Theorem hmeocldb

Description: Homeomorphisms preserve closedness. (Contributed by Jeff Hankins, 3-Jul-2009)

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

Proof

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