Theorem bj-restuni 37855

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 23393 and restuni2 23398. (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 4873	. . 3 ⊢ (𝑥 ∈ ∪ (𝑋 ↾t 𝐴) ↔ ∃𝑦(𝑥 ∈ 𝑦 ∧ 𝑦 ∈ (𝑋 ↾t 𝐴)))
2		elrest 17518	. . . . . 6 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) → (𝑦 ∈ (𝑋 ↾t 𝐴) ↔ ∃𝑧 ∈ 𝑋 𝑦 = (𝑧 ∩ 𝐴)))
3	2	anbi2d 642	. . . . 5 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) → ((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ (𝑋 ↾t 𝐴)) ↔ (𝑥 ∈ 𝑦 ∧ ∃𝑧 ∈ 𝑋 𝑦 = (𝑧 ∩ 𝐴))))
4	3	exbidv 1954	. . . 4 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) → (∃𝑦(𝑥 ∈ 𝑦 ∧ 𝑦 ∈ (𝑋 ↾t 𝐴)) ↔ ∃𝑦(𝑥 ∈ 𝑦 ∧ ∃𝑧 ∈ 𝑋 𝑦 = (𝑧 ∩ 𝐴))))
5		eluni 4873	. . . . . . . 8 ⊢ (𝑥 ∈ ∪ 𝑋 ↔ ∃𝑧(𝑥 ∈ 𝑧 ∧ 𝑧 ∈ 𝑋))
6	5	bicomi 227	. . . . . . 7 ⊢ (∃𝑧(𝑥 ∈ 𝑧 ∧ 𝑧 ∈ 𝑋) ↔ 𝑥 ∈ ∪ 𝑋)
7	6	anbi1i 636	. . . . . 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 3089	. . . . . . . . 9 ⊢ (∃𝑧 ∈ 𝑋 𝑦 = (𝑧 ∩ 𝐴) ↔ ∃𝑧(𝑧 ∈ 𝑋 ∧ 𝑦 = (𝑧 ∩ 𝐴)))
10	9	anbi2i 635	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝑦 ∧ ∃𝑧 ∈ 𝑋 𝑦 = (𝑧 ∩ 𝐴)) ↔ (𝑥 ∈ 𝑦 ∧ ∃𝑧(𝑧 ∈ 𝑋 ∧ 𝑦 = (𝑧 ∩ 𝐴))))
11		19.42v 1986	. . . . . . . . 9 ⊢ (∃𝑧(𝑥 ∈ 𝑦 ∧ (𝑧 ∈ 𝑋 ∧ 𝑦 = (𝑧 ∩ 𝐴))) ↔ (𝑥 ∈ 𝑦 ∧ ∃𝑧(𝑧 ∈ 𝑋 ∧ 𝑦 = (𝑧 ∩ 𝐴))))
12	11	bicomi 227	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝑦 ∧ ∃𝑧(𝑧 ∈ 𝑋 ∧ 𝑦 = (𝑧 ∩ 𝐴))) ↔ ∃𝑧(𝑥 ∈ 𝑦 ∧ (𝑧 ∈ 𝑋 ∧ 𝑦 = (𝑧 ∩ 𝐴))))
13	10, 12	bitri 278	. . . . . . 7 ⊢ ((𝑥 ∈ 𝑦 ∧ ∃𝑧 ∈ 𝑋 𝑦 = (𝑧 ∩ 𝐴)) ↔ ∃𝑧(𝑥 ∈ 𝑦 ∧ (𝑧 ∈ 𝑋 ∧ 𝑦 = (𝑧 ∩ 𝐴))))
14	13	exbii 1881	. . . . . 6 ⊢ (∃𝑦(𝑥 ∈ 𝑦 ∧ ∃𝑧 ∈ 𝑋 𝑦 = (𝑧 ∩ 𝐴)) ↔ ∃𝑦∃𝑧(𝑥 ∈ 𝑦 ∧ (𝑧 ∈ 𝑋 ∧ 𝑦 = (𝑧 ∩ 𝐴))))
15		excom 2199	. . . . . 6 ⊢ (∃𝑦∃𝑧(𝑥 ∈ 𝑦 ∧ (𝑧 ∈ 𝑋 ∧ 𝑦 = (𝑧 ∩ 𝐴))) ↔ ∃𝑧∃𝑦(𝑥 ∈ 𝑦 ∧ (𝑧 ∈ 𝑋 ∧ 𝑦 = (𝑧 ∩ 𝐴))))
