Theorem iunin1

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 Mario Carneiro, 30-Aug-2015)

	Ref	Expression
Assertion	iunin1	⊢ ∪ 𝑥 ∈ 𝐴 ( 𝐶 ∩ 𝐵 ) = ( ∪ 𝑥 ∈ 𝐴 𝐶 ∩ 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		iunin2	⊢ ∪ 𝑥 ∈ 𝐴 ( 𝐵 ∩ 𝐶 ) = ( 𝐵 ∩ ∪ 𝑥 ∈ 𝐴 𝐶 )
2		incom	⊢ ( 𝐶 ∩ 𝐵 ) = ( 𝐵 ∩ 𝐶 )
3	2	a1i	⊢ ( 𝑥 ∈ 𝐴 → ( 𝐶 ∩ 𝐵 ) = ( 𝐵 ∩ 𝐶 ) )
4	3	iuneq2i	⊢ ∪ 𝑥 ∈ 𝐴 ( 𝐶 ∩ 𝐵 ) = ∪ 𝑥 ∈ 𝐴 ( 𝐵 ∩ 𝐶 )
5		incom	⊢ ( ∪ 𝑥 ∈ 𝐴 𝐶 ∩ 𝐵 ) = ( 𝐵 ∩ ∪ 𝑥 ∈ 𝐴 𝐶 )
6	1 4 5	3eqtr4i	⊢ ∪ 𝑥 ∈ 𝐴 ( 𝐶 ∩ 𝐵 ) = ( ∪ 𝑥 ∈ 𝐴 𝐶 ∩ 𝐵 )