Theorem inuni 4150

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 3809	. . . . 5 ⊢ (𝑧 ∈ ∪ 𝐴 ↔ ∃𝑦 ∈ 𝐴 𝑧 ∈ 𝑦)
2	1	anbi1i 458	. . . 4 ⊢ ((𝑧 ∈ ∪ 𝐴 ∧ 𝑧 ∈ 𝐵) ↔ (∃𝑦 ∈ 𝐴 𝑧 ∈ 𝑦 ∧ 𝑧 ∈ 𝐵))
3		elin 3316	. . . 4 ⊢ (𝑧 ∈ (∪ 𝐴 ∩ 𝐵) ↔ (𝑧 ∈ ∪ 𝐴 ∧ 𝑧 ∈ 𝐵))
4		ancom 266	. . . . . . . 8 ⊢ ((𝑧 ∈ 𝑥 ∧ ∃𝑦 ∈ 𝐴 𝑥 = (𝑦 ∩ 𝐵)) ↔ (∃𝑦 ∈ 𝐴 𝑥 = (𝑦 ∩ 𝐵) ∧ 𝑧 ∈ 𝑥))
5		r19.41v 2631	. . . . . . . 8 ⊢ (∃𝑦 ∈ 𝐴 (𝑥 = (𝑦 ∩ 𝐵) ∧ 𝑧 ∈ 𝑥) ↔ (∃𝑦 ∈ 𝐴 𝑥 = (𝑦 ∩ 𝐵) ∧ 𝑧 ∈ 𝑥))
6	4, 5	bitr4i 187	. . . . . . 7 ⊢ ((𝑧 ∈ 𝑥 ∧ ∃𝑦 ∈ 𝐴 𝑥 = (𝑦 ∩ 𝐵)) ↔ ∃𝑦 ∈ 𝐴 (𝑥 = (𝑦 ∩ 𝐵) ∧ 𝑧 ∈ 𝑥))
7	6	exbii 1603	. . . . . 6 ⊢ (∃𝑥(𝑧 ∈ 𝑥 ∧ ∃𝑦 ∈ 𝐴 𝑥 = (𝑦 ∩ 𝐵)) ↔ ∃𝑥∃𝑦 ∈ 𝐴 (𝑥 = (𝑦 ∩ 𝐵) ∧ 𝑧 ∈ 𝑥))
8		rexcom4 2758	. . . . . 6 ⊢ (∃𝑦 ∈ 𝐴 ∃𝑥(𝑥 = (𝑦 ∩ 𝐵) ∧ 𝑧 ∈ 𝑥) ↔ ∃𝑥∃𝑦 ∈ 𝐴 (𝑥 = (𝑦 ∩ 𝐵) ∧ 𝑧 ∈ 𝑥))
9	7, 8	bitr4i 187	. . . . 5 ⊢ (∃𝑥(𝑧 ∈ 𝑥 ∧ ∃𝑦 ∈ 𝐴 𝑥 = (𝑦 ∩ 𝐵)) ↔ ∃𝑦 ∈ 𝐴 ∃𝑥(𝑥 = (𝑦 ∩ 𝐵) ∧ 𝑧 ∈ 𝑥))
10		vex 2738	. . . . . . . . . 10 ⊢ 𝑦 ∈ V
11	10	inex1 4132	. . . . . . . . 9 ⊢ (𝑦 ∩ 𝐵) ∈ V
12		eleq2 2239	. . . . . . . . 9 ⊢ (𝑥 = (𝑦 ∩ 𝐵) → (𝑧 ∈ 𝑥 ↔ 𝑧 ∈ (𝑦 ∩ 𝐵)))
13	11, 12	ceqsexv 2774	. . . . . . . 8 ⊢ (∃𝑥(𝑥 = (𝑦 ∩ 𝐵) ∧ 𝑧 ∈ 𝑥) ↔ 𝑧 ∈ (𝑦 ∩ 𝐵))
14		elin 3316	. . . . . . . 8 ⊢ (𝑧 ∈ (𝑦 ∩ 𝐵) ↔ (𝑧 ∈ 𝑦 ∧ 𝑧 ∈ 𝐵))
15	13, 14	bitri 184	. . . . . . 7 ⊢ (∃𝑥(𝑥 = (𝑦 ∩ 𝐵) ∧ 𝑧 ∈ 𝑥) ↔ (𝑧 ∈ 𝑦 ∧ 𝑧 ∈ 𝐵))
16	15	rexbii 2482	. . . . . 6 ⊢ (∃𝑦 ∈ 𝐴 ∃𝑥(𝑥 = (𝑦 ∩ 𝐵) ∧ 𝑧 ∈ 𝑥) ↔ ∃𝑦 ∈ 𝐴 (𝑧 ∈ 𝑦 ∧ 𝑧 ∈ 𝐵))
17		r19.41v 2631	. . . . . 6 ⊢ (∃𝑦 ∈ 𝐴 (𝑧 ∈ 𝑦 ∧ 𝑧 ∈ 𝐵) ↔ (∃𝑦 ∈ 𝐴 𝑧 ∈ 𝑦 ∧ 𝑧 ∈ 𝐵))
18	16, 17	bitri 184	. . . . 5 ⊢ (∃𝑦 ∈ 𝐴 ∃𝑥(𝑥 = (𝑦 ∩ 𝐵) ∧ 𝑧 ∈ 𝑥) ↔ (∃𝑦 ∈ 𝐴 𝑧 ∈ 𝑦 ∧ 𝑧 ∈ 𝐵))
19	9, 18	bitri 184	. . . 4 ⊢ (∃𝑥(𝑧 ∈ 𝑥 ∧ ∃𝑦 ∈ 𝐴 𝑥 = (𝑦 ∩ 𝐵)) ↔ (∃𝑦 ∈ 𝐴 𝑧 ∈ 𝑦 ∧ 𝑧 ∈ 𝐵))
20	2, 3, 19	3bitr4i 212	. . 3 ⊢ (𝑧 ∈ (∪ 𝐴 ∩ 𝐵) ↔ ∃𝑥(𝑧 ∈ 𝑥 ∧ ∃𝑦 ∈ 𝐴 𝑥 = (𝑦 ∩ 𝐵)))
21		eluniab 3817	. . 3 ⊢ (𝑧 ∈ ∪ {𝑥 ∣ ∃𝑦 ∈ 𝐴 𝑥 = (𝑦 ∩ 𝐵)} ↔ ∃𝑥(𝑧 ∈ 𝑥 ∧ ∃𝑦 ∈ 𝐴 𝑥 = (𝑦 ∩ 𝐵)))
22	20, 21	bitr4i 187	. 2 ⊢ (𝑧 ∈ (∪ 𝐴 ∩ 𝐵) ↔ 𝑧 ∈ ∪ {𝑥 ∣ ∃𝑦 ∈ 𝐴 𝑥 = (𝑦 ∩ 𝐵)})
23	22	eqriv 2172	1 ⊢ (∪ 𝐴 ∩ 𝐵) = ∪ {𝑥 ∣ ∃𝑦 ∈ 𝐴 𝑥 = (𝑦 ∩ 𝐵)}

Syntax hints:   ∧ wa 104   = wceq 1353  ∃wex 1490   ∈ wcel 2146  {cab 2161  ∃wrex 2454   ∩ cin 3126  ∪ cuni 3805

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-ext 2157  ax-sep 4116

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1459  df-sb 1761  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-rex 2459  df-v 2737  df-in 3133  df-uni 3806

This theorem is referenced by: (None)