16		an12 658	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝑦 ∧ (𝑧 ∈ 𝑋 ∧ 𝑦 = (𝑧 ∩ 𝐴))) ↔ (𝑧 ∈ 𝑋 ∧ (𝑥 ∈ 𝑦 ∧ 𝑦 = (𝑧 ∩ 𝐴))))
17	16	exbii 1881	. . . . . . . . 9 ⊢ (∃𝑦(𝑥 ∈ 𝑦 ∧ (𝑧 ∈ 𝑋 ∧ 𝑦 = (𝑧 ∩ 𝐴))) ↔ ∃𝑦(𝑧 ∈ 𝑋 ∧ (𝑥 ∈ 𝑦 ∧ 𝑦 = (𝑧 ∩ 𝐴))))
18		19.42v 1986	. . . . . . . . 9 ⊢ (∃𝑦(𝑧 ∈ 𝑋 ∧ (𝑥 ∈ 𝑦 ∧ 𝑦 = (𝑧 ∩ 𝐴))) ↔ (𝑧 ∈ 𝑋 ∧ ∃𝑦(𝑥 ∈ 𝑦 ∧ 𝑦 = (𝑧 ∩ 𝐴))))
19		eqimss 3992	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = (𝑧 ∩ 𝐴) → 𝑦 ⊆ (𝑧 ∩ 𝐴))
20	19	sseld 3933	. . . . . . . . . . . . . 14 ⊢ (𝑦 = (𝑧 ∩ 𝐴) → (𝑥 ∈ 𝑦 → 𝑥 ∈ (𝑧 ∩ 𝐴)))
21	20	imdistanri 580	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ 𝑦 ∧ 𝑦 = (𝑧 ∩ 𝐴)) → (𝑥 ∈ (𝑧 ∩ 𝐴) ∧ 𝑦 = (𝑧 ∩ 𝐴)))
22		eqimss2 3993	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = (𝑧 ∩ 𝐴) → (𝑧 ∩ 𝐴) ⊆ 𝑦)
23	22	sseld 3933	. . . . . . . . . . . . . 14 ⊢ (𝑦 = (𝑧 ∩ 𝐴) → (𝑥 ∈ (𝑧 ∩ 𝐴) → 𝑥 ∈ 𝑦))
24	23	imdistanri 580	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ (𝑧 ∩ 𝐴) ∧ 𝑦 = (𝑧 ∩ 𝐴)) → (𝑥 ∈ 𝑦 ∧ 𝑦 = (𝑧 ∩ 𝐴)))
25	21, 24	impbii 212	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ 𝑦 ∧ 𝑦 = (𝑧 ∩ 𝐴)) ↔ (𝑥 ∈ (𝑧 ∩ 𝐴) ∧ 𝑦 = (𝑧 ∩ 𝐴)))
26	25	exbii 1881	. . . . . . . . . . 11 ⊢ (∃𝑦(𝑥 ∈ 𝑦 ∧ 𝑦 = (𝑧 ∩ 𝐴)) ↔ ∃𝑦(𝑥 ∈ (𝑧 ∩ 𝐴) ∧ 𝑦 = (𝑧 ∩ 𝐴)))
27		19.42v 1986	. . . . . . . . . . 11 ⊢ (∃𝑦(𝑥 ∈ (𝑧 ∩ 𝐴) ∧ 𝑦 = (𝑧 ∩ 𝐴)) ↔ (𝑥 ∈ (𝑧 ∩ 𝐴) ∧ ∃𝑦 𝑦 = (𝑧 ∩ 𝐴)))
28		vex 3457	. . . . . . . . . . . . . . . 16 ⊢ 𝑧 ∈ V
29	28	inex1 5284	. . . . . . . . . . . . . . 15 ⊢ (𝑧 ∩ 𝐴) ∈ V
30	29	isseti 3471	. . . . . . . . . . . . . 14 ⊢ ∃𝑦 𝑦 = (𝑧 ∩ 𝐴)
31	30	biantru 539	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ (𝑧 ∩ 𝐴) ↔ (𝑥 ∈ (𝑧 ∩ 𝐴) ∧ ∃𝑦 𝑦 = (𝑧 ∩ 𝐴)))
32	31	bicomi 227	. . . . . . . . . . . 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 278	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ (𝑧 ∩ 𝐴) ∧ ∃𝑦 𝑦 = (𝑧 ∩ 𝐴)) ↔ (𝑥 ∈ 𝑧 ∧ 𝑥 ∈ 𝐴))
