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Metamath Proof Explorer

Theorem mptexf

Description: If the domain of a function given by maps-to notation is a set, the function is a set. Inference version of mptexg . (Contributed by Glauco Siliprandi, 23-Oct-2021)

Ref Expression
Hypotheses mptexf.1 ⊢ Ⅎ _ x A
mptexf.2 ⊢ A ∈ V
Assertion mptexf ⊢ x ∈ A ⟼ B ∈ V

Proof

Step Hyp Ref Expression
1 mptexf.1 ⊢ Ⅎ _ x A
2 mptexf.2 ⊢ A ∈ V
3 1 mptexgf ⊢ A ∈ V → x ∈ A ⟼ B ∈ V
4 2 3 ax-mp ⊢ x ∈ A ⟼ B ∈ V