Metamath Proof Explorer

Theorem fafv2elrn

Description: An alternate function value belongs to the codomain of the function, analogous to ffvelrn . (Contributed by AV, 2-Sep-2022)

Ref Expression
Assertion fafv2elrn ⊢ F : A ⟶ B ∧ C ∈ A → F '''' C ∈ B

Proof

Step Hyp Ref Expression
1 ffn ⊢ F : A ⟶ B → F Fn A
2 fnafv2elrn ⊢ F Fn A ∧ C ∈ A → F '''' C ∈ ran ⁡ F
3 1 2 sylan ⊢ F : A ⟶ B ∧ C ∈ A → F '''' C ∈ ran ⁡ F
4 frn ⊢ F : A ⟶ B → ran ⁡ F ⊆ B
5 4 sseld ⊢ F : A ⟶ B → F '''' C ∈ ran ⁡ F → F '''' C ∈ B
6 5 adantr ⊢ F : A ⟶ B ∧ C ∈ A → F '''' C ∈ ran ⁡ F → F '''' C ∈ B
7 3 6 mpd ⊢ F : A ⟶ B ∧ C ∈ A → F '''' C ∈ B