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 ≠ ∅ → 𝑥𝐴 𝐵 = 𝐵 )

Proof

Step Hyp Ref Expression
1 r19.9rzv ( 𝐴 ≠ ∅ → ( 𝑦𝐵 ↔ ∃ 𝑥𝐴 𝑦𝐵 ) )
2 eliun ( 𝑦 𝑥𝐴 𝐵 ↔ ∃ 𝑥𝐴 𝑦𝐵 )
3 1 2 syl6rbbr ( 𝐴 ≠ ∅ → ( 𝑦 𝑥𝐴 𝐵𝑦𝐵 ) )
4 3 eqrdv ( 𝐴 ≠ ∅ → 𝑥𝐴 𝐵 = 𝐵 )