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Theorem regsep 18779
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 18777 . . . . 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 461 . . . 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 3366 . . . . . . . 8  |-  ( y  =  U  ->  (
( ( cls `  J
) `  x )  C_  y  <->  ( ( cls `  J ) `  x
)  C_  U )
)
43anbi2d 696 . . . . . . 7  |-  ( y  =  U  ->  (
( z  e.  x  /\  ( ( cls `  J
) `  x )  C_  y )  <->  ( z  e.  x  /\  (
( cls `  J
) `  x )  C_  U ) ) )
54rexbidv 2726 . . . . . 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 2915 . . . . 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 3059 . . . 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 2493 . . . . . 6  |-  ( z  =  A  ->  (
z  e.  x  <->  A  e.  x ) )
109anbi1d 697 . . . . 5  |-  ( z  =  A  ->  (
( z  e.  x  /\  ( ( cls `  J
) `  x )  C_  U )  <->  ( A  e.  x  /\  (
( cls `  J
) `  x )  C_  U ) ) )
1110rexbidv 2726 . . . 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 3059 . . 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 1174 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 958    = wceq 1362    e. wcel 1755   A.wral 2705   E.wrex 2706    C_ wss 3316   ` cfv 5406   Topctop 18339   clsccl 18463   Regcreg 18754
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1594  ax-4 1605  ax-5 1669  ax-6 1707  ax-7 1727  ax-10 1774  ax-11 1779  ax-12 1791  ax-13 1942  ax-ext 2414
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 960  df-tru 1365  df-ex 1590  df-nf 1593  df-sb 1700  df-clab 2420  df-cleq 2426  df-clel 2429  df-nfc 2558  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2964  df-dif 3319  df-un 3321  df-in 3323  df-ss 3330  df-nul 3626  df-if 3780  df-sn 3866  df-pr 3868  df-op 3872  df-uni 4080  df-br 4281  df-iota 5369  df-fv 5414  df-reg 18761
This theorem is referenced by:  regsep2  18821  regr1lem  19153  kqreglem1  19155  kqreglem2  19156  reghmph  19207  cnextcn  19480
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