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Theorem bj-restpw 33170
Description: The elementwise intersection on a powerset is the powerset of the intersection. This allows to prove for instance that the topology induced on a subset by the discrete topology is the discrete topology on that subset. See also restdis 21030 (which uses distop 20847 and restopn2 21029). (Contributed by BJ, 27-Apr-2021.)
Assertion
Ref Expression
bj-restpw ((𝑌𝑉𝐴𝑊) → (𝒫 𝑌t 𝐴) = 𝒫 (𝑌𝐴))

Proof of Theorem bj-restpw
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 pwexg 4880 . . . 4 (𝑌𝑉 → 𝒫 𝑌 ∈ V)
2 elrest 16135 . . . 4 ((𝒫 𝑌 ∈ V ∧ 𝐴𝑊) → (𝑥 ∈ (𝒫 𝑌t 𝐴) ↔ ∃𝑦 ∈ 𝒫 𝑌𝑥 = (𝑦𝐴)))
31, 2sylan 487 . . 3 ((𝑌𝑉𝐴𝑊) → (𝑥 ∈ (𝒫 𝑌t 𝐴) ↔ ∃𝑦 ∈ 𝒫 𝑌𝑥 = (𝑦𝐴)))
4 selpw 4198 . . . . . . 7 (𝑦 ∈ 𝒫 𝑌𝑦𝑌)
54anbi1i 731 . . . . . 6 ((𝑦 ∈ 𝒫 𝑌𝑥 = (𝑦𝐴)) ↔ (𝑦𝑌𝑥 = (𝑦𝐴)))
65exbii 1814 . . . . 5 (∃𝑦(𝑦 ∈ 𝒫 𝑌𝑥 = (𝑦𝐴)) ↔ ∃𝑦(𝑦𝑌𝑥 = (𝑦𝐴)))
7 sstr2 3643 . . . . . . . . . 10 (𝑥𝑦 → (𝑦𝑌𝑥𝑌))
87com12 32 . . . . . . . . 9 (𝑦𝑌 → (𝑥𝑦𝑥𝑌))
9 inss1 3866 . . . . . . . . . 10 (𝑦𝐴) ⊆ 𝑦
10 sseq1 3659 . . . . . . . . . 10 (𝑥 = (𝑦𝐴) → (𝑥𝑦 ↔ (𝑦𝐴) ⊆ 𝑦))
119, 10mpbiri 248 . . . . . . . . 9 (𝑥 = (𝑦𝐴) → 𝑥𝑦)
128, 11impel 484 . . . . . . . 8 ((𝑦𝑌𝑥 = (𝑦𝐴)) → 𝑥𝑌)
13 inss2 3867 . . . . . . . . . 10 (𝑦𝐴) ⊆ 𝐴
14 sseq1 3659 . . . . . . . . . 10 (𝑥 = (𝑦𝐴) → (𝑥𝐴 ↔ (𝑦𝐴) ⊆ 𝐴))
1513, 14mpbiri 248 . . . . . . . . 9 (𝑥 = (𝑦𝐴) → 𝑥𝐴)
1615adantl 481 . . . . . . . 8 ((𝑦𝑌𝑥 = (𝑦𝐴)) → 𝑥𝐴)
1712, 16ssind 3870 . . . . . . 7 ((𝑦𝑌𝑥 = (𝑦𝐴)) → 𝑥 ⊆ (𝑌𝐴))
1817exlimiv 1898 . . . . . 6 (∃𝑦(𝑦𝑌𝑥 = (𝑦𝐴)) → 𝑥 ⊆ (𝑌𝐴))
19 inss1 3866 . . . . . . . 8 (𝑌𝐴) ⊆ 𝑌
20 sstr2 3643 . . . . . . . 8 (𝑥 ⊆ (𝑌𝐴) → ((𝑌𝐴) ⊆ 𝑌𝑥𝑌))
2119, 20mpi 20 . . . . . . 7 (𝑥 ⊆ (𝑌𝐴) → 𝑥𝑌)
