Theorem iinun2 4720

Description: Indexed intersection of union. Generalization of half of theorem "Distributive laws" in [Enderton] p. 30. Use intiin 4708 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 3231	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 (𝑦 ∈ 𝐵 ∨ 𝑦 ∈ 𝐶) ↔ (𝑦 ∈ 𝐵 ∨ ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝐶))
2		elun 3904	. . . . 5 ⊢ (𝑦 ∈ (𝐵 ∪ 𝐶) ↔ (𝑦 ∈ 𝐵 ∨ 𝑦 ∈ 𝐶))
3	2	ralbii 3129	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 𝑦 ∈ (𝐵 ∪ 𝐶) ↔ ∀𝑥 ∈ 𝐴 (𝑦 ∈ 𝐵 ∨ 𝑦 ∈ 𝐶))
4		vex 3354	. . . . . 6 ⊢ 𝑦 ∈ V
5		eliin 4659	. . . . . 6 ⊢ (𝑦 ∈ V → (𝑦 ∈ ∩ 𝑥 ∈ 𝐴 𝐶 ↔ ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝐶))
6	4, 5	ax-mp 5	. . . . 5 ⊢ (𝑦 ∈ ∩ 𝑥 ∈ 𝐴 𝐶 ↔ ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝐶)
7	6	orbi2i 896	. . . 4 ⊢ ((𝑦 ∈ 𝐵 ∨ 𝑦 ∈ ∩ 𝑥 ∈ 𝐴 𝐶) ↔ (𝑦 ∈ 𝐵 ∨ ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝐶))
8	1, 3, 7	3bitr4i 292	. . 3 ⊢ (∀𝑥 ∈ 𝐴 𝑦 ∈ (𝐵 ∪ 𝐶) ↔ (𝑦 ∈ 𝐵 ∨ 𝑦 ∈ ∩ 𝑥 ∈ 𝐴 𝐶))
9		eliin 4659	. . . 4 ⊢ (𝑦 ∈ V → (𝑦 ∈ ∩ 𝑥 ∈ 𝐴 (𝐵 ∪ 𝐶) ↔ ∀𝑥 ∈ 𝐴 𝑦 ∈ (𝐵 ∪ 𝐶)))
10	4, 9	ax-mp 5	. . 3 ⊢ (𝑦 ∈ ∩ 𝑥 ∈ 𝐴 (𝐵 ∪ 𝐶) ↔ ∀𝑥 ∈ 𝐴 𝑦 ∈ (𝐵 ∪ 𝐶))
11		elun 3904	. . 3 ⊢ (𝑦 ∈ (𝐵 ∪ ∩ 𝑥 ∈ 𝐴 𝐶) ↔ (𝑦 ∈ 𝐵 ∨ 𝑦 ∈ ∩ 𝑥 ∈ 𝐴 𝐶))
12	8, 10, 11	3bitr4i 292	. 2 ⊢ (𝑦 ∈ ∩ 𝑥 ∈ 𝐴 (𝐵 ∪ 𝐶) ↔ 𝑦 ∈ (𝐵 ∪ ∩ 𝑥 ∈ 𝐴 𝐶))
13	12	eqriv 2768	1 ⊢ ∩ 𝑥 ∈ 𝐴 (𝐵 ∪ 𝐶) = (𝐵 ∪ ∩ 𝑥 ∈ 𝐴 𝐶)

Syntax hints:   ↔ wb 196   ∨ wo 834   = wceq 1631   ∈ wcel 2145  ∀wral 3061  Vcvv 3351   ∪ cun 3721  ∩ ciin 4655

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751

This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-v 3353  df-un 3728  df-iin 4657

This theorem is referenced by: (None)