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Metamath Proof Explorer

Theorem eldm2

Description: Membership in a domain. Theorem 4 of Suppes p. 59. (Contributed by NM, 1-Aug-1994)

Ref Expression
Hypothesis eldm.1 ⊢ A ∈ V
Assertion eldm2 ⊢ A ∈ dom ⁡ B ↔ ∃ y A y ∈ B

Proof

Step Hyp Ref Expression
1 eldm.1 ⊢ A ∈ V
2 eldm2g ⊢ A ∈ V → A ∈ dom ⁡ B ↔ ∃ y A y ∈ B
3 1 2 ax-mp ⊢ A ∈ dom ⁡ B ↔ ∃ y A y ∈ B