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Theorem bj-restpw 34277
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 21714 (which uses distop 21531 and restopn2 21713). (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 5270 . . . 4 (𝑌𝑉 → 𝒫 𝑌 ∈ V)
2 elrest 16689 . . . 4 ((𝒫 𝑌 ∈ V ∧ 𝐴𝑊) → (𝑥 ∈ (𝒫 𝑌t 𝐴) ↔ ∃𝑦 ∈ 𝒫 𝑌𝑥 = (𝑦𝐴)))
31, 2sylan 580 . . 3 ((𝑌𝑉𝐴𝑊) → (𝑥 ∈ (𝒫 𝑌t 𝐴) ↔ ∃𝑦 ∈ 𝒫 𝑌𝑥 = (𝑦𝐴)))
4 velpw 4543 . . . . . . 7 (𝑦 ∈ 𝒫 𝑌𝑦𝑌)
54anbi1i 623 . . . . . 6 ((𝑦 ∈ 𝒫 𝑌𝑥 = (𝑦𝐴)) ↔ (𝑦𝑌𝑥 = (𝑦𝐴)))
65exbii 1839 . . . . 5 (∃𝑦(𝑦 ∈ 𝒫 𝑌𝑥 = (𝑦𝐴)) ↔ ∃𝑦(𝑦𝑌𝑥 = (𝑦𝐴)))
7 sstr2 3971 . . . . . . . . . 10 (𝑥𝑦 → (𝑦𝑌𝑥𝑌))
87com12 32 . . . . . . . . 9 (𝑦𝑌 → (𝑥𝑦𝑥𝑌))
9 inss1 4202 . . . . . . . . . 10 (𝑦𝐴) ⊆ 𝑦
10 sseq1 3989 . . . . . . . . . 10 (𝑥 = (𝑦𝐴) → (𝑥𝑦 ↔ (𝑦𝐴) ⊆ 𝑦))
119, 10mpbiri 259 . . . . . . . . 9 (𝑥 = (𝑦𝐴) → 𝑥𝑦)
128, 11impel 506 . . . . . . . 8 ((𝑦𝑌𝑥 = (𝑦𝐴)) → 𝑥𝑌)
13 inss2 4203 . . . . . . . . . 10 (𝑦𝐴) ⊆ 𝐴
14 sseq1 3989 . . . . . . . . . 10 (𝑥 = (𝑦𝐴) → (𝑥𝐴 ↔ (𝑦𝐴) ⊆ 𝐴))
1513, 14mpbiri 259 . . . . . . . . 9 (𝑥 = (𝑦𝐴) → 𝑥𝐴)
1615adantl 482 . . . . . . . 8 ((𝑦𝑌𝑥 = (𝑦𝐴)) → 𝑥𝐴)
1712, 16ssind 4206 . . . . . . 7 ((𝑦𝑌𝑥 = (𝑦𝐴)) → 𝑥 ⊆ (𝑌𝐴))
1817exlimiv 1922 . . . . . 6 (∃𝑦(𝑦𝑌𝑥 = (𝑦𝐴)) → 𝑥 ⊆ (𝑌𝐴))
19 inss1 4202 . . . . . . . 8 (𝑌𝐴) ⊆ 𝑌
20 sstr2 3971 . . . . . . . 8 (𝑥 ⊆ (𝑌𝐴) → ((𝑌𝐴) ⊆ 𝑌𝑥𝑌))
2119, 20mpi 20 . . . . . . 7 (𝑥 ⊆ (𝑌𝐴) → 𝑥𝑌)
22 inss2 4203 . . . . . . . 8 (𝑌𝐴) ⊆ 𝐴
23 sstr2 3971 . . . . . . . 8 (𝑥 ⊆ (𝑌𝐴) → ((𝑌𝐴) ⊆ 𝐴𝑥𝐴))
2422, 23mpi 20 . . . . . . 7 (𝑥 ⊆ (𝑌𝐴) → 𝑥𝐴)
25 ssidd 3987 . . . . . . . . . 10 (𝑥𝐴𝑥𝑥)
