Theorem isreg 21136

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.

Step	Hyp	Ref	Expression
1		fveq2 6191	. . . . . . . 8 |- ( j = J -> ( cls ` j ) = ( cls ` J ) )
2	1	fveq1d 6193	. . . . . . 7 |- ( j = J -> ( ( cls ` j ) ` z ) = ( ( cls ` J ) ` z ) )
3	2	sseq1d 3632	. . . . . 6 |- ( j = J -> ( ( ( cls ` j ) ` z ) C_ x <-> ( ( cls ` J ) ` z ) C_ x ) )
4	3	anbi2d 740	. . . . 5 |- ( j = J -> ( ( y e. z /\ ( ( cls ` j ) ` z ) C_ x ) <-> ( y e. z /\ ( ( cls ` J ) ` z ) C_ x ) ) )
5	4	rexeqbi1dv 3147	. . . 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 ) ) )
6	5	ralbidv 2986	. . 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 ) ) )
7	6	raleqbi1dv 3146	. 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 21120	. 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 ) }
9	7, 8	elrab2 3366	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 ) ) )

Syntax hints:   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   E.wrex 2913   C_ wss 3574   ` cfv 5888   Topctop 20698   clsccl 20822   Regcreg 21113

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-iota 5851  df-fv 5896  df-reg 21120

This theorem is referenced by:  regtop  21137  regsep  21138  isreg2  21181  kqreglem1  21544  kqreglem2  21545  nrmr0reg  21552  reghmph  21596  utopreg  22056