Theorem iunin2

Description: Indexed union of intersection. Generalization of half of theorem "Distributive laws" in Enderton p. 30. Use uniiun to recover Enderton's theorem. (Contributed by NM, 26-Mar-2004)

	Ref	Expression
Assertion	iunin2	⊢ ∪ 𝑥 ∈ 𝐴 ( 𝐵 ∩ 𝐶 ) = ( 𝐵 ∩ ∪ 𝑥 ∈ 𝐴 𝐶 )

Proof

Step	Hyp	Ref	Expression
1		r19.42v	⊢ ( ∃ 𝑥 ∈ 𝐴 ( 𝑦 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶 ) ↔ ( 𝑦 ∈ 𝐵 ∧ ∃ 𝑥 ∈ 𝐴 𝑦 ∈ 𝐶 ) )
2		elin	⊢ ( 𝑦 ∈ ( 𝐵 ∩ 𝐶 ) ↔ ( 𝑦 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶 ) )
3	2	rexbii	⊢ ( ∃ 𝑥 ∈ 𝐴 𝑦 ∈ ( 𝐵 ∩ 𝐶 ) ↔ ∃ 𝑥 ∈ 𝐴 ( 𝑦 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶 ) )
4		eliun	⊢ ( 𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐶 ↔ ∃ 𝑥 ∈ 𝐴 𝑦 ∈ 𝐶 )
5	4	anbi2i	⊢ ( ( 𝑦 ∈ 𝐵 ∧ 𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐶 ) ↔ ( 𝑦 ∈ 𝐵 ∧ ∃ 𝑥 ∈ 𝐴 𝑦 ∈ 𝐶 ) )
6	1 3 5	3bitr4i	⊢ ( ∃ 𝑥 ∈ 𝐴 𝑦 ∈ ( 𝐵 ∩ 𝐶 ) ↔ ( 𝑦 ∈ 𝐵 ∧ 𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐶 ) )
7		eliun	⊢ ( 𝑦 ∈ ∪ 𝑥 ∈ 𝐴 ( 𝐵 ∩ 𝐶 ) ↔ ∃ 𝑥 ∈ 𝐴 𝑦 ∈ ( 𝐵 ∩ 𝐶 ) )
8		elin	⊢ ( 𝑦 ∈ ( 𝐵 ∩ ∪ 𝑥 ∈ 𝐴 𝐶 ) ↔ ( 𝑦 ∈ 𝐵 ∧ 𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐶 ) )
9	6 7 8	3bitr4i	⊢ ( 𝑦 ∈ ∪ 𝑥 ∈ 𝐴 ( 𝐵 ∩ 𝐶 ) ↔ 𝑦 ∈ ( 𝐵 ∩ ∪ 𝑥 ∈ 𝐴 𝐶 ) )
10	9	eqriv	⊢ ∪ 𝑥 ∈ 𝐴 ( 𝐵 ∩ 𝐶 ) = ( 𝐵 ∩ ∪ 𝑥 ∈ 𝐴 𝐶 )