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Theorem bj-restuni 36281
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 22886 and restuni2 22891. (Contributed by BJ, 27-Apr-2021.)
Assertion
Ref Expression
bj-restuni ((𝑋𝑉𝐴𝑊) → (𝑋t 𝐴) = ( 𝑋𝐴))

Proof of Theorem bj-restuni
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eluni 4910 . . 3 (𝑥 (𝑋t 𝐴) ↔ ∃𝑦(𝑥𝑦𝑦 ∈ (𝑋t 𝐴)))
2 elrest 17377 . . . . . 6 ((𝑋𝑉𝐴𝑊) → (𝑦 ∈ (𝑋t 𝐴) ↔ ∃𝑧𝑋 𝑦 = (𝑧𝐴)))
32anbi2d 627 . . . . 5 ((𝑋𝑉𝐴𝑊) → ((𝑥𝑦𝑦 ∈ (𝑋t 𝐴)) ↔ (𝑥𝑦 ∧ ∃𝑧𝑋 𝑦 = (𝑧𝐴))))
43exbidv 1922 . . . 4 ((𝑋𝑉𝐴𝑊) → (∃𝑦(𝑥𝑦𝑦 ∈ (𝑋t 𝐴)) ↔ ∃𝑦(𝑥𝑦 ∧ ∃𝑧𝑋 𝑦 = (𝑧𝐴))))
5 eluni 4910 . . . . . . . 8 (𝑥 𝑋 ↔ ∃𝑧(𝑥𝑧𝑧𝑋))
65bicomi 223 . . . . . . 7 (∃𝑧(𝑥𝑧𝑧𝑋) ↔ 𝑥 𝑋)
76anbi1i 622 . . . . . 6 ((∃𝑧(𝑥𝑧𝑧𝑋) ∧ 𝑥𝐴) ↔ (𝑥 𝑋𝑥𝐴))
87a1i 11 . . . . 5 ((𝑋𝑉𝐴𝑊) → ((∃𝑧(𝑥𝑧𝑧𝑋) ∧ 𝑥𝐴) ↔ (𝑥 𝑋𝑥𝐴)))
9 df-rex 3069 . . . . . . . . 9 (∃𝑧𝑋 𝑦 = (𝑧𝐴) ↔ ∃𝑧(𝑧𝑋𝑦 = (𝑧𝐴)))
109anbi2i 621 . . . . . . . 8 ((𝑥𝑦 ∧ ∃𝑧𝑋 𝑦 = (𝑧𝐴)) ↔ (𝑥𝑦 ∧ ∃𝑧(𝑧𝑋𝑦 = (𝑧𝐴))))
11 19.42v 1955 . . . . . . . . 9 (∃𝑧(𝑥𝑦 ∧ (𝑧𝑋𝑦 = (𝑧𝐴))) ↔ (𝑥𝑦 ∧ ∃𝑧(𝑧𝑋𝑦 = (𝑧𝐴))))
1211bicomi 223 . . . . . . . 8 ((𝑥𝑦 ∧ ∃𝑧(𝑧𝑋𝑦 = (𝑧𝐴))) ↔ ∃𝑧(𝑥𝑦 ∧ (𝑧𝑋𝑦 = (𝑧𝐴))))
1310, 12bitri 274 . . . . . . 7 ((𝑥𝑦 ∧ ∃𝑧𝑋 𝑦 = (𝑧𝐴)) ↔ ∃𝑧(𝑥𝑦 ∧ (𝑧𝑋𝑦 = (𝑧𝐴))))
1413exbii 1848 . . . . . 6 (∃𝑦(𝑥𝑦 ∧ ∃𝑧𝑋 𝑦 = (𝑧𝐴)) ↔ ∃𝑦𝑧(𝑥𝑦 ∧ (𝑧𝑋𝑦 = (𝑧𝐴))))
15 excom 2160 . . . . . 6 (∃𝑦𝑧(𝑥𝑦 ∧ (𝑧𝑋𝑦 = (𝑧𝐴))) ↔ ∃𝑧𝑦(𝑥𝑦 ∧ (𝑧𝑋𝑦 = (𝑧𝐴))))
16 an12 641 . . . . . . . . . 10 ((𝑥𝑦 ∧ (𝑧𝑋𝑦 = (𝑧𝐴))) ↔ (𝑧𝑋 ∧ (𝑥𝑦𝑦 = (𝑧𝐴))))
1716exbii 1848 . . . . . . . . 9 (∃𝑦(𝑥𝑦 ∧ (𝑧𝑋𝑦 = (𝑧𝐴))) ↔ ∃𝑦(𝑧𝑋 ∧ (𝑥𝑦𝑦 = (𝑧𝐴))))
18 19.42v 1955 . . . . . . . . 9 (∃𝑦(𝑧𝑋 ∧ (𝑥𝑦𝑦 = (𝑧𝐴))) ↔ (𝑧𝑋 ∧ ∃𝑦(𝑥𝑦𝑦 = (𝑧𝐴))))
19 eqimss 4039 . . . . . . . . . . . . . . 15 (𝑦 = (𝑧𝐴) → 𝑦 ⊆ (𝑧𝐴))
2019sseld 3980 . . . . . . . . . . . . . 14 (𝑦 = (𝑧𝐴) → (𝑥𝑦𝑥 ∈ (𝑧𝐴)))
2120imdistanri 568 . . . . . . . . . . . . 13 ((𝑥𝑦𝑦 = (𝑧𝐴)) → (𝑥 ∈ (𝑧𝐴) ∧ 𝑦 = (𝑧𝐴)))
22 eqimss2 4040 . . . . . . . . . . . . . . 15 (𝑦 = (𝑧𝐴) → (𝑧𝐴) ⊆ 𝑦)
2322sseld 3980 . . . . . . . . . . . . . 14 (𝑦 = (𝑧𝐴) → (𝑥 ∈ (𝑧𝐴) → 𝑥𝑦))
2423imdistanri 568 . . . . . . . . . . . . 13 ((𝑥 ∈ (𝑧𝐴) ∧ 𝑦 = (𝑧𝐴)) → (𝑥𝑦𝑦 = (𝑧𝐴)))
