Theorem ntrelmap

Description: The interior function is a map from the powerset of the base set to itself. (Contributed by RP, 22-Apr-2021)

	Ref	Expression
Hypotheses	ntrrn.x	|- X = U. J
	ntrrn.i	|- I = ( int ` J )
Assertion	ntrelmap	|- ( J e. Top -> I e. ( ~P X ^m ~P X ) )

Proof

Step	Hyp	Ref	Expression
1		ntrrn.x	|- X = U. J
2		ntrrn.i	|- I = ( int ` J )
3	1 2	ntrf2	|- ( J e. Top -> I : ~P X --> ~P X )
4	1	topopn	|- ( J e. Top -> X e. J )
5	4	pwexd	|- ( J e. Top -> ~P X e. _V )
6	5 5	elmapd	|- ( J e. Top -> ( I e. ( ~P X ^m ~P X ) <-> I : ~P X --> ~P X ) )
7	3 6	mpbird	|- ( J e. Top -> I e. ( ~P X ^m ~P X ) )