Theorem lautcl

Description: A lattice automorphism value belongs to the base set. (Contributed by NM, 20-May-2012)

	Ref	Expression
Hypotheses	laut1o.b	|- B = ( Base ` K )
	laut1o.i	|- I = ( LAut ` K )
Assertion	lautcl	|- ( ( ( K e. V /\ F e. I ) /\ X e. B ) -> ( F ` X ) e. B )

Proof

Step	Hyp	Ref	Expression
1		laut1o.b	|- B = ( Base ` K )
2		laut1o.i	|- I = ( LAut ` K )
3	1 2	laut1o	|- ( ( K e. V /\ F e. I ) -> F : B -1-1-onto-> B )
4		f1of	|- ( F : B -1-1-onto-> B -> F : B --> B )
5	3 4	syl	|- ( ( K e. V /\ F e. I ) -> F : B --> B )
6	5	ffvelcdmda	|- ( ( ( K e. V /\ F e. I ) /\ X e. B ) -> ( F ` X ) e. B )