Theorem bj-restuni 37130

Description: The union of an elementwise intersection by a set is equal to the intersection with that set of the union of the family. See also restuni 23075 and restuni2 23080. (Contributed by BJ, 27-Apr-2021.)

Assertion

Ref	Expression
bj-restuni	⊢ ((𝑋 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) → ∪ (𝑋 ↾t 𝐴) = (∪ 𝑋 ∩ 𝐴))

Proof of Theorem bj-restuni

Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eluni 4862	. . 3 ⊢ (𝑥 ∈ ∪ (𝑋 ↾t 𝐴) ↔ ∃𝑦(𝑥 ∈ 𝑦 ∧ 𝑦 ∈ (𝑋 ↾t 𝐴)))
2		elrest 17328	. . . . . 6 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) → (𝑦 ∈ (𝑋 ↾t 𝐴) ↔ ∃𝑧 ∈ 𝑋 𝑦 = (𝑧 ∩ 𝐴)))
3	2	anbi2d 630	. . . . 5 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) → ((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ (𝑋 ↾t 𝐴)) ↔ (𝑥 ∈ 𝑦 ∧ ∃𝑧 ∈ 𝑋 𝑦 = (𝑧 ∩ 𝐴))))
4	3	exbidv 1922	. . . 4 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) → (∃𝑦(𝑥 ∈ 𝑦 ∧ 𝑦 ∈ (𝑋 ↾t 𝐴)) ↔ ∃𝑦(𝑥 ∈ 𝑦 ∧ ∃𝑧 ∈ 𝑋 𝑦 = (𝑧 ∩ 𝐴))))
5		eluni 4862	. . . . . . . 8 ⊢ (𝑥 ∈ ∪ 𝑋 ↔ ∃𝑧(𝑥 ∈ 𝑧 ∧ 𝑧 ∈ 𝑋))
6	5	bicomi 224	. . . . . . 7 ⊢ (∃𝑧(𝑥 ∈ 𝑧 ∧ 𝑧 ∈ 𝑋) ↔ 𝑥 ∈ ∪ 𝑋)
7	6	anbi1i 624	. . . . . 6 ⊢ ((∃𝑧(𝑥 ∈ 𝑧 ∧ 𝑧 ∈ 𝑋) ∧ 𝑥 ∈ 𝐴) ↔ (𝑥 ∈ ∪ 𝑋 ∧ 𝑥 ∈ 𝐴))
8	7	a1i 11	. . . . 5 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) → ((∃𝑧(𝑥 ∈ 𝑧 ∧ 𝑧 ∈ 𝑋) ∧ 𝑥 ∈ 𝐴) ↔ (𝑥 ∈ ∪ 𝑋 ∧ 𝑥 ∈ 𝐴)))
9		df-rex 3057	. . . . . . . . 9 ⊢ (∃𝑧 ∈ 𝑋 𝑦 = (𝑧 ∩ 𝐴) ↔ ∃𝑧(𝑧 ∈ 𝑋 ∧ 𝑦 = (𝑧 ∩ 𝐴)))
10	9	anbi2i 623	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝑦 ∧ ∃𝑧 ∈ 𝑋 𝑦 = (𝑧 ∩ 𝐴)) ↔ (𝑥 ∈ 𝑦 ∧ ∃𝑧(𝑧 ∈ 𝑋 ∧ 𝑦 = (𝑧 ∩ 𝐴))))
11		19.42v 1954	. . . . . . . . 9 ⊢ (∃𝑧(𝑥 ∈ 𝑦 ∧ (𝑧 ∈ 𝑋 ∧ 𝑦 = (𝑧 ∩ 𝐴))) ↔ (𝑥 ∈ 𝑦 ∧ ∃𝑧(𝑧 ∈ 𝑋 ∧ 𝑦 = (𝑧 ∩ 𝐴))))
12	11	bicomi 224	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝑦 ∧ ∃𝑧(𝑧 ∈ 𝑋 ∧ 𝑦 = (𝑧 ∩ 𝐴))) ↔ ∃𝑧(𝑥 ∈ 𝑦 ∧ (𝑧 ∈ 𝑋 ∧ 𝑦 = (𝑧 ∩ 𝐴))))
13	10, 12	bitri 275	. . . . . . 7 ⊢ ((𝑥 ∈ 𝑦 ∧ ∃𝑧 ∈ 𝑋 𝑦 = (𝑧 ∩ 𝐴)) ↔ ∃𝑧(𝑥 ∈ 𝑦 ∧ (𝑧 ∈ 𝑋 ∧ 𝑦 = (𝑧 ∩ 𝐴))))
14	13	exbii 1849	. . . . . 6 ⊢ (∃𝑦(𝑥 ∈ 𝑦 ∧ ∃𝑧 ∈ 𝑋 𝑦 = (𝑧 ∩ 𝐴)) ↔ ∃𝑦∃𝑧(𝑥 ∈ 𝑦 ∧ (𝑧 ∈ 𝑋 ∧ 𝑦 = (𝑧 ∩ 𝐴))))