26 id 22 . . . . . . . . . 10 (𝑥𝐴𝑥𝐴)
2725, 26ssind 4206 . . . . . . . . 9 (𝑥𝐴𝑥 ⊆ (𝑥𝐴))
28 inss1 4202 . . . . . . . . . 10 (𝑥𝐴) ⊆ 𝑥
2928a1i 11 . . . . . . . . 9 (𝑥𝐴 → (𝑥𝐴) ⊆ 𝑥)
3027, 29eqssd 3981 . . . . . . . 8 (𝑥𝐴𝑥 = (𝑥𝐴))
31 vex 3495 . . . . . . . . 9 𝑥 ∈ V
32 sseq1 3989 . . . . . . . . . 10 (𝑦 = 𝑥 → (𝑦𝑌𝑥𝑌))
33 ineq1 4178 . . . . . . . . . . 11 (𝑦 = 𝑥 → (𝑦𝐴) = (𝑥𝐴))
3433eqeq2d 2829 . . . . . . . . . 10 (𝑦 = 𝑥 → (𝑥 = (𝑦𝐴) ↔ 𝑥 = (𝑥𝐴)))
3532, 34anbi12d 630 . . . . . . . . 9 (𝑦 = 𝑥 → ((𝑦𝑌𝑥 = (𝑦𝐴)) ↔ (𝑥𝑌𝑥 = (𝑥𝐴))))
3631, 35spcev 3604 . . . . . . . 8 ((𝑥𝑌𝑥 = (𝑥𝐴)) → ∃𝑦(𝑦𝑌𝑥 = (𝑦𝐴)))
3730, 36sylan2 592 . . . . . . 7 ((𝑥𝑌𝑥𝐴) → ∃𝑦(𝑦𝑌𝑥 = (𝑦𝐴)))
3821, 24, 37syl2anc 584 . . . . . 6 (𝑥 ⊆ (𝑌𝐴) → ∃𝑦(𝑦𝑌𝑥 = (𝑦𝐴)))
3918, 38impbii 210 . . . . 5 (∃𝑦(𝑦𝑌𝑥 = (𝑦𝐴)) ↔ 𝑥 ⊆ (𝑌𝐴))
406, 39bitri 276 . . . 4 (∃𝑦(𝑦 ∈ 𝒫 𝑌𝑥 = (𝑦𝐴)) ↔ 𝑥 ⊆ (𝑌𝐴))
41 df-rex 3141 . . . 4 (∃𝑦 ∈ 𝒫 𝑌𝑥 = (𝑦𝐴) ↔ ∃𝑦(𝑦 ∈ 𝒫 𝑌𝑥 = (𝑦𝐴)))
42 velpw 4543 . . . 4 (𝑥 ∈ 𝒫 (𝑌𝐴) ↔ 𝑥 ⊆ (𝑌𝐴))
4340, 41, 423bitr4i 304 . . 3 (∃𝑦 ∈ 𝒫 𝑌𝑥 = (𝑦𝐴) ↔ 𝑥 ∈ 𝒫 (𝑌𝐴))
443, 43syl6bb 288 . 2 ((𝑌𝑉𝐴𝑊) → (𝑥 ∈ (𝒫 𝑌t 𝐴) ↔ 𝑥 ∈ 𝒫 (𝑌𝐴)))
4544eqrdv 2816 1 ((𝑌𝑉𝐴𝑊) → (𝒫 𝑌t 𝐴) = 𝒫 (𝑌𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396   = wceq 1528  wex 1771  wcel 2105  wrex 3136  Vcvv 3492  cin 3932  wss 3933  𝒫 cpw 4535  (class class class)co 7145  t crest 16682
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-ov 7148  df-oprab 7149  df-mpo 7150  df-rest 16684
This theorem is referenced by: (None)
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