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Theorem bj-restpw 34003
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 21474 (which uses distop 21291 and restopn2 21473). (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 5177 . . . 4 (𝑌𝑉 → 𝒫 𝑌 ∈ V)
2 elrest 16534 . . . 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 selpw 4466 . . . . . . 7 (𝑦 ∈ 𝒫 𝑌𝑦𝑌)
54anbi1i 623 . . . . . 6 ((𝑦 ∈ 𝒫 𝑌𝑥 = (𝑦𝐴)) ↔ (𝑦𝑌𝑥 = (𝑦𝐴)))
65exbii 1833 . . . . 5 (∃𝑦(𝑦 ∈ 𝒫 𝑌𝑥 = (𝑦𝐴)) ↔ ∃𝑦(𝑦𝑌𝑥 = (𝑦𝐴)))
7 sstr2 3902 . . . . . . . . . 10 (𝑥𝑦 → (𝑦𝑌𝑥𝑌))
87com12 32 . . . . . . . . 9 (𝑦𝑌 → (𝑥𝑦𝑥𝑌))
9 inss1 4131 . . . . . . . . . 10 (𝑦𝐴) ⊆ 𝑦
10 sseq1 3919 . . . . . . . . . 10 (𝑥 = (𝑦𝐴) → (𝑥𝑦 ↔ (𝑦𝐴) ⊆ 𝑦))
119, 10mpbiri 259 . . . . . . . . 9 (𝑥 = (𝑦𝐴) → 𝑥𝑦)
128, 11impel 506 . . . . . . . 8 ((𝑦𝑌𝑥 = (𝑦𝐴)) → 𝑥𝑌)
13 inss2 4132 . . . . . . . . . 10 (𝑦𝐴) ⊆ 𝐴
14 sseq1 3919 . . . . . . . . . 10 (𝑥 = (𝑦𝐴) → (𝑥𝐴 ↔ (𝑦𝐴) ⊆ 𝐴))
1513, 14mpbiri 259 . . . . . . . . 9 (𝑥 = (𝑦𝐴) → 𝑥𝐴)
1615adantl 482 . . . . . . . 8 ((𝑦𝑌𝑥 = (𝑦𝐴)) → 𝑥𝐴)
1712, 16ssind 4135 . . . . . . 7 ((𝑦𝑌𝑥 = (𝑦𝐴)) → 𝑥 ⊆ (𝑌𝐴))
1817exlimiv 1912 . . . . . 6 (∃𝑦(𝑦𝑌𝑥 = (𝑦𝐴)) → 𝑥 ⊆ (𝑌𝐴))
19 inss1 4131 . . . . . . . 8 (𝑌𝐴) ⊆ 𝑌
20 sstr2 3902 . . . . . . . 8 (𝑥 ⊆ (𝑌𝐴) → ((𝑌𝐴) ⊆ 𝑌𝑥𝑌))
2119, 20mpi 20 . . . . . . 7 (𝑥 ⊆ (𝑌𝐴) → 𝑥𝑌)
22 inss2 4132 . . . . . . . 8 (𝑌𝐴) ⊆ 𝐴
23 sstr2 3902 . . . . . . . 8 (𝑥 ⊆ (𝑌𝐴) → ((𝑌𝐴) ⊆ 𝐴𝑥𝐴))
2422, 23mpi 20 . . . . . . 7 (𝑥 ⊆ (𝑌𝐴) → 𝑥𝐴)
25 ssidd 3917 . . . . . . . . . 10 (𝑥𝐴𝑥𝑥)
26 id 22 . . . . . . . . . 10 (𝑥𝐴𝑥𝐴)
2725, 26ssind 4135 . . . . . . . . 9 (𝑥𝐴𝑥 ⊆ (𝑥𝐴))
28 inss1 4131 . . . . . . . . . 10 (𝑥𝐴) ⊆ 𝑥
2928a1i 11 . . . . . . . . 9 (𝑥𝐴 → (𝑥𝐴) ⊆ 𝑥)
3027, 29eqssd 3912 . . . . . . . 8 (𝑥𝐴𝑥 = (𝑥𝐴))
31 vex 3443 . . . . . . . . 9 𝑥 ∈ V
32 sseq1 3919 . . . . . . . . . 10 (𝑦 = 𝑥 → (𝑦𝑌𝑥𝑌))
33 ineq1 4107 . . . . . . . . . . 11 (𝑦 = 𝑥 → (𝑦𝐴) = (𝑥𝐴))
3433eqeq2d 2807 . . . . . . . . . 10 (𝑦 = 𝑥 → (𝑥 = (𝑦𝐴) ↔ 𝑥 = (𝑥𝐴)))
3532, 34anbi12d 630 . . . . . . . . 9 (𝑦 = 𝑥 → ((𝑦𝑌𝑥 = (𝑦𝐴)) ↔ (𝑥𝑌𝑥 = (𝑥𝐴))))
3631, 35spcev 3551 . . . . . . . 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 3113 . . . 4 (∃𝑦 ∈ 𝒫 𝑌𝑥 = (𝑦𝐴) ↔ ∃𝑦(𝑦 ∈ 𝒫 𝑌𝑥 = (𝑦𝐴)))
42 selpw 4466 . . . 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 2795 1 ((𝑌𝑉𝐴𝑊) → (𝒫 𝑌t 𝐴) = 𝒫 (𝑌𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396   = wceq 1525  wex 1765  wcel 2083  wrex 3108  Vcvv 3440  cin 3864  wss 3865  𝒫 cpw 4459  (class class class)co 7023  t crest 16527
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1781  ax-4 1795  ax-5 1892  ax-6 1951  ax-7 1996  ax-8 2085  ax-9 2093  ax-10 2114  ax-11 2128  ax-12 2143  ax-13 2346  ax-ext 2771  ax-rep 5088  ax-sep 5101  ax-nul 5108  ax-pow 5164  ax-pr 5228  ax-un 7326
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1528  df-ex 1766  df-nf 1770  df-sb 2045  df-mo 2578  df-eu 2614  df-clab 2778  df-cleq 2790  df-clel 2865  df-nfc 2937  df-ne 2987  df-ral 3112  df-rex 3113  df-reu 3114  df-rab 3116  df-v 3442  df-sbc 3712  df-csb 3818  df-dif 3868  df-un 3870  df-in 3872  df-ss 3880  df-nul 4218  df-if 4388  df-pw 4461  df-sn 4479  df-pr 4481  df-op 4485  df-uni 4752  df-iun 4833  df-br 4969  df-opab 5031  df-mpt 5048  df-id 5355  df-xp 5456  df-rel 5457  df-cnv 5458  df-co 5459  df-dm 5460  df-rn 5461  df-res 5462  df-ima 5463  df-iota 6196  df-fun 6234  df-fn 6235  df-f 6236  df-f1 6237  df-fo 6238  df-f1o 6239  df-fv 6240  df-ov 7026  df-oprab 7027  df-mpo 7028  df-rest 16529
This theorem is referenced by: (None)
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