Theorem iscnrm

Description: The property of being completely or hereditarily normal. (Contributed by Mario Carneiro, 26-Aug-2015)

	Ref	Expression
Hypothesis	ist0.1	|- X = U. J
Assertion	iscnrm	|- ( J e. CNrm <-> ( J e. Top /\ A. x e. ~P X ( J |`t x ) e. Nrm ) )

Proof

Step	Hyp	Ref	Expression
1		ist0.1	|- X = U. J
2		unieq	|- ( j = J -> U. j = U. J )
3	2 1	eqtr4di	|- ( j = J -> U. j = X )
4	3	pweqd	|- ( j = J -> ~P U. j = ~P X )
5		oveq1	|- ( j = J -> ( j |`t x ) = ( J |`t x ) )
6	5	eleq1d	|- ( j = J -> ( ( j |`t x ) e. Nrm <-> ( J |`t x ) e. Nrm ) )
7	4 6	raleqbidv	|- ( j = J -> ( A. x e. ~P U. j ( j |`t x ) e. Nrm <-> A. x e. ~P X ( J |`t x ) e. Nrm ) )
8		df-cnrm	|- CNrm = { j e. Top | A. x e. ~P U. j ( j |`t x ) e. Nrm }
9	7 8	elrab2	|- ( J e. CNrm <-> ( J e. Top /\ A. x e. ~P X ( J |`t x ) e. Nrm ) )