Theorem inuni 4826

Description: The intersection of a union ∪ 𝐴 with a class 𝐵 is equal to the union of the intersections of each element of 𝐴 with 𝐵. (Contributed by FL, 24-Mar-2007.)

Assertion

Ref	Expression
inuni	⊢ (∪ 𝐴 ∩ 𝐵) = ∪ {𝑥 ∣ ∃𝑦 ∈ 𝐴 𝑥 = (𝑦 ∩ 𝐵)}

Distinct variable groups:   𝑥,𝐴,𝑦   𝑥,𝐵,𝑦

Proof of Theorem inuni

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eluni2 4440	. . . . 5 ⊢ (𝑧 ∈ ∪ 𝐴 ↔ ∃𝑦 ∈ 𝐴 𝑧 ∈ 𝑦)
2	1	anbi1i 731	. . . 4 ⊢ ((𝑧 ∈ ∪ 𝐴 ∧ 𝑧 ∈ 𝐵) ↔ (∃𝑦 ∈ 𝐴 𝑧 ∈ 𝑦 ∧ 𝑧 ∈ 𝐵))
3		elin 3796	. . . 4 ⊢ (𝑧 ∈ (∪ 𝐴 ∩ 𝐵) ↔ (𝑧 ∈ ∪ 𝐴 ∧ 𝑧 ∈ 𝐵))
4		ancom 466	. . . . . . . 8 ⊢ ((𝑧 ∈ 𝑥 ∧ ∃𝑦 ∈ 𝐴 𝑥 = (𝑦 ∩ 𝐵)) ↔ (∃𝑦 ∈ 𝐴 𝑥 = (𝑦 ∩ 𝐵) ∧ 𝑧 ∈ 𝑥))
5		r19.41v 3089	. . . . . . . 8 ⊢ (∃𝑦 ∈ 𝐴 (𝑥 = (𝑦 ∩ 𝐵) ∧ 𝑧 ∈ 𝑥) ↔ (∃𝑦 ∈ 𝐴 𝑥 = (𝑦 ∩ 𝐵) ∧ 𝑧 ∈ 𝑥))
6	4, 5	bitr4i 267	. . . . . . 7 ⊢ ((𝑧 ∈ 𝑥 ∧ ∃𝑦 ∈ 𝐴 𝑥 = (𝑦 ∩ 𝐵)) ↔ ∃𝑦 ∈ 𝐴 (𝑥 = (𝑦 ∩ 𝐵) ∧ 𝑧 ∈ 𝑥))
7	6	exbii 1774	. . . . . 6 ⊢ (∃𝑥(𝑧 ∈ 𝑥 ∧ ∃𝑦 ∈ 𝐴 𝑥 = (𝑦 ∩ 𝐵)) ↔ ∃𝑥∃𝑦 ∈ 𝐴 (𝑥 = (𝑦 ∩ 𝐵) ∧ 𝑧 ∈ 𝑥))
8		rexcom4 3225	. . . . . 6 ⊢ (∃𝑦 ∈ 𝐴 ∃𝑥(𝑥 = (𝑦 ∩ 𝐵) ∧ 𝑧 ∈ 𝑥) ↔ ∃𝑥∃𝑦 ∈ 𝐴 (𝑥 = (𝑦 ∩ 𝐵) ∧ 𝑧 ∈ 𝑥))
9	7, 8	bitr4i 267	. . . . 5 ⊢ (∃𝑥(𝑧 ∈ 𝑥 ∧ ∃𝑦 ∈ 𝐴 𝑥 = (𝑦 ∩ 𝐵)) ↔ ∃𝑦 ∈ 𝐴 ∃𝑥(𝑥 = (𝑦 ∩ 𝐵) ∧ 𝑧 ∈ 𝑥))
10		vex 3203	. . . . . . . . . 10 ⊢ 𝑦 ∈ V
11	10	inex1 4799	. . . . . . . . 9 ⊢ (𝑦 ∩ 𝐵) ∈ V
