  cnextcn  20295
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