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Theorem bj-restuni 33543
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 21295 and restuni2 21300. (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 4631 . . 3 (𝑥 (𝑋t 𝐴) ↔ ∃𝑦(𝑥𝑦𝑦 ∈ (𝑋t 𝐴)))
2 elrest 16403 . . . . . 6 ((𝑋𝑉𝐴𝑊) → (𝑦 ∈ (𝑋t 𝐴) ↔ ∃𝑧𝑋 𝑦 = (𝑧𝐴)))
32anbi2d 623 . . . . 5 ((𝑋𝑉𝐴𝑊) → ((𝑥𝑦𝑦 ∈ (𝑋t 𝐴)) ↔ (𝑥𝑦 ∧ ∃𝑧𝑋 𝑦 = (𝑧𝐴))))
43exbidv 2017 . . . 4 ((𝑋𝑉𝐴𝑊) → (∃𝑦(𝑥𝑦𝑦 ∈ (𝑋t 𝐴)) ↔ ∃𝑦(𝑥𝑦 ∧ ∃𝑧𝑋 𝑦 = (𝑧𝐴))))
5 eluni 4631 . . . . . . . 8 (𝑥 𝑋 ↔ ∃𝑧(𝑥𝑧𝑧𝑋))
65bicomi 216 . . . . . . 7 (∃𝑧(𝑥𝑧𝑧𝑋) ↔ 𝑥 𝑋)
76anbi1i 618 . . . . . 6 ((∃𝑧(𝑥𝑧𝑧𝑋) ∧ 𝑥𝐴) ↔ (𝑥 𝑋𝑥𝐴))
87a1i 11 . . . . 5 ((𝑋𝑉𝐴𝑊) → ((∃𝑧(𝑥𝑧𝑧𝑋) ∧ 𝑥𝐴) ↔ (𝑥 𝑋𝑥𝐴)))
9 df-rex 3095 . . . . . . . . 9 (∃𝑧𝑋 𝑦 = (𝑧𝐴) ↔ ∃𝑧(𝑧𝑋𝑦 = (𝑧𝐴)))
109anbi2i 617 . . . . . . . 8 ((𝑥𝑦 ∧ ∃𝑧𝑋 𝑦 = (𝑧𝐴)) ↔ (𝑥𝑦 ∧ ∃𝑧(𝑧𝑋𝑦 = (𝑧𝐴))))
11 19.42v 2049 . . . . . . . . 9 (∃𝑧(𝑥𝑦 ∧ (𝑧𝑋𝑦 = (𝑧𝐴))) ↔ (𝑥𝑦 ∧ ∃𝑧(𝑧𝑋𝑦 = (𝑧𝐴))))
1211bicomi 216 . . . . . . . 8 ((𝑥𝑦 ∧ ∃𝑧(𝑧𝑋𝑦 = (𝑧𝐴))) ↔ ∃𝑧(𝑥𝑦 ∧ (𝑧𝑋𝑦 = (𝑧𝐴))))
1310, 12bitri 267 . . . . . . 7 ((𝑥𝑦 ∧ ∃𝑧𝑋 𝑦 = (𝑧𝐴)) ↔ ∃𝑧(𝑥𝑦 ∧ (𝑧𝑋𝑦 = (𝑧𝐴))))
1413exbii 1944 . . . . . 6 (∃𝑦(𝑥𝑦 ∧ ∃𝑧𝑋 𝑦 = (𝑧𝐴)) ↔ ∃𝑦𝑧(𝑥𝑦 ∧ (𝑧𝑋𝑦 = (𝑧𝐴))))
15 excom 2205 . . . . . 6 (∃𝑦𝑧(𝑥𝑦 ∧ (𝑧𝑋𝑦 = (𝑧𝐴))) ↔ ∃𝑧𝑦(𝑥𝑦 ∧ (𝑧𝑋𝑦 = (𝑧𝐴))))
16 an12 636 . . . . . . . . . 10 ((𝑥𝑦 ∧ (𝑧𝑋𝑦 = (𝑧𝐴))) ↔ (𝑧𝑋 ∧ (𝑥𝑦𝑦 = (𝑧𝐴))))
1716exbii 1944 . . . . . . . . 9 (∃𝑦(𝑥𝑦 ∧ (𝑧𝑋𝑦 = (𝑧𝐴))) ↔ ∃𝑦(𝑧𝑋 ∧ (𝑥𝑦𝑦 = (𝑧𝐴))))
18 19.42v 2049 . . . . . . . . 9 (∃𝑦(𝑧𝑋 ∧ (𝑥𝑦𝑦 = (𝑧𝐴))) ↔ (𝑧𝑋 ∧ ∃𝑦(𝑥𝑦𝑦 = (𝑧𝐴))))
19 eqimss 3853 . . . . . . . . . . . . . . . 16 (𝑦 = (𝑧𝐴) → 𝑦 ⊆ (𝑧𝐴))
2019sseld 3797 . . . . . . . . . . . . . . 15 (𝑦 = (𝑧𝐴) → (𝑥𝑦𝑥 ∈ (𝑧𝐴)))
