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Metamath Proof Explorer

Theorem ffvelcdmd

Description: A function's value belongs to its codomain. (Contributed by Mario Carneiro, 29-Dec-2016)

Ref Expression
Hypotheses ffvelcdmd.1 ⊢ ( 𝜑 → 𝐹 : 𝐴 ⟶ 𝐵 )
ffvelcdmd.2 ⊢ ( 𝜑 → 𝐶 ∈ 𝐴 )
Assertion ffvelcdmd ( 𝜑 → ( 𝐹 ‘ 𝐶 ) ∈ 𝐵 )

Proof

Step Hyp Ref Expression
1 ffvelcdmd.1 ⊢ ( 𝜑 → 𝐹 : 𝐴 ⟶ 𝐵 )
2 ffvelcdmd.2 ⊢ ( 𝜑 → 𝐶 ∈ 𝐴 )
3 1 ffvelcdmda ⊢ ( ( 𝜑 ∧ 𝐶 ∈ 𝐴 ) → ( 𝐹 ‘ 𝐶 ) ∈ 𝐵 )
4 2 3 mpdan ⊢ ( 𝜑 → ( 𝐹 ‘ 𝐶 ) ∈ 𝐵 )