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 ∈ 𝑉 → ( 𝐹 ∈ 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 ∈ 𝐵 ↔ 𝐹 ∈ ( 𝐴 ↑m 𝐴 ) )
5		id	⊢ ( 𝐴 ∈ 𝑉 → 𝐴 ∈ 𝑉 )
6	5 5	elmapd	⊢ ( 𝐴 ∈ 𝑉 → ( 𝐹 ∈ ( 𝐴 ↑m 𝐴 ) ↔ 𝐹 : 𝐴 ⟶ 𝐴 ) )
7	4 6	bitrid	⊢ ( 𝐴 ∈ 𝑉 → ( 𝐹 ∈ 𝐵 ↔ 𝐹 : 𝐴 ⟶ 𝐴 ) )