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Metamath Proof Explorer

Theorem imaexg

Description: The image of a set is a set. Theorem 3.17 of Monk1 p. 39. (Contributed by NM, 24-Jul-1995)

Ref Expression
Assertion imaexg ( 𝐴 ∈ 𝑉 → ( 𝐴 “ 𝐵 ) ∈ V )

Proof

Step Hyp Ref Expression
1 imassrn ⊢ ( 𝐴 “ 𝐵 ) ⊆ ran 𝐴
2 rnexg ⊢ ( 𝐴 ∈ 𝑉 → ran 𝐴 ∈ V )
3 ssexg ⊢ ( ( ( 𝐴 “ 𝐵 ) ⊆ ran 𝐴 ∧ ran 𝐴 ∈ V ) → ( 𝐴 “ 𝐵 ) ∈ V )
4 1 2 3 sylancr ⊢ ( 𝐴 ∈ 𝑉 → ( 𝐴 “ 𝐵 ) ∈ V )