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

Metamath Proof Explorer

Theorem elmaprd

Description: Deduction associated with elmapd . Reverse direction of elmapdd . (Contributed by Thierry Arnoux, 13-Oct-2025)

Ref Expression
Hypotheses elmaprd.1 ⊢ φ → A ∈ V
elmaprd.2 ⊢ φ → B ∈ W
elmaprd.3 ⊢ φ → F ∈ B A
Assertion elmaprd ⊢ φ → F : A ⟶ B

Proof

Step Hyp Ref Expression
1 elmaprd.1 ⊢ φ → A ∈ V
2 elmaprd.2 ⊢ φ → B ∈ W
3 elmaprd.3 ⊢ φ → F ∈ B A
4 2 1 elmapd ⊢ φ → F ∈ B A ↔ F : A ⟶ B
5 3 4 mpbid ⊢ φ → F : A ⟶ B