Metamath Proof Explorer


Theorem frnvafv2v

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 : 𝐴𝐵𝐵𝑉 ) → ( 𝐹 '''' 𝐶 ) ∈ V )

Proof

Step Hyp Ref Expression
1 df-f ( 𝐹 : 𝐴𝐵 ↔ ( 𝐹 Fn 𝐴 ∧ ran 𝐹𝐵 ) )
2 ssexg ( ( ran 𝐹𝐵𝐵𝑉 ) → ran 𝐹 ∈ V )
3 2 ex ( ran 𝐹𝐵 → ( 𝐵𝑉 → ran 𝐹 ∈ V ) )
4 1 3 simplbiim ( 𝐹 : 𝐴𝐵 → ( 𝐵𝑉 → ran 𝐹 ∈ V ) )
5 4 imp ( ( 𝐹 : 𝐴𝐵𝐵𝑉 ) → ran 𝐹 ∈ V )
6 afv2ex ( ran 𝐹 ∈ V → ( 𝐹 '''' 𝐶 ) ∈ V )
7 5 6 syl ( ( 𝐹 : 𝐴𝐵𝐵𝑉 ) → ( 𝐹 '''' 𝐶 ) ∈ V )