15		excom 2165	. . . . . 6 ⊢ (∃𝑦∃𝑧(𝑥 ∈ 𝑦 ∧ (𝑧 ∈ 𝑋 ∧ 𝑦 = (𝑧 ∩ 𝐴))) ↔ ∃𝑧∃𝑦(𝑥 ∈ 𝑦 ∧ (𝑧 ∈ 𝑋 ∧ 𝑦 = (𝑧 ∩ 𝐴))))
16		an12 645	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝑦 ∧ (𝑧 ∈ 𝑋 ∧ 𝑦 = (𝑧 ∩ 𝐴))) ↔ (𝑧 ∈ 𝑋 ∧ (𝑥 ∈ 𝑦 ∧ 𝑦 = (𝑧 ∩ 𝐴))))
17	16	exbii 1849	. . . . . . . . 9 ⊢ (∃𝑦(𝑥 ∈ 𝑦 ∧ (𝑧 ∈ 𝑋 ∧ 𝑦 = (𝑧 ∩ 𝐴))) ↔ ∃𝑦(𝑧 ∈ 𝑋 ∧ (𝑥 ∈ 𝑦 ∧ 𝑦 = (𝑧 ∩ 𝐴))))
18		19.42v 1954	. . . . . . . . 9 ⊢ (∃𝑦(𝑧 ∈ 𝑋 ∧ (𝑥 ∈ 𝑦 ∧ 𝑦 = (𝑧 ∩ 𝐴))) ↔ (𝑧 ∈ 𝑋 ∧ ∃𝑦(𝑥 ∈ 𝑦 ∧ 𝑦 = (𝑧 ∩ 𝐴))))
19		eqimss 3993	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = (𝑧 ∩ 𝐴) → 𝑦 ⊆ (𝑧 ∩ 𝐴))
20	19	sseld 3933	. . . . . . . . . . . . . 14 ⊢ (𝑦 = (𝑧 ∩ 𝐴) → (𝑥 ∈ 𝑦 → 𝑥 ∈ (𝑧 ∩ 𝐴)))
21	20	imdistanri 569	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ 𝑦 ∧ 𝑦 = (𝑧 ∩ 𝐴)) → (𝑥 ∈ (𝑧 ∩ 𝐴) ∧ 𝑦 = (𝑧 ∩ 𝐴)))
22		eqimss2 3994	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = (𝑧 ∩ 𝐴) → (𝑧 ∩ 𝐴) ⊆ 𝑦)
23	22	sseld 3933	. . . . . . . . . . . . . 14 ⊢ (𝑦 = (𝑧 ∩ 𝐴) → (𝑥 ∈ (𝑧 ∩ 𝐴) → 𝑥 ∈ 𝑦))
24	23	imdistanri 569	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ (𝑧 ∩ 𝐴) ∧ 𝑦 = (𝑧 ∩ 𝐴)) → (𝑥 ∈ 𝑦 ∧ 𝑦 = (𝑧 ∩ 𝐴)))
25	21, 24	impbii 209	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ 𝑦 ∧ 𝑦 = (𝑧 ∩ 𝐴)) ↔ (𝑥 ∈ (𝑧 ∩ 𝐴) ∧ 𝑦 = (𝑧 ∩ 𝐴)))
26	25	exbii 1849	. . . . . . . . . . 11 ⊢ (∃𝑦(𝑥 ∈ 𝑦 ∧ 𝑦 = (𝑧 ∩ 𝐴)) ↔ ∃𝑦(𝑥 ∈ (𝑧 ∩ 𝐴) ∧ 𝑦 = (𝑧 ∩ 𝐴)))
27		19.42v 1954	. . . . . . . . . . 11 ⊢ (∃𝑦(𝑥 ∈ (𝑧 ∩ 𝐴) ∧ 𝑦 = (𝑧 ∩ 𝐴)) ↔ (𝑥 ∈ (𝑧 ∩ 𝐴) ∧ ∃𝑦 𝑦 = (𝑧 ∩ 𝐴)))
28		vex 3440	. . . . . . . . . . . . . . . 16 ⊢ 𝑧 ∈ V
29	28	inex1 5255	. . . . . . . . . . . . . . 15 ⊢ (𝑧 ∩ 𝐴) ∈ V
30	29	isseti 3454	. . . . . . . . . . . . . 14 ⊢ ∃𝑦 𝑦 = (𝑧 ∩ 𝐴)
31	30	biantru 529	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ (𝑧 ∩ 𝐴) ↔ (𝑥 ∈ (𝑧 ∩ 𝐴) ∧ ∃𝑦 𝑦 = (𝑧 ∩ 𝐴)))
32	31	bicomi 224	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ (𝑧 ∩ 𝐴) ∧ ∃𝑦 𝑦 = (𝑧 ∩ 𝐴)) ↔ 𝑥 ∈ (𝑧 ∩ 𝐴))
33		elin 3918	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ (𝑧 ∩ 𝐴) ↔ (𝑥 ∈ 𝑧 ∧ 𝑥 ∈ 𝐴))
34	32, 33	bitri 275	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ (𝑧 ∩ 𝐴) ∧ ∃𝑦 𝑦 = (𝑧 ∩ 𝐴)) ↔ (𝑥 ∈ 𝑧 ∧ 𝑥 ∈ 𝐴))
