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Theorem bj-restuni 33175
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 21014 and restuni2 21019. (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 4471 . . 3 (𝑥 (𝑋t 𝐴) ↔ ∃𝑦(𝑥𝑦𝑦 ∈ (𝑋t 𝐴)))
2 elrest 16135 . . . . . 6 ((𝑋𝑉𝐴𝑊) → (𝑦 ∈ (𝑋t 𝐴) ↔ ∃𝑧𝑋 𝑦 = (𝑧𝐴)))
32anbi2d 740 . . . . 5 ((𝑋𝑉𝐴𝑊) → ((𝑥𝑦𝑦 ∈ (𝑋t 𝐴)) ↔ (𝑥𝑦 ∧ ∃𝑧𝑋 𝑦 = (𝑧𝐴))))
43exbidv 1890 . . . 4 ((𝑋𝑉𝐴𝑊) → (∃𝑦(𝑥𝑦𝑦 ∈ (𝑋t 𝐴)) ↔ ∃𝑦(𝑥𝑦 ∧ ∃𝑧𝑋 𝑦 = (𝑧𝐴))))
5 eluni 4471 . . . . . . . 8 (𝑥 𝑋 ↔ ∃𝑧(𝑥𝑧𝑧𝑋))
65bicomi 214 . . . . . . 7 (∃𝑧(𝑥𝑧𝑧𝑋) ↔ 𝑥 𝑋)
76anbi1i 731 . . . . . 6 ((∃𝑧(𝑥𝑧𝑧𝑋) ∧ 𝑥𝐴) ↔ (𝑥 𝑋𝑥𝐴))
87a1i 11 . . . . 5 ((𝑋𝑉𝐴𝑊) → ((∃𝑧(𝑥𝑧𝑧𝑋) ∧ 𝑥𝐴) ↔ (𝑥 𝑋𝑥𝐴)))
9 df-rex 2947 . . . . . . . . 9 (∃𝑧𝑋 𝑦 = (𝑧𝐴) ↔ ∃𝑧(𝑧𝑋𝑦 = (𝑧𝐴)))
109anbi2i 730 . . . . . . . 8 ((𝑥𝑦 ∧ ∃𝑧𝑋 𝑦 = (𝑧𝐴)) ↔ (𝑥𝑦 ∧ ∃𝑧(𝑧𝑋𝑦 = (𝑧𝐴))))
11 19.42v 1921 . . . . . . . . 9 (∃𝑧(𝑥𝑦 ∧ (𝑧𝑋𝑦 = (𝑧𝐴))) ↔ (𝑥𝑦 ∧ ∃𝑧(𝑧𝑋𝑦 = (𝑧𝐴))))
1211bicomi 214 . . . . . . . 8 ((𝑥𝑦 ∧ ∃𝑧(𝑧𝑋𝑦 = (𝑧𝐴))) ↔ ∃𝑧(𝑥𝑦 ∧ (𝑧𝑋𝑦 = (𝑧𝐴))))
1310, 12bitri 264 . . . . . . 7 ((𝑥𝑦 ∧ ∃𝑧𝑋 𝑦 = (𝑧𝐴)) ↔ ∃𝑧(𝑥𝑦 ∧ (𝑧𝑋𝑦 = (𝑧𝐴))))
1413exbii 1814 . . . . . 6 (∃𝑦(𝑥𝑦 ∧ ∃𝑧𝑋 𝑦 = (𝑧𝐴)) ↔ ∃𝑦𝑧(𝑥𝑦 ∧ (𝑧𝑋𝑦 = (𝑧𝐴))))
15 excom 2082 . . . . . 6 (∃𝑦𝑧(𝑥𝑦 ∧ (𝑧𝑋𝑦 = (𝑧𝐴))) ↔ ∃𝑧𝑦(𝑥𝑦 ∧ (𝑧𝑋𝑦 = (𝑧𝐴))))
16 an12 855 . . . . . . . . . 10 ((𝑥𝑦 ∧ (𝑧𝑋𝑦 = (𝑧𝐴))) ↔ (𝑧𝑋 ∧ (𝑥𝑦𝑦 = (𝑧𝐴))))
1716exbii 1814 . . . . . . . . 9 (∃𝑦(𝑥𝑦 ∧ (𝑧𝑋𝑦 = (𝑧𝐴))) ↔ ∃𝑦(𝑧𝑋 ∧ (𝑥𝑦𝑦 = (𝑧𝐴))))
