Description: If the codomain of a function is a set, the alternate function value is always also a set. (Contributed by AV, 4-Sep-2022)
Ref | Expression | ||
---|---|---|---|
Assertion | frnvafv2v | ⊢ ( ( 𝐹 : 𝐴 ⟶ 𝐵 ∧ 𝐵 ∈ 𝑉 ) → ( 𝐹 '''' 𝐶 ) ∈ V ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-f | ⊢ ( 𝐹 : 𝐴 ⟶ 𝐵 ↔ ( 𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝐵 ) ) | |
2 | ssexg | ⊢ ( ( ran 𝐹 ⊆ 𝐵 ∧ 𝐵 ∈ 𝑉 ) → ran 𝐹 ∈ V ) | |
3 | 2 | ex | ⊢ ( ran 𝐹 ⊆ 𝐵 → ( 𝐵 ∈ 𝑉 → ran 𝐹 ∈ V ) ) |
4 | 1 3 | simplbiim | ⊢ ( 𝐹 : 𝐴 ⟶ 𝐵 → ( 𝐵 ∈ 𝑉 → ran 𝐹 ∈ V ) ) |
5 | 4 | imp | ⊢ ( ( 𝐹 : 𝐴 ⟶ 𝐵 ∧ 𝐵 ∈ 𝑉 ) → ran 𝐹 ∈ V ) |
6 | afv2ex | ⊢ ( ran 𝐹 ∈ V → ( 𝐹 '''' 𝐶 ) ∈ V ) | |
7 | 5 6 | syl | ⊢ ( ( 𝐹 : 𝐴 ⟶ 𝐵 ∧ 𝐵 ∈ 𝑉 ) → ( 𝐹 '''' 𝐶 ) ∈ V ) |