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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 }