2120imdistanri 566 . . . . . . . . . . . . . 14 ((𝑥𝑦𝑦 = (𝑧𝐴)) → (𝑥 ∈ (𝑧𝐴) ∧ 𝑦 = (𝑧𝐴)))
22 eqimss2 3854 . . . . . . . . . . . . . . . 16 (𝑦 = (𝑧𝐴) → (𝑧𝐴) ⊆ 𝑦)
2322sseld 3797 . . . . . . . . . . . . . . 15 (𝑦 = (𝑧𝐴) → (𝑥 ∈ (𝑧𝐴) → 𝑥𝑦))
2423imdistanri 566 . . . . . . . . . . . . . 14 ((𝑥 ∈ (𝑧𝐴) ∧ 𝑦 = (𝑧𝐴)) → (𝑥𝑦𝑦 = (𝑧𝐴)))
2521, 24impbii 201 . . . . . . . . . . . . 13 ((𝑥𝑦𝑦 = (𝑧𝐴)) ↔ (𝑥 ∈ (𝑧𝐴) ∧ 𝑦 = (𝑧𝐴)))
2625exbii 1944 . . . . . . . . . . . 12 (∃𝑦(𝑥𝑦𝑦 = (𝑧𝐴)) ↔ ∃𝑦(𝑥 ∈ (𝑧𝐴) ∧ 𝑦 = (𝑧𝐴)))
27 19.42v 2049 . . . . . . . . . . . 12 (∃𝑦(𝑥 ∈ (𝑧𝐴) ∧ 𝑦 = (𝑧𝐴)) ↔ (𝑥 ∈ (𝑧𝐴) ∧ ∃𝑦 𝑦 = (𝑧𝐴)))
28 vex 3388 . . . . . . . . . . . . . . . . 17 𝑧 ∈ V
2928inex1 4994 . . . . . . . . . . . . . . . 16 (𝑧𝐴) ∈ V
3029isseti 3397 . . . . . . . . . . . . . . 15 𝑦 𝑦 = (𝑧𝐴)
3130biantru 526 . . . . . . . . . . . . . 14 (𝑥 ∈ (𝑧𝐴) ↔ (𝑥 ∈ (𝑧𝐴) ∧ ∃𝑦 𝑦 = (𝑧𝐴)))
3231bicomi 216 . . . . . . . . . . . . 13 ((𝑥 ∈ (𝑧𝐴) ∧ ∃𝑦 𝑦 = (𝑧𝐴)) ↔ 𝑥 ∈ (𝑧𝐴))
33 elin 3994 . . . . . . . . . . . . 13 (𝑥 ∈ (𝑧𝐴) ↔ (𝑥𝑧𝑥𝐴))
3432, 33bitri 267 . . . . . . . . . . . 12 ((𝑥 ∈ (𝑧𝐴) ∧ ∃𝑦 𝑦 = (𝑧𝐴)) ↔ (𝑥𝑧𝑥𝐴))
3526, 27, 343bitri 289 . . . . . . . . . . 11 (∃𝑦(𝑥𝑦𝑦 = (𝑧𝐴)) ↔ (𝑥𝑧𝑥𝐴))
3635anbi2i 617 . . . . . . . . . 10 ((𝑧𝑋 ∧ ∃𝑦(𝑥𝑦𝑦 = (𝑧𝐴))) ↔ (𝑧𝑋 ∧ (𝑥𝑧𝑥𝐴)))
37 biid 253 . . . . . . . . . . 11 ((𝑥𝑧𝑥𝐴) ↔ (𝑥𝑧𝑥𝐴))
3837bianass 633 . . . . . . . . . 10 ((𝑧𝑋 ∧ (𝑥𝑧𝑥𝐴)) ↔ ((𝑧𝑋𝑥𝑧) ∧ 𝑥𝐴))
39 ancom 453 . . . . . . . . . . 11 ((𝑧𝑋𝑥𝑧) ↔ (𝑥𝑧𝑧𝑋))
4039anbi1i 618 . . . . . . . . . 10 (((𝑧𝑋𝑥𝑧) ∧ 𝑥𝐴) ↔ ((𝑥𝑧𝑧𝑋) ∧ 𝑥𝐴))
4136, 38, 403bitri 289 . . . . . . . . 9 ((𝑧𝑋 ∧ ∃𝑦(𝑥𝑦𝑦 = (𝑧𝐴))) ↔ ((𝑥𝑧𝑧𝑋) ∧ 𝑥𝐴))
