Theorem bj-restuni 37092

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 23056 and restuni2 23061. (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 4877	. . 3 ⊢ (𝑥 ∈ ∪ (𝑋 ↾t 𝐴) ↔ ∃𝑦(𝑥 ∈ 𝑦 ∧ 𝑦 ∈ (𝑋 ↾t 𝐴)))
2		elrest 17397	. . . . . 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 1921	. . . 4 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) → (∃𝑦(𝑥 ∈ 𝑦 ∧ 𝑦 ∈ (𝑋 ↾t 𝐴)) ↔ ∃𝑦(𝑥 ∈ 𝑦 ∧ ∃𝑧 ∈ 𝑋 𝑦 = (𝑧 ∩ 𝐴))))
5		eluni 4877	. . . . . . . 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 3055	. . . . . . . . 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 1953	. . . . . . . . 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 1848	. . . . . 6 ⊢ (∃𝑦(𝑥 ∈ 𝑦 ∧ ∃𝑧 ∈ 𝑋 𝑦 = (𝑧 ∩ 𝐴)) ↔ ∃𝑦∃𝑧(𝑥 ∈ 𝑦 ∧ (𝑧 ∈ 𝑋 ∧ 𝑦 = (𝑧 ∩ 𝐴))))
15		excom 2163	. . . . . 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 1848	. . . . . . . . 9 ⊢ (∃𝑦(𝑥 ∈ 𝑦 ∧ (𝑧 ∈ 𝑋 ∧ 𝑦 = (𝑧 ∩ 𝐴))) ↔ ∃𝑦(𝑧 ∈ 𝑋 ∧ (𝑥 ∈ 𝑦 ∧ 𝑦 = (𝑧 ∩ 𝐴))))
18		19.42v 1953	. . . . . . . . 9 ⊢ (∃𝑦(𝑧 ∈ 𝑋 ∧ (𝑥 ∈ 𝑦 ∧ 𝑦 = (𝑧 ∩ 𝐴))) ↔ (𝑧 ∈ 𝑋 ∧ ∃𝑦(𝑥 ∈ 𝑦 ∧ 𝑦 = (𝑧 ∩ 𝐴))))
19		eqimss 4008	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = (𝑧 ∩ 𝐴) → 𝑦 ⊆ (𝑧 ∩ 𝐴))
20	19	sseld 3948	. . . . . . . . . . . . . 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 4009	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = (𝑧 ∩ 𝐴) → (𝑧 ∩ 𝐴) ⊆ 𝑦)
23	22	sseld 3948	. . . . . . . . . . . . . 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 1848	. . . . . . . . . . 11 ⊢ (∃𝑦(𝑥 ∈ 𝑦 ∧ 𝑦 = (𝑧 ∩ 𝐴)) ↔ ∃𝑦(𝑥 ∈ (𝑧 ∩ 𝐴) ∧ 𝑦 = (𝑧 ∩ 𝐴)))
27		19.42v 1953	. . . . . . . . . . 11 ⊢ (∃𝑦(𝑥 ∈ (𝑧 ∩ 𝐴) ∧ 𝑦 = (𝑧 ∩ 𝐴)) ↔ (𝑥 ∈ (𝑧 ∩ 𝐴) ∧ ∃𝑦 𝑦 = (𝑧 ∩ 𝐴)))
28		vex 3454	. . . . . . . . . . . . . . . 16 ⊢ 𝑧 ∈ V
29	28	inex1 5275	. . . . . . . . . . . . . . 15 ⊢ (𝑧 ∩ 𝐴) ∈ V
30	29	isseti 3468	. . . . . . . . . . . . . 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 3933	. . . . . . . . . . . 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 1848	. . . . . . 7 ⊢ (∃𝑧∃𝑦(𝑥 ∈ 𝑦 ∧ (𝑧 ∈ 𝑋 ∧ 𝑦 = (𝑧 ∩ 𝐴))) ↔ ∃𝑧((𝑥 ∈ 𝑧 ∧ 𝑧 ∈ 𝑋) ∧ 𝑥 ∈ 𝐴))
39		19.41v 1949	. . . . . . 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 3933	. . . . 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 2728	1 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) → ∪ (𝑋 ↾t 𝐴) = (∪ 𝑋 ∩ 𝐴))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540  ∃wex 1779   ∈ wcel 2109  ∃wrex 3054   ∩ cin 3916  ∪ cuni 4874  (class class class)co 7390   ↾_t crest 17390

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-rep 5237  ax-sep 5254  ax-nul 5264  ax-pr 5390  ax-un 7714

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-id 5536  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-ov 7393  df-oprab 7394  df-mpo 7395  df-rest 17392

This theorem is referenced by:  bj-restuni2  37093