Theorem efmndbas0

Description: The base set of the monoid of endofunctions on the empty set is the singleton containing the empty set. (Contributed by AV, 27-Jan-2024) (Proof shortened by AV, 31-Mar-2024)

	Ref	Expression
Assertion	efmndbas0	|- ( Base ` ( EndoFMnd ` (/) ) ) = { (/) }

Proof

Step	Hyp	Ref	Expression
1		eqid	|- ( EndoFMnd ` (/) ) = ( EndoFMnd ` (/) )
2		eqid	|- ( Base ` ( EndoFMnd ` (/) ) ) = ( Base ` ( EndoFMnd ` (/) ) )
3	1 2	efmndbas	|- ( Base ` ( EndoFMnd ` (/) ) ) = ( (/) ^m (/) )
4		0map0sn0	|- ( (/) ^m (/) ) = { (/) }
5	3 4	eqtri	|- ( Base ` ( EndoFMnd ` (/) ) ) = { (/) }