Theorem ntrfval

Description: The interior function on the subsets of a topology's base set. (Contributed by NM, 10-Sep-2006) (Revised by Mario Carneiro, 11-Nov-2013)

	Ref	Expression
Hypothesis	cldval.1	|- X = U. J
Assertion	ntrfval	|- ( J e. Top -> ( int ` J ) = ( x e. ~P X |-> U. ( J i^i ~P x ) ) )

Proof

Step	Hyp	Ref	Expression
1		cldval.1	|- X = U. J
2	1	topopn	|- ( J e. Top -> X e. J )
3		pwexg	|- ( X e. J -> ~P X e. _V )
4		mptexg	|- ( ~P X e. _V -> ( x e. ~P X |-> U. ( J i^i ~P x ) ) e. _V )
5	2 3 4	3syl	|- ( J e. Top -> ( x e. ~P X |-> U. ( J i^i ~P x ) ) e. _V )
6		unieq	|- ( j = J -> U. j = U. J )
7	6 1	eqtr4di	|- ( j = J -> U. j = X )
8	7	pweqd	|- ( j = J -> ~P U. j = ~P X )
9		ineq1	|- ( j = J -> ( j i^i ~P x ) = ( J i^i ~P x ) )
10	9	unieqd	|- ( j = J -> U. ( j i^i ~P x ) = U. ( J i^i ~P x ) )
11	8 10	mpteq12dv	|- ( j = J -> ( x e. ~P U. j |-> U. ( j i^i ~P x ) ) = ( x e. ~P X |-> U. ( J i^i ~P x ) ) )
12		df-ntr	|- int = ( j e. Top |-> ( x e. ~P U. j |-> U. ( j i^i ~P x ) ) )
13	11 12	fvmptg	|- ( ( J e. Top /\ ( x e. ~P X |-> U. ( J i^i ~P x ) ) e. _V ) -> ( int ` J ) = ( x e. ~P X |-> U. ( J i^i ~P x ) ) )
14	5 13	mpdan	|- ( J e. Top -> ( int ` J ) = ( x e. ~P X |-> U. ( J i^i ~P x ) ) )