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Metamath Proof Explorer

Theorem iunconst

Description: Indexed union of a constant class, i.e. where B does not depend on x . (Contributed by NM, 5-Sep-2004) (Proof shortened by Andrew Salmon, 25-Jul-2011)

		Ref	Expression
	Assertion	iunconst	⊢ ( 𝐴 ≠ ∅ → ∪ 𝑥 ∈ 𝐴 𝐵 = 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		eliun	⊢ ( 𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ↔ ∃ 𝑥 ∈ 𝐴 𝑦 ∈ 𝐵 )
2		r19.9rzv	⊢ ( 𝐴 ≠ ∅ → ( 𝑦 ∈ 𝐵 ↔ ∃ 𝑥 ∈ 𝐴 𝑦 ∈ 𝐵 ) )
3	1 2	bitr4id	⊢ ( 𝐴 ≠ ∅ → ( 𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ↔ 𝑦 ∈ 𝐵 ) )
4	3	eqrdv	⊢ ( 𝐴 ≠ ∅ → ∪ 𝑥 ∈ 𝐴 𝐵 = 𝐵 )