18 19.42v 1921 . . . . . . . . 9 (∃𝑦(𝑧𝑋 ∧ (𝑥𝑦𝑦 = (𝑧𝐴))) ↔ (𝑧𝑋 ∧ ∃𝑦(𝑥𝑦𝑦 = (𝑧𝐴))))
19 eqimss 3690 . . . . . . . . . . . . . . . 16 (𝑦 = (𝑧𝐴) → 𝑦 ⊆ (𝑧𝐴))
2019sseld 3635 . . . . . . . . . . . . . . 15 (𝑦 = (𝑧𝐴) → (𝑥𝑦𝑥 ∈ (𝑧𝐴)))
2120imdistanri 727 . . . . . . . . . . . . . 14 ((𝑥𝑦𝑦 = (𝑧𝐴)) → (𝑥 ∈ (𝑧𝐴) ∧ 𝑦 = (𝑧𝐴)))
22 eqimss2 3691 . . . . . . . . . . . . . . . 16 (𝑦 = (𝑧𝐴) → (𝑧𝐴) ⊆ 𝑦)
2322sseld 3635 . . . . . . . . . . . . . . 15 (𝑦 = (𝑧𝐴) → (𝑥 ∈ (𝑧𝐴) → 𝑥𝑦))
2423imdistanri 727 . . . . . . . . . . . . . 14 ((𝑥 ∈ (𝑧𝐴) ∧ 𝑦 = (𝑧𝐴)) → (𝑥𝑦𝑦 = (𝑧𝐴)))
2521, 24impbii 199 . . . . . . . . . . . . 13 ((𝑥𝑦𝑦 = (𝑧𝐴)) ↔ (𝑥 ∈ (𝑧𝐴) ∧ 𝑦 = (𝑧𝐴)))
2625exbii 1814 . . . . . . . . . . . 12 (∃𝑦(𝑥𝑦𝑦 = (𝑧𝐴)) ↔ ∃𝑦(𝑥 ∈ (𝑧𝐴) ∧ 𝑦 = (𝑧𝐴)))
27 19.42v 1921 . . . . . . . . . . . 12 (∃𝑦(𝑥 ∈ (𝑧𝐴) ∧ 𝑦 = (𝑧𝐴)) ↔ (𝑥 ∈ (𝑧𝐴) ∧ ∃𝑦 𝑦 = (𝑧𝐴)))
28 vex 3234 . . . . . . . . . . . . . . . . 17 𝑧 ∈ V
2928inex1 4832 . . . . . . . . . . . . . . . 16 (𝑧𝐴) ∈ V
3029isseti 3240 . . . . . . . . . . . . . . 15 𝑦 𝑦 = (𝑧𝐴)
3130biantru 525 . . . . . . . . . . . . . 14 (𝑥 ∈ (𝑧𝐴) ↔ (𝑥 ∈ (𝑧𝐴) ∧ ∃𝑦 𝑦 = (𝑧𝐴)))
3231bicomi 214 . . . . . . . . . . . . 13 ((𝑥 ∈ (𝑧𝐴) ∧ ∃𝑦 𝑦 = (𝑧𝐴)) ↔ 𝑥 ∈ (𝑧𝐴))
33 elin 3829 . . . . . . . . . . . . 13 (𝑥 ∈ (𝑧𝐴) ↔ (𝑥𝑧𝑥𝐴))
3432, 33bitri 264 . . . . . . . . . . . 12 ((𝑥 ∈ (𝑧𝐴) ∧ ∃𝑦 𝑦 = (𝑧𝐴)) ↔ (𝑥𝑧𝑥𝐴))
3526, 27, 343bitri 286 . . . . . . . . . . 11 (∃𝑦(𝑥𝑦𝑦 = (𝑧𝐴)) ↔ (𝑥𝑧𝑥𝐴))
