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Theorem regsep 19056
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 19054 . . . . 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 3478 . . . . . . . 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 2848 . . . . . 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 3023 . . . . 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 3168 . . . 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 2523 . . . . . 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 2848 . . . 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 3168 . . 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 1182 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 965    = wceq 1370    e. wcel 1758   A.wral 2795   E.wrex 2796    C_ wss 3428   ` cfv 5518   Topctop 18616   clsccl 18740   Regcreg 19031
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-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430
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-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3072  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-nul 3738  df-if 3892  df-sn 3978  df-pr 3980  df-op 3984  df-uni 4192  df-br 4393  df-iota 5481  df-fv 5526  df-reg 19038
This theorem is referenced by:  regsep2  19098  regr1lem  19430  kqreglem1  19432  kqreglem2  19433  reghmph  19484  cnextcn  19757
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