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Metamath Proof Explorer

Theorem elmapdd

Description: Deduction associated with elmapd . (Contributed by SN, 29-Jul-2024)

Ref Expression
Hypotheses elmapdd.a ⊢ ( 𝜑 → 𝐴 ∈ 𝑉 )
elmapdd.b ⊢ ( 𝜑 → 𝐵 ∈ 𝑊 )
elmapdd.c ⊢ ( 𝜑 → 𝐶 : 𝐵 ⟶ 𝐴 )
Assertion elmapdd ( 𝜑 → 𝐶 ∈ ( 𝐴 ↑m 𝐵 ) )

Proof

Step Hyp Ref Expression
1 elmapdd.a ⊢ ( 𝜑 → 𝐴 ∈ 𝑉 )
2 elmapdd.b ⊢ ( 𝜑 → 𝐵 ∈ 𝑊 )
3 elmapdd.c ⊢ ( 𝜑 → 𝐶 : 𝐵 ⟶ 𝐴 )
4 1 2 elmapd ⊢ ( 𝜑 → ( 𝐶 ∈ ( 𝐴 ↑m 𝐵 ) ↔ 𝐶 : 𝐵 ⟶ 𝐴 ) )
5 3 4 mpbird ⊢ ( 𝜑 → 𝐶 ∈ ( 𝐴 ↑m 𝐵 ) )