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Metamath Proof Explorer

Theorem dfiun2

Description: Alternate definition of indexed union when B is a set. Definition 15(a) of Suppes p. 44. (Contributed by NM, 27-Jun-1998) (Revised by David Abernethy, 19-Jun-2012)

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

Proof

Step Hyp Ref Expression
1 dfiun2.1 ⊢ B ∈ V
2 dfiun2g ⊢ ∀ 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