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Theorem regsep 20128
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 20126 . . . . 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 462 . . . 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 3464 . . . . . . . 8  |-  ( y  =  U  ->  (
( ( cls `  J
) `  x )  C_  y  <->  ( ( cls `  J ) `  x
)  C_  U )
)
43anbi2d 702 . . . . . . 7  |-  ( y  =  U  ->  (
( z  e.  x  /\  ( ( cls `  J
) `  x )  C_  y )  <->  ( z  e.  x  /\  (
( cls `  J
) `  x )  C_  U ) ) )
54rexbidv 2918 . . . . . 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 3012 . . . . 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 3157 . . . 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 17 . . 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 2474 . . . . . 6  |-  ( z  =  A  ->  (
z  e.  x  <->  A  e.  x ) )
109anbi1d 703 . . . . 5  |-  ( z  =  A  ->  (
( z  e.  x  /\  ( ( cls `  J
) `  x )  C_  U )  <->  ( A  e.  x  /\  (
( cls `  J
) `  x )  C_  U ) ) )
1110rexbidv 2918 . . . 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 3157 . . 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 31 . 2  |-  ( J  e.  Reg  ->  ( U  e.  J  ->  ( A  e.  U  ->  E. x  e.  J  ( A  e.  x  /\  ( ( cls `  J
) `  x )  C_  U ) ) ) )
14133imp 1191 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 367    /\ w3a 974    = wceq 1405    e. wcel 1842   A.wral 2754   E.wrex 2755    C_ wss 3414   ` cfv 5569   Topctop 19686   clsccl 19811   Regcreg 20103
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ral 2759  df-rex 2760  df-rab 2763  df-v 3061  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-nul 3739  df-if 3886  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4192  df-br 4396  df-iota 5533  df-fv 5577  df-reg 20110
This theorem is referenced by:  regsep2  20170  regr1lem  20532  kqreglem1  20534  kqreglem2  20535  reghmph  20586  cnextcn  20859
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