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Theorem isreg 19960
Description: The predicate "is a regular space." In a regular space, any open neighborhood has a closed subneighborhood. Note that some authors require the space to be Hausdorff (which would make it the same as T3), but we reserve the phrase "regular Hausdorff" for that as many topologists do. (Contributed by Jeff Hankins, 1-Feb-2010.) (Revised by Mario Carneiro, 25-Aug-2015.)
Assertion
Ref Expression
isreg  |-  ( J  e.  Reg  <->  ( J  e.  Top  /\  A. x  e.  J  A. y  e.  x  E. z  e.  J  ( y  e.  z  /\  (
( cls `  J
) `  z )  C_  x ) ) )
Distinct variable group:    x, y, z, J

Proof of Theorem isreg
Dummy variable  j is distinct from all other variables.
StepHypRef Expression
1 fveq2 5872 . . . . . . . 8  |-  ( j  =  J  ->  ( cls `  j )  =  ( cls `  J
) )
21fveq1d 5874 . . . . . . 7  |-  ( j  =  J  ->  (
( cls `  j
) `  z )  =  ( ( cls `  J ) `  z
) )
32sseq1d 3526 . . . . . 6  |-  ( j  =  J  ->  (
( ( cls `  j
) `  z )  C_  x  <->  ( ( cls `  J ) `  z
)  C_  x )
)
43anbi2d 703 . . . . 5  |-  ( j  =  J  ->  (
( y  e.  z  /\  ( ( cls `  j ) `  z
)  C_  x )  <->  ( y  e.  z  /\  ( ( cls `  J
) `  z )  C_  x ) ) )
54rexeqbi1dv 3063 . . . 4  |-  ( j  =  J  ->  ( E. z  e.  j 
( y  e.  z  /\  ( ( cls `  j ) `  z
)  C_  x )  <->  E. z  e.  J  ( y  e.  z  /\  ( ( cls `  J
) `  z )  C_  x ) ) )
65ralbidv 2896 . . 3  |-  ( j  =  J  ->  ( A. y  e.  x  E. z  e.  j 
( y  e.  z  /\  ( ( cls `  j ) `  z
)  C_  x )  <->  A. y  e.  x  E. z  e.  J  (
y  e.  z  /\  ( ( cls `  J
) `  z )  C_  x ) ) )
76raleqbi1dv 3062 . 2  |-  ( j  =  J  ->  ( A. x  e.  j  A. y  e.  x  E. z  e.  j 
( y  e.  z  /\  ( ( cls `  j ) `  z
)  C_  x )  <->  A. x  e.  J  A. y  e.  x  E. z  e.  J  (
y  e.  z  /\  ( ( cls `  J
) `  z )  C_  x ) ) )
8 df-reg 19944 . 2  |-  Reg  =  { j  e.  Top  | 
A. x  e.  j 
A. y  e.  x  E. z  e.  j 
( y  e.  z  /\  ( ( cls `  j ) `  z
)  C_  x ) }
97, 8elrab2 3259 1  |-  ( J  e.  Reg  <->  ( J  e.  Top  /\  A. x  e.  J  A. y  e.  x  E. z  e.  J  ( y  e.  z  /\  (
( cls `  J
) `  z )  C_  x ) ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 369    = wceq 1395    e. wcel 1819   A.wral 2807   E.wrex 2808    C_ wss 3471   ` cfv 5594   Topctop 19521   clsccl 19646   Regcreg 19937
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-iota 5557  df-fv 5602  df-reg 19944
This theorem is referenced by:  regtop  19961  regsep  19962  isreg2  20005  kqreglem1  20368  kqreglem2  20369  nrmr0reg  20376  reghmph  20420  utopreg  20881
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