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Theorem bj-restuni 36764
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 23149 and restuni2 23154. (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 4915 . . 3 (𝑥 (𝑋t 𝐴) ↔ ∃𝑦(𝑥𝑦𝑦 ∈ (𝑋t 𝐴)))
2 elrest 17437 . . . . . 6 ((𝑋𝑉𝐴𝑊) → (𝑦 ∈ (𝑋t 𝐴) ↔ ∃𝑧𝑋 𝑦 = (𝑧𝐴)))
32anbi2d 628 . . . . 5 ((𝑋𝑉𝐴𝑊) → ((𝑥𝑦𝑦 ∈ (𝑋t 𝐴)) ↔ (𝑥𝑦 ∧ ∃𝑧𝑋 𝑦 = (𝑧𝐴))))
43exbidv 1916 . . . 4 ((𝑋𝑉𝐴𝑊) → (∃𝑦(𝑥𝑦𝑦 ∈ (𝑋t 𝐴)) ↔ ∃𝑦(𝑥𝑦 ∧ ∃𝑧𝑋 𝑦 = (𝑧𝐴))))
5 eluni 4915 . . . . . . . 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 3060 . . . . . . . . 9 (∃𝑧𝑋 𝑦 = (𝑧𝐴) ↔ ∃𝑧(𝑧𝑋𝑦 = (𝑧𝐴)))
109anbi2i 621 . . . . . . . 8 ((𝑥𝑦 ∧ ∃𝑧𝑋 𝑦 = (𝑧𝐴)) ↔ (𝑥𝑦 ∧ ∃𝑧(𝑧𝑋𝑦 = (𝑧𝐴))))
11 19.42v 1949 . . . . . . . . 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 1842 . . . . . 6 (∃𝑦(𝑥𝑦 ∧ ∃𝑧𝑋 𝑦 = (𝑧𝐴)) ↔ ∃𝑦𝑧(𝑥𝑦 ∧ (𝑧𝑋𝑦 = (𝑧𝐴))))
15 excom 2151 . . . . . 6 (∃𝑦𝑧(𝑥𝑦 ∧ (𝑧𝑋𝑦 = (𝑧𝐴))) ↔ ∃𝑧𝑦(𝑥𝑦 ∧ (𝑧𝑋𝑦 = (𝑧𝐴))))
16 an12 643 . . . . . . . . . 10 ((𝑥𝑦 ∧ (𝑧𝑋𝑦 = (𝑧𝐴))) ↔ (𝑧𝑋 ∧ (𝑥𝑦𝑦 = (𝑧𝐴))))
1716exbii 1842 . . . . . . . . 9 (∃𝑦(𝑥𝑦 ∧ (𝑧𝑋𝑦 = (𝑧𝐴))) ↔ ∃𝑦(𝑧𝑋 ∧ (𝑥𝑦𝑦 = (𝑧𝐴))))
18 19.42v 1949 . . . . . . . . 9 (∃𝑦(𝑧𝑋 ∧ (𝑥𝑦𝑦 = (𝑧𝐴))) ↔ (𝑧𝑋 ∧ ∃𝑦(𝑥𝑦𝑦 = (𝑧𝐴))))
19 eqimss 4037 . . . . . . . . . . . . . . 15 (𝑦 = (𝑧𝐴) → 𝑦 ⊆ (𝑧𝐴))
2019sseld 3977 . . . . . . . . . . . . . 14 (𝑦 = (𝑧𝐴) → (𝑥𝑦𝑥 ∈ (𝑧𝐴)))
2120imdistanri 568 . . . . . . . . . . . . 13 ((𝑥𝑦𝑦 = (𝑧𝐴)) → (𝑥 ∈ (𝑧𝐴) ∧ 𝑦 = (𝑧𝐴)))
22 eqimss2 4038 . . . . . . . . . . . . . . 15 (𝑦 = (𝑧𝐴) → (𝑧𝐴) ⊆ 𝑦)
2322sseld 3977 . . . . . . . . . . . . . 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 1842 . . . . . . . . . . 11 (∃𝑦(𝑥𝑦𝑦 = (𝑧𝐴)) ↔ ∃𝑦(𝑥 ∈ (𝑧𝐴) ∧ 𝑦 = (𝑧𝐴)))
27 19.42v 1949 . . . . . . . . . . 11 (∃𝑦(𝑥 ∈ (𝑧𝐴) ∧ 𝑦 = (𝑧𝐴)) ↔ (𝑥 ∈ (𝑧𝐴) ∧ ∃𝑦 𝑦 = (𝑧𝐴)))
28 vex 3465 . . . . . . . . . . . . . . . 16 𝑧 ∈ V
2928inex1 5321 . . . . . . . . . . . . . . 15 (𝑧𝐴) ∈ V
3029isseti 3478 . . . . . . . . . . . . . 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 3962 . . . . . . . . . . . 12 (𝑥 ∈ (𝑧𝐴) ↔ (𝑥𝑧𝑥𝐴))
3432, 33bitri 274 . . . . . . . . . . 11 ((𝑥 ∈ (𝑧𝐴) ∧ ∃𝑦 𝑦 = (𝑧𝐴)) ↔ (𝑥𝑧𝑥𝐴))
3526, 27, 343bitri 296 . . . . . . . . . 10 (∃𝑦(𝑥𝑦𝑦 = (𝑧𝐴)) ↔ (𝑥𝑧𝑥𝐴))
3635bianassc 641 . . . . . . . . 9 ((𝑧𝑋 ∧ ∃𝑦(𝑥𝑦𝑦 = (𝑧𝐴))) ↔ ((𝑥𝑧𝑧𝑋) ∧ 𝑥𝐴))
3717, 18, 363bitri 296 . . . . . . . 8 (∃𝑦(𝑥𝑦 ∧ (𝑧𝑋𝑦 = (𝑧𝐴))) ↔ ((𝑥𝑧𝑧𝑋) ∧ 𝑥𝐴))
3837exbii 1842 . . . . . . 7 (∃𝑧𝑦(𝑥𝑦 ∧ (𝑧𝑋𝑦 = (𝑧𝐴))) ↔ ∃𝑧((𝑥𝑧𝑧𝑋) ∧ 𝑥𝐴))
39 19.41v 1945 . . . . . . 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 3962 . . . . 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 2723 1 ((𝑋𝑉𝐴𝑊) → (𝑋t 𝐴) = ( 𝑋𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394   = wceq 1533  wex 1773  wcel 2098  wrex 3059  cin 3945   cuni 4912  (class class class)co 7423  t crest 17430
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5289  ax-sep 5303  ax-nul 5310  ax-pr 5432  ax-un 7745
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-reu 3364  df-rab 3419  df-v 3463  df-sbc 3776  df-csb 3892  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-nul 4325  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-id 5579  df-xp 5687  df-rel 5688  df-cnv 5689  df-co 5690  df-dm 5691  df-rn 5692  df-res 5693  df-ima 5694  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-ov 7426  df-oprab 7427  df-mpo 7428  df-rest 17432
This theorem is referenced by:  bj-restuni2  36765
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