2521, 24impbii 208 . . . . . . . . . . . 12 ((𝑥𝑦𝑦 = (𝑧𝐴)) ↔ (𝑥 ∈ (𝑧𝐴) ∧ 𝑦 = (𝑧𝐴)))
2625exbii 1848 . . . . . . . . . . 11 (∃𝑦(𝑥𝑦𝑦 = (𝑧𝐴)) ↔ ∃𝑦(𝑥 ∈ (𝑧𝐴) ∧ 𝑦 = (𝑧𝐴)))
27 19.42v 1955 . . . . . . . . . . 11 (∃𝑦(𝑥 ∈ (𝑧𝐴) ∧ 𝑦 = (𝑧𝐴)) ↔ (𝑥 ∈ (𝑧𝐴) ∧ ∃𝑦 𝑦 = (𝑧𝐴)))
28 vex 3476 . . . . . . . . . . . . . . . 16 𝑧 ∈ V
2928inex1 5316 . . . . . . . . . . . . . . 15 (𝑧𝐴) ∈ V
3029isseti 3488 . . . . . . . . . . . . . 14 𝑦 𝑦 = (𝑧𝐴)
3130biantru 528 . . . . . . . . . . . . 13 (𝑥 ∈ (𝑧𝐴) ↔ (𝑥 ∈ (𝑧𝐴) ∧ ∃𝑦 𝑦 = (𝑧𝐴)))
3231bicomi 223 . . . . . . . . . . . 12 ((𝑥 ∈ (𝑧𝐴) ∧ ∃𝑦 𝑦 = (𝑧𝐴)) ↔ 𝑥 ∈ (𝑧𝐴))
33 elin 3963 . . . . . . . . . . . 12 (𝑥 ∈ (𝑧𝐴) ↔ (𝑥𝑧𝑥𝐴))
3432, 33bitri 274 . . . . . . . . . . 11 ((𝑥 ∈ (𝑧𝐴) ∧ ∃𝑦 𝑦 = (𝑧𝐴)) ↔ (𝑥𝑧𝑥𝐴))
3526, 27, 343bitri 296 . . . . . . . . . 10 (∃𝑦(𝑥𝑦𝑦 = (𝑧𝐴)) ↔ (𝑥𝑧𝑥𝐴))
3635bianassc 639 . . . . . . . . 9 ((𝑧𝑋 ∧ ∃𝑦(𝑥𝑦𝑦 = (𝑧𝐴))) ↔ ((𝑥𝑧𝑧𝑋) ∧ 𝑥𝐴))
3717, 18, 363bitri 296 . . . . . . . 8 (∃𝑦(𝑥𝑦 ∧ (𝑧𝑋𝑦 = (𝑧𝐴))) ↔ ((𝑥𝑧𝑧𝑋) ∧ 𝑥𝐴))
3837exbii 1848 . . . . . . 7 (∃𝑧𝑦(𝑥𝑦 ∧ (𝑧𝑋𝑦 = (𝑧𝐴))) ↔ ∃𝑧((𝑥𝑧𝑧𝑋) ∧ 𝑥𝐴))
39 19.41v 1951 . . . . . . 7 (∃𝑧((𝑥𝑧𝑧𝑋) ∧ 𝑥𝐴) ↔ (∃𝑧(𝑥𝑧𝑧𝑋) ∧ 𝑥𝐴))
4038, 39bitri 274 . . . . . 6 (∃𝑧𝑦(𝑥𝑦 ∧ (𝑧𝑋𝑦 = (𝑧𝐴))) ↔ (∃𝑧(𝑥𝑧𝑧𝑋) ∧ 𝑥𝐴))
4114, 15, 403bitri 296 . . . . 5 (∃𝑦(𝑥𝑦 ∧ ∃𝑧𝑋 𝑦 = (𝑧𝐴)) ↔ (∃𝑧(𝑥𝑧𝑧𝑋) ∧ 𝑥𝐴))
42 elin 3963 . . . . 5 (𝑥 ∈ ( 𝑋𝐴) ↔ (𝑥 𝑋𝑥𝐴))
438, 41, 423bitr4g 313 . . . 4 ((𝑋𝑉𝐴𝑊) → (∃𝑦(𝑥𝑦 ∧ ∃𝑧𝑋 𝑦 = (𝑧𝐴)) ↔ 𝑥 ∈ ( 𝑋𝐴)))
444, 43bitrd 278 . . 3 ((𝑋𝑉𝐴𝑊) → (∃𝑦(𝑥𝑦𝑦 ∈ (𝑋t 𝐴)) ↔ 𝑥 ∈ ( 𝑋𝐴)))
451, 44bitrid 282 . 2 ((𝑋𝑉𝐴𝑊) → (𝑥 (𝑋t 𝐴) ↔ 𝑥 ∈ ( 𝑋𝐴)))
4645eqrdv 2728 1 ((𝑋𝑉𝐴𝑊) → (𝑋t 𝐴) = ( 𝑋𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394   = wceq 1539  wex 1779  wcel 2104  wrex 3068  cin 3946   cuni 4907  (class class class)co 7411  t crest 17370
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 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pr 5426  ax-un 7727
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3375  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-ov 7414  df-oprab 7415  df-mpo 7416  df-rest 17372
This theorem is referenced by:  bj-restuni2  36282
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