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 ) |
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 ) |