Theorem iinun2 5016

Description: Indexed intersection of union. Generalization of half of theorem "Distributive laws" in [Enderton] p. 30. Use intiin 5003 to recover Enderton's theorem. (Contributed by NM, 19-Aug-2004.)

Assertion

Ref	Expression
iinun2	⊢ ∩ 𝑥 ∈ 𝐴 (𝐵 ∪ 𝐶) = (𝐵 ∪ ∩ 𝑥 ∈ 𝐴 𝐶)

Distinct variable group:   𝑥,𝐵

Allowed substitution hints:   𝐴(𝑥)   𝐶(𝑥)

Proof of Theorem iinun2

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		r19.32v 3165	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 (𝑦 ∈ 𝐵 ∨ 𝑦 ∈ 𝐶) ↔ (𝑦 ∈ 𝐵 ∨ ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝐶))
2		elun 4098	. . . . 5 ⊢ (𝑦 ∈ (𝐵 ∪ 𝐶) ↔ (𝑦 ∈ 𝐵 ∨ 𝑦 ∈ 𝐶))
3	2	ralbii 3078	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 𝑦 ∈ (𝐵 ∪ 𝐶) ↔ ∀𝑥 ∈ 𝐴 (𝑦 ∈ 𝐵 ∨ 𝑦 ∈ 𝐶))
4		eliin 4941	. . . . . 6 ⊢ (𝑦 ∈ V → (𝑦 ∈ ∩ 𝑥 ∈ 𝐴 𝐶 ↔ ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝐶))
5	4	elv 3441	. . . . 5 ⊢ (𝑦 ∈ ∩ 𝑥 ∈ 𝐴 𝐶 ↔ ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝐶)
6	5	orbi2i 912	. . . 4 ⊢ ((𝑦 ∈ 𝐵 ∨ 𝑦 ∈ ∩ 𝑥 ∈ 𝐴 𝐶) ↔ (𝑦 ∈ 𝐵 ∨ ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝐶))
7	1, 3, 6	3bitr4i 303	. . 3 ⊢ (∀𝑥 ∈ 𝐴 𝑦 ∈ (𝐵 ∪ 𝐶) ↔ (𝑦 ∈ 𝐵 ∨ 𝑦 ∈ ∩ 𝑥 ∈ 𝐴 𝐶))
8		eliin 4941	. . . 4 ⊢ (𝑦 ∈ V → (𝑦 ∈ ∩ 𝑥 ∈ 𝐴 (𝐵 ∪ 𝐶) ↔ ∀𝑥 ∈ 𝐴 𝑦 ∈ (𝐵 ∪ 𝐶)))
9	8	elv 3441	. . 3 ⊢ (𝑦 ∈ ∩ 𝑥 ∈ 𝐴 (𝐵 ∪ 𝐶) ↔ ∀𝑥 ∈ 𝐴 𝑦 ∈ (𝐵 ∪ 𝐶))
10		elun 4098	. . 3 ⊢ (𝑦 ∈ (𝐵 ∪ ∩ 𝑥 ∈ 𝐴 𝐶) ↔ (𝑦 ∈ 𝐵 ∨ 𝑦 ∈ ∩ 𝑥 ∈ 𝐴 𝐶))
11	7, 9, 10	3bitr4i 303	. 2 ⊢ (𝑦 ∈ ∩ 𝑥 ∈ 𝐴 (𝐵 ∪ 𝐶) ↔ 𝑦 ∈ (𝐵 ∪ ∩ 𝑥 ∈ 𝐴 𝐶))
12	11	eqriv 2728	1 ⊢ ∩ 𝑥 ∈ 𝐴 (𝐵 ∪ 𝐶) = (𝐵 ∪ ∩ 𝑥 ∈ 𝐴 𝐶)

Syntax hints:   ↔ wb 206   ∨ wo 847   = wceq 1541   ∈ wcel 2111  ∀wral 3047  Vcvv 3436   ∪ cun 3895  ∩ ciin 4937

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1544  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-ral 3048  df-v 3438  df-un 3902  df-iin 4939

This theorem is referenced by: (None)