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 ( 𝜑𝐹 : 𝐴𝐵 )
ffvelcdmd.2 ( 𝜑𝐶𝐴 )
Assertion ffvelcdmd ( 𝜑 → ( 𝐹𝐶 ) ∈ 𝐵 )

Proof

Step Hyp Ref Expression
1 ffvelcdmd.1 ( 𝜑𝐹 : 𝐴𝐵 )
2 ffvelcdmd.2 ( 𝜑𝐶𝐴 )
3 1 ffvelcdmda ( ( 𝜑𝐶𝐴 ) → ( 𝐹𝐶 ) ∈ 𝐵 )
4 2 3 mpdan ( 𝜑 → ( 𝐹𝐶 ) ∈ 𝐵 )