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Metamath Proof Explorer

Theorem fvbigcup

Description: For sets, Bigcup yields union. (Contributed by Scott Fenton, 11-Apr-2012)

Ref Expression
Hypothesis fvbigcup.1 ⊢ 𝐴 ∈ V
Assertion fvbigcup ( Bigcup ‘ 𝐴 ) = ∪ 𝐴

Proof

Step Hyp Ref Expression
1 fvbigcup.1 ⊢ 𝐴 ∈ V
2 eqid ⊢ ∪ 𝐴 = ∪ 𝐴
3 1 uniex ⊢ ∪ 𝐴 ∈ V
4 3 brbigcup ⊢ ( 𝐴 Bigcup ∪ 𝐴 ↔ ∪ 𝐴 = ∪ 𝐴 )
5 2 4 mpbir ⊢ 𝐴 Bigcup ∪ 𝐴
6 fnbigcup ⊢ Bigcup Fn V
7 fnbrfvb ⊢ ( ( Bigcup Fn V ∧ 𝐴 ∈ V ) → ( ( Bigcup ‘ 𝐴 ) = ∪ 𝐴 ↔ 𝐴 Bigcup ∪ 𝐴 ) )
8 6 1 7 mp2an ⊢ ( ( Bigcup ‘ 𝐴 ) = ∪ 𝐴 ↔ 𝐴 Bigcup ∪ 𝐴 )
9 5 8 mpbir ⊢ ( Bigcup ‘ 𝐴 ) = ∪ 𝐴