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	$$⊢G=\mathrm{EndoFMnd}\left(A\right)$$
	efmndbas.b	$$⊢B={\mathrm{Base}}_G$$
Assertion	elefmndbas	$$⊢A\in V\to \left(F\in B\leftrightarrow F:A⟶A\right)$$

Proof

Step	Hyp	Ref	Expression
1		efmndbas.g	$$⊢G=\mathrm{EndoFMnd}\left(A\right)$$
2		efmndbas.b	$$⊢B={\mathrm{Base}}_G$$
3	1 2	efmndbas	$$⊢B=A^A$$
4	3	eleq2i	$$⊢F\in B\leftrightarrow F\in \left(A^A\right)$$
5		id	$$⊢A\in V\to A\in V$$
6	5 5	elmapd	$$⊢A\in V\to \left(F\in \left(A^A\right)\leftrightarrow F:A⟶A\right)$$
7	4 6	bitrid	$$⊢A\in V\to \left(F\in B\leftrightarrow F:A⟶A\right)$$