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