4217, 18, 413bitri 289 . . . . . . . 8 (∃𝑦(𝑥𝑦 ∧ (𝑧𝑋𝑦 = (𝑧𝐴))) ↔ ((𝑥𝑧𝑧𝑋) ∧ 𝑥𝐴))
4342exbii 1944 . . . . . . 7 (∃𝑧𝑦(𝑥𝑦 ∧ (𝑧𝑋𝑦 = (𝑧𝐴))) ↔ ∃𝑧((𝑥𝑧𝑧𝑋) ∧ 𝑥𝐴))
44 19.41v 2045 . . . . . . 7 (∃𝑧((𝑥𝑧𝑧𝑋) ∧ 𝑥𝐴) ↔ (∃𝑧(𝑥𝑧𝑧𝑋) ∧ 𝑥𝐴))
4543, 44bitri 267 . . . . . 6 (∃𝑧𝑦(𝑥𝑦 ∧ (𝑧𝑋𝑦 = (𝑧𝐴))) ↔ (∃𝑧(𝑥𝑧𝑧𝑋) ∧ 𝑥𝐴))
4614, 15, 453bitri 289 . . . . 5 (∃𝑦(𝑥𝑦 ∧ ∃𝑧𝑋 𝑦 = (𝑧𝐴)) ↔ (∃𝑧(𝑥𝑧𝑧𝑋) ∧ 𝑥𝐴))
47 elin 3994 . . . . 5 (𝑥 ∈ ( 𝑋𝐴) ↔ (𝑥 𝑋𝑥𝐴))
488, 46, 473bitr4g 306 . . . 4 ((𝑋𝑉𝐴𝑊) → (∃𝑦(𝑥𝑦 ∧ ∃𝑧𝑋 𝑦 = (𝑧𝐴)) ↔ 𝑥 ∈ ( 𝑋𝐴)))
494, 48bitrd 271 . . 3 ((𝑋𝑉𝐴𝑊) → (∃𝑦(𝑥𝑦𝑦 ∈ (𝑋t 𝐴)) ↔ 𝑥 ∈ ( 𝑋𝐴)))
501, 49syl5bb 275 . 2 ((𝑋𝑉𝐴𝑊) → (𝑥 (𝑋t 𝐴) ↔ 𝑥 ∈ ( 𝑋𝐴)))
5150eqrdv 2797 1 ((𝑋𝑉𝐴𝑊) → (𝑋t 𝐴) = ( 𝑋𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 385   = wceq 1653  wex 1875  wcel 2157  wrex 3090  cin 3768   cuni 4628  (class class class)co 6878  t crest 16396
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-rep 4964  ax-sep 4975  ax-nul 4983  ax-pr 5097  ax-un 7183
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ne 2972  df-ral 3094  df-rex 3095  df-reu 3096  df-rab 3098  df-v 3387  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-sn 4369  df-pr 4371  df-op 4375  df-uni 4629  df-iun 4712  df-br 4844  df-opab 4906  df-mpt 4923  df-id 5220  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-rn 5323  df-res 5324  df-ima 5325  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-f1 6106  df-fo 6107  df-f1o 6108  df-fv 6109  df-ov 6881  df-oprab 6882  df-mpt2 6883  df-rest 16398
This theorem is referenced by:  bj-restuni2  33544
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