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Theorem bj-restuni 32687
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 20876 and restuni2 20881. (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 4405 . . 3 (𝑥 (𝑋t 𝐴) ↔ ∃𝑦(𝑥𝑦𝑦 ∈ (𝑋t 𝐴)))
2 elrest 16009 . . . . . 6 ((𝑋𝑉𝐴𝑊) → (𝑦 ∈ (𝑋t 𝐴) ↔ ∃𝑧𝑋 𝑦 = (𝑧𝐴)))
32anbi2d 739 . . . . 5 ((𝑋𝑉𝐴𝑊) → ((𝑥𝑦𝑦 ∈ (𝑋t 𝐴)) ↔ (𝑥𝑦 ∧ ∃𝑧𝑋 𝑦 = (𝑧𝐴))))
43exbidv 1847 . . . 4 ((𝑋𝑉𝐴𝑊) → (∃𝑦(𝑥𝑦𝑦 ∈ (𝑋t 𝐴)) ↔ ∃𝑦(𝑥𝑦 ∧ ∃𝑧𝑋 𝑦 = (𝑧𝐴))))
5 eluni 4405 . . . . . . . 8 (𝑥 𝑋 ↔ ∃𝑧(𝑥𝑧𝑧𝑋))
65bicomi 214 . . . . . . 7 (∃𝑧(𝑥𝑧𝑧𝑋) ↔ 𝑥 𝑋)
76anbi1i 730 . . . . . 6 ((∃𝑧(𝑥𝑧𝑧𝑋) ∧ 𝑥𝐴) ↔ (𝑥 𝑋𝑥𝐴))
87a1i 11 . . . . 5 ((𝑋𝑉𝐴𝑊) → ((∃𝑧(𝑥𝑧𝑧𝑋) ∧ 𝑥𝐴) ↔ (𝑥 𝑋𝑥𝐴)))
9 df-rex 2913 . . . . . . . . 9 (∃𝑧𝑋 𝑦 = (𝑧𝐴) ↔ ∃𝑧(𝑧𝑋𝑦 = (𝑧𝐴)))
109anbi2i 729 . . . . . . . 8 ((𝑥𝑦 ∧ ∃𝑧𝑋 𝑦 = (𝑧𝐴)) ↔ (𝑥𝑦 ∧ ∃𝑧(𝑧𝑋𝑦 = (𝑧𝐴))))
11 19.42v 1915 . . . . . . . . 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 1771 . . . . . 6 (∃𝑦(𝑥𝑦 ∧ ∃𝑧𝑋 𝑦 = (𝑧𝐴)) ↔ ∃𝑦𝑧(𝑥𝑦 ∧ (𝑧𝑋𝑦 = (𝑧𝐴))))
15 excom 2039 . . . . . 6 (∃𝑦𝑧(𝑥𝑦 ∧ (𝑧𝑋𝑦 = (𝑧𝐴))) ↔ ∃𝑧𝑦(𝑥𝑦 ∧ (𝑧𝑋𝑦 = (𝑧𝐴))))
16 an12 837 . . . . . . . . . 10 ((𝑥𝑦 ∧ (𝑧𝑋𝑦 = (𝑧𝐴))) ↔ (𝑧𝑋 ∧ (𝑥𝑦𝑦 = (𝑧𝐴))))
1716exbii 1771 . . . . . . . . 9 (∃𝑦(𝑥𝑦 ∧ (𝑧𝑋𝑦 = (𝑧𝐴))) ↔ ∃𝑦(𝑧𝑋 ∧ (𝑥𝑦𝑦 = (𝑧𝐴))))
18 19.42v 1915 . . . . . . . . 9 (∃𝑦(𝑧𝑋 ∧ (𝑥𝑦𝑦 = (𝑧𝐴))) ↔ (𝑧𝑋 ∧ ∃𝑦(𝑥𝑦𝑦 = (𝑧𝐴))))
19 eqimss 3636 . . . . . . . . . . . . . . . 16 (𝑦 = (𝑧𝐴) → 𝑦 ⊆ (𝑧𝐴))
2019sseld 3582 . . . . . . . . . . . . . . 15 (𝑦 = (𝑧𝐴) → (𝑥𝑦𝑥 ∈ (𝑧𝐴)))
2120imdistanri 726 . . . . . . . . . . . . . 14 ((𝑥𝑦𝑦 = (𝑧𝐴)) → (𝑥 ∈ (𝑧𝐴) ∧ 𝑦 = (𝑧𝐴)))
22 eqimss2 3637 . . . . . . . . . . . . . . . 16 (𝑦 = (𝑧𝐴) → (𝑧𝐴) ⊆ 𝑦)
2322sseld 3582 . . . . . . . . . . . . . . 15 (𝑦 = (𝑧𝐴) → (𝑥 ∈ (𝑧𝐴) → 𝑥𝑦))
2423imdistanri 726 . . . . . . . . . . . . . 14 ((𝑥 ∈ (𝑧𝐴) ∧ 𝑦 = (𝑧𝐴)) → (𝑥𝑦𝑦 = (𝑧𝐴)))
