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Metamath Proof Explorer

Theorem dfiin2

Description: Alternate definition of indexed intersection when B is a set. Definition 15(b) of Suppes p. 44. (Contributed by NM, 28-Jun-1998) (Proof shortened by Andrew Salmon, 25-Jul-2011)

Ref Expression
Hypothesis dfiun2.1 ⊢ B ∈ V
Assertion dfiin2 ⊢ ⋂ x ∈ A B = ⋂ y | ∃ x ∈ A y = B

Proof

Step Hyp Ref Expression
1 dfiun2.1 ⊢ B ∈ V
2 dfiin2g ⊢ ∀ x ∈ A B ∈ V → ⋂ x ∈ A B = ⋂ y | ∃ x ∈ A y = B
3 1 a1i ⊢ x ∈ A → B ∈ V
4 2 3 mprg ⊢ ⋂ x ∈ A B = ⋂ y | ∃ x ∈ A y = B