Theorem isreg

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 T_3), 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)

	Ref	Expression
Assertion	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 ) ) )

Proof

Step	Hyp	Ref	Expression
1		fveq2	|- ( j = J -> ( cls ` j ) = ( cls ` J ) )
2	1	fveq1d	|- ( j = J -> ( ( cls ` j ) ` z ) = ( ( cls ` J ) ` z ) )
3	2	sseq1d	|- ( j = J -> ( ( ( cls ` j ) ` z ) C_ x <-> ( ( cls ` J ) ` z ) C_ x ) )
4	3	anbi2d	|- ( j = J -> ( ( y e. z /\ ( ( cls ` j ) ` z ) C_ x ) <-> ( y e. z /\ ( ( cls ` J ) ` z ) C_ x ) ) )
5	4	rexeqbi1dv	|- ( 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	|- ( 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	|- ( 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	|- 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	|- ( 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 ) ) )