2521, 24impbii 199 . . . . . . . . . . . . 13 ((𝑥𝑦𝑦 = (𝑧𝐴)) ↔ (𝑥 ∈ (𝑧𝐴) ∧ 𝑦 = (𝑧𝐴)))
2625exbii 1771 . . . . . . . . . . . 12 (∃𝑦(𝑥𝑦𝑦 = (𝑧𝐴)) ↔ ∃𝑦(𝑥 ∈ (𝑧𝐴) ∧ 𝑦 = (𝑧𝐴)))
27 19.42v 1915 . . . . . . . . . . . 12 (∃𝑦(𝑥 ∈ (𝑧𝐴) ∧ 𝑦 = (𝑧𝐴)) ↔ (𝑥 ∈ (𝑧𝐴) ∧ ∃𝑦 𝑦 = (𝑧𝐴)))
28 vex 3189 . . . . . . . . . . . . . . . . 17 𝑧 ∈ V
2928inex1 4759 . . . . . . . . . . . . . . . 16 (𝑧𝐴) ∈ V
3029isseti 3195 . . . . . . . . . . . . . . 15 𝑦 𝑦 = (𝑧𝐴)
3130biantru 526 . . . . . . . . . . . . . 14 (𝑥 ∈ (𝑧𝐴) ↔ (𝑥 ∈ (𝑧𝐴) ∧ ∃𝑦 𝑦 = (𝑧𝐴)))
3231bicomi 214 . . . . . . . . . . . . 13 ((𝑥 ∈ (𝑧𝐴) ∧ ∃𝑦 𝑦 = (𝑧𝐴)) ↔ 𝑥 ∈ (𝑧𝐴))
33 elin 3774 . . . . . . . . . . . . 13 (𝑥 ∈ (𝑧𝐴) ↔ (𝑥𝑧𝑥𝐴))
3432, 33bitri 264 . . . . . . . . . . . 12 ((𝑥 ∈ (𝑧𝐴) ∧ ∃𝑦 𝑦 = (𝑧𝐴)) ↔ (𝑥𝑧𝑥𝐴))
3526, 27, 343bitri 286 . . . . . . . . . . 11 (∃𝑦(𝑥𝑦𝑦 = (𝑧𝐴)) ↔ (𝑥𝑧𝑥𝐴))
3635anbi2i 729 . . . . . . . . . 10 ((𝑧𝑋 ∧ ∃𝑦(𝑥𝑦𝑦 = (𝑧𝐴))) ↔ (𝑧𝑋 ∧ (𝑥𝑧𝑥𝐴)))
37 biid 251 . . . . . . . . . . 11 ((𝑥𝑧𝑥𝐴) ↔ (𝑥𝑧𝑥𝐴))
3837bianass 841 . . . . . . . . . 10 ((𝑧𝑋 ∧ (𝑥𝑧𝑥𝐴)) ↔ ((𝑧𝑋𝑥𝑧) ∧ 𝑥𝐴))
39 ancom 466 . . . . . . . . . . 11 ((𝑧𝑋𝑥𝑧) ↔ (𝑥𝑧𝑧𝑋))
4039anbi1i 730 . . . . . . . . . 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 1771 . . . . . . 7 (∃𝑧𝑦(𝑥𝑦 ∧ (𝑧𝑋𝑦 = (𝑧𝐴))) ↔ ∃𝑧((𝑥𝑧𝑧𝑋) ∧ 𝑥𝐴))
44 19.41v 1911 . . . . . . 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 3774 . . . . 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 2619 1 ((𝑋𝑉𝐴𝑊) → (𝑋t 𝐴) = ( 𝑋𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1480  wex 1701  wcel 1987  wrex 2908  cin 3554   cuni 4402  (class class class)co 6604  t crest 16002
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-rest 16004
This theorem is referenced by:  bj-restuni2  32688
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