This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem elefmndbas2

Description: Two ways of saying a function is a mapping of A to itself. (Contributed by AV, 27-Jan-2024) (Proof shortened by AV, 29-Mar-2024)

Ref Expression
Hypotheses efmndbas.g ⊢ G = EndoFMnd ⁡ A
efmndbas.b ⊢ B = Base G
Assertion elefmndbas2 ⊢ F ∈ 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 efmndbasabf ⊢ B = f | f : A ⟶ A
4 3 a1i ⊢ F ∈ V → B = f | f : A ⟶ A
5 4 eleq2d ⊢ F ∈ V → F ∈ B ↔ F ∈ f | f : A ⟶ A
6 feq1 ⊢ f = F → f : A ⟶ A ↔ F : A ⟶ A
7 eqid ⊢ f | f : A ⟶ A = f | f : A ⟶ A
8 6 7 elab2g ⊢ F ∈ V → F ∈ f | f : A ⟶ A ↔ F : A ⟶ A
9 5 8 bitrd ⊢ F ∈ V → F ∈ B ↔ F : A ⟶ A