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Metamath Proof Explorer

Theorem fnfvelrnd

Description: A function's value belongs to its range. (Contributed by Glauco Siliprandi, 2-Jan-2022)

Ref Expression
Hypotheses fnfvelrnd.1 ⊢ ( 𝜑 → 𝐹 Fn 𝐴 )
fnfvelrnd.2 ⊢ ( 𝜑 → 𝐵 ∈ 𝐴 )
Assertion fnfvelrnd ( 𝜑 → ( 𝐹 ‘ 𝐵 ) ∈ ran 𝐹 )

Proof

Step Hyp Ref Expression
1 fnfvelrnd.1 ⊢ ( 𝜑 → 𝐹 Fn 𝐴 )
2 fnfvelrnd.2 ⊢ ( 𝜑 → 𝐵 ∈ 𝐴 )
3 fnfvelrn ⊢ ( ( 𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴 ) → ( 𝐹 ‘ 𝐵 ) ∈ ran 𝐹 )
4 1 2 3 syl2anc ⊢ ( 𝜑 → ( 𝐹 ‘ 𝐵 ) ∈ ran 𝐹 )