35	26, 27, 34	3bitri 300	. . . . . . . . . 10 ⊢ (∃𝑦(𝑥 ∈ 𝑦 ∧ 𝑦 = (𝑧 ∩ 𝐴)) ↔ (𝑥 ∈ 𝑧 ∧ 𝑥 ∈ 𝐴))
36	35	bianassc 656	. . . . . . . . 9 ⊢ ((𝑧 ∈ 𝑋 ∧ ∃𝑦(𝑥 ∈ 𝑦 ∧ 𝑦 = (𝑧 ∩ 𝐴))) ↔ ((𝑥 ∈ 𝑧 ∧ 𝑧 ∈ 𝑋) ∧ 𝑥 ∈ 𝐴))
37	17, 18, 36	3bitri 300	. . . . . . . 8 ⊢ (∃𝑦(𝑥 ∈ 𝑦 ∧ (𝑧 ∈ 𝑋 ∧ 𝑦 = (𝑧 ∩ 𝐴))) ↔ ((𝑥 ∈ 𝑧 ∧ 𝑧 ∈ 𝑋) ∧ 𝑥 ∈ 𝐴))
38	37	exbii 1881	. . . . . . 7 ⊢ (∃𝑧∃𝑦(𝑥 ∈ 𝑦 ∧ (𝑧 ∈ 𝑋 ∧ 𝑦 = (𝑧 ∩ 𝐴))) ↔ ∃𝑧((𝑥 ∈ 𝑧 ∧ 𝑧 ∈ 𝑋) ∧ 𝑥 ∈ 𝐴))
39		19.41v 1982	. . . . . . 7 ⊢ (∃𝑧((𝑥 ∈ 𝑧 ∧ 𝑧 ∈ 𝑋) ∧ 𝑥 ∈ 𝐴) ↔ (∃𝑧(𝑥 ∈ 𝑧 ∧ 𝑧 ∈ 𝑋) ∧ 𝑥 ∈ 𝐴))
40	38, 39	bitri 278	. . . . . 6 ⊢ (∃𝑧∃𝑦(𝑥 ∈ 𝑦 ∧ (𝑧 ∈ 𝑋 ∧ 𝑦 = (𝑧 ∩ 𝐴))) ↔ (∃𝑧(𝑥 ∈ 𝑧 ∧ 𝑧 ∈ 𝑋) ∧ 𝑥 ∈ 𝐴))
41	14, 15, 40	3bitri 300	. . . . 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 317	. . . 4 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) → (∃𝑦(𝑥 ∈ 𝑦 ∧ ∃𝑧 ∈ 𝑋 𝑦 = (𝑧 ∩ 𝐴)) ↔ 𝑥 ∈ (∪ 𝑋 ∩ 𝐴)))
44	4, 43	bitrd 282	. . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) → (∃𝑦(𝑥 ∈ 𝑦 ∧ 𝑦 ∈ (𝑋 ↾t 𝐴)) ↔ 𝑥 ∈ (∪ 𝑋 ∩ 𝐴)))
45	1, 44	bitrid 286	. 2 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) → (𝑥 ∈ ∪ (𝑋 ↾t 𝐴) ↔ 𝑥 ∈ (∪ 𝑋 ∩ 𝐴)))
46	45	eqrdv 2760	1 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) → ∪ (𝑋 ↾t 𝐴) = (∪ 𝑋 ∩ 𝐴))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570  ∃wex 1812   ∈ wcel 2145  ∃wrex 3088   ∩ cin 3901  ∪ cuni 4870  (class class class)co 7417   ↾_t crest 17511

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-rep 5236  ax-sep 5255  ax-nul 5267  ax-pr 5402  ax-un 7740

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-iun 4956  df-br 5108  df-opab 5172  df-mpt 5191  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7420  df-oprab 7421  df-mpo 7422  df-rest 17513

This theorem is used by:  bj-restuni2  37856