Theorem isnrm

Description: The predicate "is a normal space." Much like the case for regular spaces, normal does not imply Hausdorff or even regular. (Contributed by Jeff Hankins, 1-Feb-2010) (Revised by Mario Carneiro, 24-Aug-2015)

	Ref	Expression
Assertion	isnrm	|- ( J e. Nrm <-> ( J e. Top /\ A. x e. J A. y e. ( ( Clsd ` J ) i^i ~P x ) E. z e. J ( y C_ z /\ ( ( cls ` J ) ` z ) C_ x ) ) )

Proof

Step	Hyp	Ref	Expression
1		fveq2	|- ( j = J -> ( Clsd ` j ) = ( Clsd ` J ) )
2	1	ineq1d	|- ( j = J -> ( ( Clsd ` j ) i^i ~P x ) = ( ( Clsd ` J ) i^i ~P x ) )
3		fveq2	|- ( j = J -> ( cls ` j ) = ( cls ` J ) )
4	3	fveq1d	|- ( j = J -> ( ( cls ` j ) ` z ) = ( ( cls ` J ) ` z ) )
5	4	sseq1d	|- ( j = J -> ( ( ( cls ` j ) ` z ) C_ x <-> ( ( cls ` J ) ` z ) C_ x ) )
6	5	anbi2d	|- ( j = J -> ( ( y C_ z /\ ( ( cls ` j ) ` z ) C_ x ) <-> ( y C_ z /\ ( ( cls ` J ) ` z ) C_ x ) ) )
7	6	rexeqbi1dv	|- ( j = J -> ( E. z e. j ( y C_ z /\ ( ( cls ` j ) ` z ) C_ x ) <-> E. z e. J ( y C_ z /\ ( ( cls ` J ) ` z ) C_ x ) ) )
8	2 7	raleqbidv	|- ( j = J -> ( A. y e. ( ( Clsd ` j ) i^i ~P x ) E. z e. j ( y C_ z /\ ( ( cls ` j ) ` z ) C_ x ) <-> A. y e. ( ( Clsd ` J ) i^i ~P x ) E. z e. J ( y C_ z /\ ( ( cls ` J ) ` z ) C_ x ) ) )
9	8	raleqbi1dv	|- ( j = J -> ( A. x e. j A. y e. ( ( Clsd ` j ) i^i ~P x ) E. z e. j ( y C_ z /\ ( ( cls ` j ) ` z ) C_ x ) <-> A. x e. J A. y e. ( ( Clsd ` J ) i^i ~P x ) E. z e. J ( y C_ z /\ ( ( cls ` J ) ` z ) C_ x ) ) )
10		df-nrm	|- Nrm = { j e. Top | A. x e. j A. y e. ( ( Clsd ` j ) i^i ~P x ) E. z e. j ( y C_ z /\ ( ( cls ` j ) ` z ) C_ x ) }
11	9 10	elrab2	|- ( J e. Nrm <-> ( J e. Top /\ A. x e. J A. y e. ( ( Clsd ` J ) i^i ~P x ) E. z e. J ( y C_ z /\ ( ( cls ` J ) ` z ) C_ x ) ) )