Theorem hmeoclda

Description: Homeomorphisms preserve closedness. (Contributed by Jeff Hankins, 3-Jul-2009) (Revised by Mario Carneiro, 3-Jun-2014)

	Ref	Expression
Assertion	hmeoclda	|- ( ( ( J e. Top /\ K e. Top /\ F e. ( J Homeo K ) ) /\ S e. ( Clsd ` J ) ) -> ( F " S ) e. ( Clsd ` K ) )

Proof

Step	Hyp	Ref	Expression
1		hmeocnvcn	|- ( F e. ( J Homeo K ) -> `' F e. ( K Cn J ) )
2	1	3ad2ant3	|- ( ( J e. Top /\ K e. Top /\ F e. ( J Homeo K ) ) -> `' F e. ( K Cn J ) )
3		imacnvcnv	|- ( `' `' F " S ) = ( F " S )
4		cnclima	|- ( ( `' F e. ( K Cn J ) /\ S e. ( Clsd ` J ) ) -> ( `' `' F " S ) e. ( Clsd ` K ) )
5	3 4	eqeltrrid	|- ( ( `' F e. ( K Cn J ) /\ S e. ( Clsd ` J ) ) -> ( F " S ) e. ( Clsd ` K ) )
6	2 5	sylan	|- ( ( ( J e. Top /\ K e. Top /\ F e. ( J Homeo K ) ) /\ S e. ( Clsd ` J ) ) -> ( F " S ) e. ( Clsd ` K ) )