3635anbi2i 730 . . . . . . . . . 10 ((𝑧𝑋 ∧ ∃𝑦(𝑥𝑦𝑦 = (𝑧𝐴))) ↔ (𝑧𝑋 ∧ (𝑥𝑧𝑥𝐴)))
37 biid 251 . . . . . . . . . . 11 ((𝑥𝑧𝑥𝐴) ↔ (𝑥𝑧𝑥𝐴))
3837bianass 859 . . . . . . . . . 10 ((𝑧𝑋 ∧ (𝑥𝑧𝑥𝐴)) ↔ ((𝑧𝑋𝑥𝑧) ∧ 𝑥𝐴))
39 ancom 465 . . . . . . . . . . 11 ((𝑧𝑋𝑥𝑧) ↔ (𝑥𝑧𝑧𝑋))
4039anbi1i 731 . . . . . . . . . 10 (((𝑧𝑋𝑥𝑧) ∧ 𝑥𝐴) ↔ ((𝑥𝑧𝑧𝑋) ∧ 𝑥𝐴))
4136, 38, 403bitri 286 . . . . . . . . 9 ((𝑧𝑋 ∧ ∃𝑦(𝑥𝑦𝑦 = (𝑧𝐴))) ↔ ((𝑥𝑧𝑧𝑋) ∧ 𝑥𝐴))
4217, 18, 413bitri 286 . . . . . . . 8 (∃𝑦(𝑥𝑦 ∧ (𝑧𝑋𝑦 = (𝑧𝐴))) ↔ ((𝑥𝑧𝑧𝑋) ∧ 𝑥𝐴))
4342exbii 1814 . . . . . . 7 (∃𝑧𝑦(𝑥𝑦 ∧ (𝑧𝑋𝑦 = (𝑧𝐴))) ↔ ∃𝑧((𝑥𝑧𝑧𝑋) ∧ 𝑥𝐴))
44 19.41v 1917 . . . . . . 7 (∃𝑧((𝑥𝑧𝑧𝑋) ∧ 𝑥𝐴) ↔ (∃𝑧(𝑥𝑧𝑧𝑋) ∧ 𝑥𝐴))
4543, 44bitri 264 . . . . . 6 (∃𝑧𝑦(𝑥𝑦 ∧ (𝑧𝑋𝑦 = (𝑧𝐴))) ↔ (∃𝑧(𝑥𝑧𝑧𝑋) ∧ 𝑥𝐴))
4614, 15, 453bitri 286 . . . . 5 (∃𝑦(𝑥𝑦 ∧ ∃𝑧𝑋 𝑦 = (𝑧𝐴)) ↔ (∃𝑧(𝑥𝑧𝑧𝑋) ∧ 𝑥𝐴))
47 elin 3829 . . . . 5 (𝑥 ∈ ( 𝑋𝐴) ↔ (𝑥 𝑋𝑥𝐴))
488, 46, 473bitr4g 303 . . . 4 ((𝑋𝑉𝐴𝑊) → (∃𝑦(𝑥𝑦 ∧ ∃𝑧𝑋 𝑦 = (𝑧𝐴)) ↔ 𝑥 ∈ ( 𝑋𝐴)))
494, 48bitrd 268 . . 3 ((𝑋𝑉𝐴𝑊) → (∃𝑦(𝑥𝑦𝑦 ∈ (𝑋t 𝐴)) ↔ 𝑥 ∈ ( 𝑋𝐴)))
501, 49syl5bb 272 . 2 ((𝑋𝑉𝐴𝑊) → (𝑥 (𝑋t 𝐴) ↔ 𝑥 ∈ ( 𝑋𝐴)))
5150eqrdv 2649 1 ((𝑋𝑉𝐴𝑊) → (𝑋t 𝐴) = ( 𝑋𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1523  wex 1744  wcel 2030  wrex 2942  cin 3606   cuni 4468  (class class class)co 6690  t crest 16128
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-rest 16130
This theorem is referenced by:  bj-restuni2  33176
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