35	26, 27, 34	3bitri 297	. . . . . . . . . 10 ⊢ (∃𝑦(𝑥 ∈ 𝑦 ∧ 𝑦 = (𝑧 ∩ 𝐴)) ↔ (𝑥 ∈ 𝑧 ∧ 𝑥 ∈ 𝐴))
36	35	bianassc 643	. . . . . . . . 9 ⊢ ((𝑧 ∈ 𝑋 ∧ ∃𝑦(𝑥 ∈ 𝑦 ∧ 𝑦 = (𝑧 ∩ 𝐴))) ↔ ((𝑥 ∈ 𝑧 ∧ 𝑧 ∈ 𝑋) ∧ 𝑥 ∈ 𝐴))
37	17, 18, 36	3bitri 297	. . . . . . . 8 ⊢ (∃𝑦(𝑥 ∈ 𝑦 ∧ (𝑧 ∈ 𝑋 ∧ 𝑦 = (𝑧 ∩ 𝐴))) ↔ ((𝑥 ∈ 𝑧 ∧ 𝑧 ∈ 𝑋) ∧ 𝑥 ∈ 𝐴))
38	37	exbii 1849	. . . . . . 7 ⊢ (∃𝑧∃𝑦(𝑥 ∈ 𝑦 ∧ (𝑧 ∈ 𝑋 ∧ 𝑦 = (𝑧 ∩ 𝐴))) ↔ ∃𝑧((𝑥 ∈ 𝑧 ∧ 𝑧 ∈ 𝑋) ∧ 𝑥 ∈ 𝐴))
39		19.41v 1950	. . . . . . 7 ⊢ (∃𝑧((𝑥 ∈ 𝑧 ∧ 𝑧 ∈ 𝑋) ∧ 𝑥 ∈ 𝐴) ↔ (∃𝑧(𝑥 ∈ 𝑧 ∧ 𝑧 ∈ 𝑋) ∧ 𝑥 ∈ 𝐴))
40	38, 39	bitri 275	. . . . . 6 ⊢ (∃𝑧∃𝑦(𝑥 ∈ 𝑦 ∧ (𝑧 ∈ 𝑋 ∧ 𝑦 = (𝑧 ∩ 𝐴))) ↔ (∃𝑧(𝑥 ∈ 𝑧 ∧ 𝑧 ∈ 𝑋) ∧ 𝑥 ∈ 𝐴))
41	14, 15, 40	3bitri 297	. . . . 5 ⊢ (∃𝑦(𝑥 ∈ 𝑦 ∧ ∃𝑧 ∈ 𝑋 𝑦 = (𝑧 ∩ 𝐴)) ↔ (∃𝑧(𝑥 ∈ 𝑧 ∧ 𝑧 ∈ 𝑋) ∧ 𝑥 ∈ 𝐴))
42		elin 3918	. . . . 5 ⊢ (𝑥 ∈ (∪ 𝑋 ∩ 𝐴) ↔ (𝑥 ∈ ∪ 𝑋 ∧ 𝑥 ∈ 𝐴))
43	8, 41, 42	3bitr4g 314	. . . 4 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) → (∃𝑦(𝑥 ∈ 𝑦 ∧ ∃𝑧 ∈ 𝑋 𝑦 = (𝑧 ∩ 𝐴)) ↔ 𝑥 ∈ (∪ 𝑋 ∩ 𝐴)))
44	4, 43	bitrd 279	. . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) → (∃𝑦(𝑥 ∈ 𝑦 ∧ 𝑦 ∈ (𝑋 ↾t 𝐴)) ↔ 𝑥 ∈ (∪ 𝑋 ∩ 𝐴)))
45	1, 44	bitrid 283	. 2 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) → (𝑥 ∈ ∪ (𝑋 ↾t 𝐴) ↔ 𝑥 ∈ (∪ 𝑋 ∩ 𝐴)))
46	45	eqrdv 2729	1 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) → ∪ (𝑋 ↾t 𝐴) = (∪ 𝑋 ∩ 𝐴))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1541  ∃wex 1780   ∈ wcel 2111  ∃wrex 3056   ∩ cin 3901  ∪ cuni 4859  (class class class)co 7346   ↾_t crest 17321

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-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5217  ax-sep 5234  ax-nul 5244  ax-pr 5370  ax-un 7668

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4476  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-iun 4943  df-br 5092  df-opab 5154  df-mpt 5173  df-id 5511  df-xp 5622  df-rel 5623  df-cnv 5624  df-co 5625  df-dm 5626  df-rn 5627  df-res 5628  df-ima 5629  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-ov 7349  df-oprab 7350  df-mpo 7351  df-rest 17323

This theorem is referenced by:  bj-restuni2  37131