Theorem iunin1f 30897

Description: Indexed union of intersection. Generalization of half of theorem "Distributive laws" in [Enderton] p. 30. Use uniiun 4988 to recover Enderton's theorem. (Contributed by NM, 26-Mar-2004.) (Revised by Thierry Arnoux, 2-May-2020.)

Hypothesis

Ref	Expression
iunin1f.1	⊢ Ⅎ𝑥𝐶

Assertion

Ref	Expression
iunin1f	⊢ ∪ 𝑥 ∈ 𝐴 (𝐵 ∩ 𝐶) = (∪ 𝑥 ∈ 𝐴 𝐵 ∩ 𝐶)

Proof of Theorem iunin1f

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		iunin1f.1	. . . . . 6 ⊢ Ⅎ𝑥𝐶
2	1	nfcri 2894	. . . . 5 ⊢ Ⅎ𝑥 𝑦 ∈ 𝐶
3	2	r19.41 3277	. . . 4 ⊢ (∃𝑥 ∈ 𝐴 (𝑦 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶) ↔ (∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶))
4		elin 3903	. . . . 5 ⊢ (𝑦 ∈ (𝐵 ∩ 𝐶) ↔ (𝑦 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶))
5	4	rexbii 3181	. . . 4 ⊢ (∃𝑥 ∈ 𝐴 𝑦 ∈ (𝐵 ∩ 𝐶) ↔ ∃𝑥 ∈ 𝐴 (𝑦 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶))
6		eliun 4928	. . . . 5 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ↔ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵)
7	6	anbi1i 624	. . . 4 ⊢ ((𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑦 ∈ 𝐶) ↔ (∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶))
8	3, 5, 7	3bitr4i 303	. . 3 ⊢ (∃𝑥 ∈ 𝐴 𝑦 ∈ (𝐵 ∩ 𝐶) ↔ (𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑦 ∈ 𝐶))
9		eliun 4928	. . 3 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ 𝐴 (𝐵 ∩ 𝐶) ↔ ∃𝑥 ∈ 𝐴 𝑦 ∈ (𝐵 ∩ 𝐶))
10		elin 3903	. . 3 ⊢ (𝑦 ∈ (∪ 𝑥 ∈ 𝐴 𝐵 ∩ 𝐶) ↔ (𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑦 ∈ 𝐶))
11	8, 9, 10	3bitr4i 303	. 2 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ 𝐴 (𝐵 ∩ 𝐶) ↔ 𝑦 ∈ (∪ 𝑥 ∈ 𝐴 𝐵 ∩ 𝐶))
12	11	eqriv 2735	1 ⊢ ∪ 𝑥 ∈ 𝐴 (𝐵 ∩ 𝐶) = (∪ 𝑥 ∈ 𝐴 𝐵 ∩ 𝐶)

Syntax hints:   ∧ wa 396   = wceq 1539   ∈ wcel 2106  Ⅎwnfc 2887  ∃wrex 3065   ∩ cin 3886  ∪ ciun 4924

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-12 2171  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1542  df-ex 1783  df-nf 1787  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rex 3070  df-v 3434  df-in 3894  df-iun 4926

This theorem is referenced by:  esum2dlem  32060