Theorem elefmndbas

Description: Two ways of saying a function is a mapping of A to itself. (Contributed by AV, 27-Jan-2024)

	Ref	Expression
Hypotheses	efmndbas.g	|- G = ( EndoFMnd ` A )
	efmndbas.b	|- B = ( Base ` G )
Assertion	elefmndbas	|- ( A e. V -> ( F e. B <-> F : A --> A ) )

Proof

Step	Hyp	Ref	Expression
1		efmndbas.g	|- G = ( EndoFMnd ` A )
2		efmndbas.b	|- B = ( Base ` G )
3	1 2	efmndbas	|- B = ( A ^m A )
4	3	eleq2i	|- ( F e. B <-> F e. ( A ^m A ) )
5		id	|- ( A e. V -> A e. V )
6	5 5	elmapd	|- ( A e. V -> ( F e. ( A ^m A ) <-> F : A --> A ) )
7	4 6	syl5bb	|- ( A e. V -> ( F e. B <-> F : A --> A ) )