22 inss2 3867 . . . . . . . 8 (𝑌𝐴) ⊆ 𝐴
23 sstr2 3643 . . . . . . . 8 (𝑥 ⊆ (𝑌𝐴) → ((𝑌𝐴) ⊆ 𝐴𝑥𝐴))
2422, 23mpi 20 . . . . . . 7 (𝑥 ⊆ (𝑌𝐴) → 𝑥𝐴)
25 ssid 3657 . . . . . . . . . . 11 𝑥𝑥
2625a1i 11 . . . . . . . . . 10 (𝑥𝐴𝑥𝑥)
27 id 22 . . . . . . . . . 10 (𝑥𝐴𝑥𝐴)
2826, 27ssind 3870 . . . . . . . . 9 (𝑥𝐴𝑥 ⊆ (𝑥𝐴))
29 inss1 3866 . . . . . . . . . 10 (𝑥𝐴) ⊆ 𝑥
3029a1i 11 . . . . . . . . 9 (𝑥𝐴 → (𝑥𝐴) ⊆ 𝑥)
3128, 30eqssd 3653 . . . . . . . 8 (𝑥𝐴𝑥 = (𝑥𝐴))
32 vex 3234 . . . . . . . . 9 𝑥 ∈ V
33 sseq1 3659 . . . . . . . . . 10 (𝑦 = 𝑥 → (𝑦𝑌𝑥𝑌))
34 ineq1 3840 . . . . . . . . . . 11 (𝑦 = 𝑥 → (𝑦𝐴) = (𝑥𝐴))
3534eqeq2d 2661 . . . . . . . . . 10 (𝑦 = 𝑥 → (𝑥 = (𝑦𝐴) ↔ 𝑥 = (𝑥𝐴)))
3633, 35anbi12d 747 . . . . . . . . 9 (𝑦 = 𝑥 → ((𝑦𝑌𝑥 = (𝑦𝐴)) ↔ (𝑥𝑌𝑥 = (𝑥𝐴))))
3732, 36spcev 3331 . . . . . . . 8 ((𝑥𝑌𝑥 = (𝑥𝐴)) → ∃𝑦(𝑦𝑌𝑥 = (𝑦𝐴)))
3831, 37sylan2 490 . . . . . . 7 ((𝑥𝑌𝑥𝐴) → ∃𝑦(𝑦𝑌𝑥 = (𝑦𝐴)))
3921, 24, 38syl2anc 694 . . . . . 6 (𝑥 ⊆ (𝑌𝐴) → ∃𝑦(𝑦𝑌𝑥 = (𝑦𝐴)))
4018, 39impbii 199 . . . . 5 (∃𝑦(𝑦𝑌𝑥 = (𝑦𝐴)) ↔ 𝑥 ⊆ (𝑌𝐴))
416, 40bitri 264 . . . 4 (∃𝑦(𝑦 ∈ 𝒫 𝑌𝑥 = (𝑦𝐴)) ↔ 𝑥 ⊆ (𝑌𝐴))
42 df-rex 2947 . . . 4 (∃𝑦 ∈ 𝒫 𝑌𝑥 = (𝑦𝐴) ↔ ∃𝑦(𝑦 ∈ 𝒫 𝑌𝑥 = (𝑦𝐴)))
43 selpw 4198 . . . 4 (𝑥 ∈ 𝒫 (𝑌𝐴) ↔ 𝑥 ⊆ (𝑌𝐴))
4441, 42, 433bitr4i 292 . . 3 (∃𝑦 ∈ 𝒫 𝑌𝑥 = (𝑦𝐴) ↔ 𝑥 ∈ 𝒫 (𝑌𝐴))
453, 44syl6bb 276 . 2 ((𝑌𝑉𝐴𝑊) → (𝑥 ∈ (𝒫 𝑌t 𝐴) ↔ 𝑥 ∈ 𝒫 (𝑌𝐴)))
4645eqrdv 2649 1 ((𝑌𝑉𝐴𝑊) → (𝒫 𝑌t 𝐴) = 𝒫 (𝑌𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1523  wex 1744  wcel 2030  wrex 2942  Vcvv 3231  cin 3606  wss 3607  𝒫 cpw 4191  (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-pow 4873  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-pw 4193  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: (None)
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