12		eleq2 2690	. . . . . . . . 9 ⊢ (𝑥 = (𝑦 ∩ 𝐵) → (𝑧 ∈ 𝑥 ↔ 𝑧 ∈ (𝑦 ∩ 𝐵)))
13	11, 12	ceqsexv 3242	. . . . . . . 8 ⊢ (∃𝑥(𝑥 = (𝑦 ∩ 𝐵) ∧ 𝑧 ∈ 𝑥) ↔ 𝑧 ∈ (𝑦 ∩ 𝐵))
14		elin 3796	. . . . . . . 8 ⊢ (𝑧 ∈ (𝑦 ∩ 𝐵) ↔ (𝑧 ∈ 𝑦 ∧ 𝑧 ∈ 𝐵))
15	13, 14	bitri 264	. . . . . . 7 ⊢ (∃𝑥(𝑥 = (𝑦 ∩ 𝐵) ∧ 𝑧 ∈ 𝑥) ↔ (𝑧 ∈ 𝑦 ∧ 𝑧 ∈ 𝐵))
16	15	rexbii 3041	. . . . . 6 ⊢ (∃𝑦 ∈ 𝐴 ∃𝑥(𝑥 = (𝑦 ∩ 𝐵) ∧ 𝑧 ∈ 𝑥) ↔ ∃𝑦 ∈ 𝐴 (𝑧 ∈ 𝑦 ∧ 𝑧 ∈ 𝐵))
17		r19.41v 3089	. . . . . 6 ⊢ (∃𝑦 ∈ 𝐴 (𝑧 ∈ 𝑦 ∧ 𝑧 ∈ 𝐵) ↔ (∃𝑦 ∈ 𝐴 𝑧 ∈ 𝑦 ∧ 𝑧 ∈ 𝐵))
18	16, 17	bitri 264	. . . . 5 ⊢ (∃𝑦 ∈ 𝐴 ∃𝑥(𝑥 = (𝑦 ∩ 𝐵) ∧ 𝑧 ∈ 𝑥) ↔ (∃𝑦 ∈ 𝐴 𝑧 ∈ 𝑦 ∧ 𝑧 ∈ 𝐵))
19	9, 18	bitri 264	. . . 4 ⊢ (∃𝑥(𝑧 ∈ 𝑥 ∧ ∃𝑦 ∈ 𝐴 𝑥 = (𝑦 ∩ 𝐵)) ↔ (∃𝑦 ∈ 𝐴 𝑧 ∈ 𝑦 ∧ 𝑧 ∈ 𝐵))
20	2, 3, 19	3bitr4i 292	. . 3 ⊢ (𝑧 ∈ (∪ 𝐴 ∩ 𝐵) ↔ ∃𝑥(𝑧 ∈ 𝑥 ∧ ∃𝑦 ∈ 𝐴 𝑥 = (𝑦 ∩ 𝐵)))
21		eluniab 4447	. . 3 ⊢ (𝑧 ∈ ∪ {𝑥 ∣ ∃𝑦 ∈ 𝐴 𝑥 = (𝑦 ∩ 𝐵)} ↔ ∃𝑥(𝑧 ∈ 𝑥 ∧ ∃𝑦 ∈ 𝐴 𝑥 = (𝑦 ∩ 𝐵)))
22	20, 21	bitr4i 267	. 2 ⊢ (𝑧 ∈ (∪ 𝐴 ∩ 𝐵) ↔ 𝑧 ∈ ∪ {𝑥 ∣ ∃𝑦 ∈ 𝐴 𝑥 = (𝑦 ∩ 𝐵)})
23	22	eqriv 2619	1 ⊢ (∪ 𝐴 ∩ 𝐵) = ∪ {𝑥 ∣ ∃𝑦 ∈ 𝐴 𝑥 = (𝑦 ∩ 𝐵)}

Syntax hints:   ∧ wa 384   = wceq 1483  ∃wex 1704   ∈ wcel 1990  {cab 2608  ∃wrex 2913   ∩ cin 3573  ∪ cuni 4436

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-v 3202  df-in 3581  df-uni 4437

This theorem is referenced by: (None)