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Theorem bj-restpw 34526
 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 21793 (which uses distop 21610 and restopn2 21792). (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 5245 . . . 4 (𝑌𝑉 → 𝒫 𝑌 ∈ V)
2 elrest 16696 . . . 4 ((𝒫 𝑌 ∈ V ∧ 𝐴𝑊) → (𝑥 ∈ (𝒫 𝑌t 𝐴) ↔ ∃𝑦 ∈ 𝒫 𝑌𝑥 = (𝑦𝐴)))
31, 2sylan 583 . . 3 ((𝑌𝑉𝐴𝑊) → (𝑥 ∈ (𝒫 𝑌t 𝐴) ↔ ∃𝑦 ∈ 𝒫 𝑌𝑥 = (𝑦𝐴)))
4 velpw 4502 . . . . . . 7 (𝑦 ∈ 𝒫 𝑌𝑦𝑌)
54anbi1i 626 . . . . . 6 ((𝑦 ∈ 𝒫 𝑌𝑥 = (𝑦𝐴)) ↔ (𝑦𝑌𝑥 = (𝑦𝐴)))
65exbii 1849 . . . . 5 (∃𝑦(𝑦 ∈ 𝒫 𝑌𝑥 = (𝑦𝐴)) ↔ ∃𝑦(𝑦𝑌𝑥 = (𝑦𝐴)))
7 sstr2 3922 . . . . . . . . . 10 (𝑥𝑦 → (𝑦𝑌𝑥𝑌))
87com12 32 . . . . . . . . 9 (𝑦𝑌 → (𝑥𝑦𝑥𝑌))
9 inss1 4155 . . . . . . . . . 10 (𝑦𝐴) ⊆ 𝑦
10 sseq1 3940 . . . . . . . . . 10 (𝑥 = (𝑦𝐴) → (𝑥𝑦 ↔ (𝑦𝐴) ⊆ 𝑦))
119, 10mpbiri 261 . . . . . . . . 9 (𝑥 = (𝑦𝐴) → 𝑥𝑦)
128, 11impel 509 . . . . . . . 8 ((𝑦𝑌𝑥 = (𝑦𝐴)) → 𝑥𝑌)
13 inss2 4156 . . . . . . . . . 10 (𝑦𝐴) ⊆ 𝐴
14 sseq1 3940 . . . . . . . . . 10 (𝑥 = (𝑦𝐴) → (𝑥𝐴 ↔ (𝑦𝐴) ⊆ 𝐴))
1513, 14mpbiri 261 . . . . . . . . 9 (𝑥 = (𝑦𝐴) → 𝑥𝐴)
1615adantl 485 . . . . . . . 8 ((𝑦𝑌𝑥 = (𝑦𝐴)) → 𝑥𝐴)
1712, 16ssind 4159 . . . . . . 7 ((𝑦𝑌𝑥 = (𝑦𝐴)) → 𝑥 ⊆ (𝑌𝐴))
1817exlimiv 1931 . . . . . 6 (∃𝑦(𝑦𝑌𝑥 = (𝑦𝐴)) → 𝑥 ⊆ (𝑌𝐴))
19 inss1 4155 . . . . . . . 8 (𝑌𝐴) ⊆ 𝑌
20 sstr2 3922 . . . . . . . 8 (𝑥 ⊆ (𝑌𝐴) → ((𝑌𝐴) ⊆ 𝑌𝑥𝑌))
2119, 20mpi 20 . . . . . . 7 (𝑥 ⊆ (𝑌𝐴) → 𝑥𝑌)
22 inss2 4156 . . . . . . . 8 (𝑌𝐴) ⊆ 𝐴
23 sstr2 3922 . . . . . . . 8 (𝑥 ⊆ (𝑌𝐴) → ((𝑌𝐴) ⊆ 𝐴𝑥𝐴))
2422, 23mpi 20 . . . . . . 7 (𝑥 ⊆ (𝑌𝐴) → 𝑥𝐴)
25 ssidd 3938 . . . . . . . . . 10 (𝑥𝐴𝑥𝑥)
26 id 22 . . . . . . . . . 10 (𝑥𝐴𝑥𝐴)
2725, 26ssind 4159 . . . . . . . . 9 (𝑥𝐴𝑥 ⊆ (𝑥𝐴))
28 inss1 4155 . . . . . . . . . 10 (𝑥𝐴) ⊆ 𝑥
2928a1i 11 . . . . . . . . 9 (𝑥𝐴 → (𝑥𝐴) ⊆ 𝑥)
3027, 29eqssd 3932 . . . . . . . 8 (𝑥𝐴𝑥 = (𝑥𝐴))
31 vex 3444 . . . . . . . . 9 𝑥 ∈ V
32 sseq1 3940 . . . . . . . . . 10 (𝑦 = 𝑥 → (𝑦𝑌𝑥𝑌))
33 ineq1 4131 . . . . . . . . . . 11 (𝑦 = 𝑥 → (𝑦𝐴) = (𝑥𝐴))
3433eqeq2d 2809 . . . . . . . . . 10 (𝑦 = 𝑥 → (𝑥 = (𝑦𝐴) ↔ 𝑥 = (𝑥𝐴)))
3532, 34anbi12d 633 . . . . . . . . 9 (𝑦 = 𝑥 → ((𝑦𝑌𝑥 = (𝑦𝐴)) ↔ (𝑥𝑌𝑥 = (𝑥𝐴))))
3631, 35spcev 3555 . . . . . . . 8 ((𝑥𝑌𝑥 = (𝑥𝐴)) → ∃𝑦(𝑦𝑌𝑥 = (𝑦𝐴)))
3730, 36sylan2 595 . . . . . . 7 ((𝑥𝑌𝑥𝐴) → ∃𝑦(𝑦𝑌𝑥 = (𝑦𝐴)))
3821, 24, 37syl2anc 587 . . . . . 6 (𝑥 ⊆ (𝑌𝐴) → ∃𝑦(𝑦𝑌𝑥 = (𝑦𝐴)))
3918, 38impbii 212 . . . . 5 (∃𝑦(𝑦𝑌𝑥 = (𝑦𝐴)) ↔ 𝑥 ⊆ (𝑌𝐴))
406, 39bitri 278 . . . 4 (∃𝑦(𝑦 ∈ 𝒫 𝑌𝑥 = (𝑦𝐴)) ↔ 𝑥 ⊆ (𝑌𝐴))
41 df-rex 3112 . . . 4 (∃𝑦 ∈ 𝒫 𝑌𝑥 = (𝑦𝐴) ↔ ∃𝑦(𝑦 ∈ 𝒫 𝑌𝑥 = (𝑦𝐴)))
42 velpw 4502 . . . 4 (𝑥 ∈ 𝒫 (𝑌𝐴) ↔ 𝑥 ⊆ (𝑌𝐴))
4340, 41, 423bitr4i 306 . . 3 (∃𝑦 ∈ 𝒫 𝑌𝑥 = (𝑦𝐴) ↔ 𝑥 ∈ 𝒫 (𝑌𝐴))
443, 43syl6bb 290 . 2 ((𝑌𝑉𝐴𝑊) → (𝑥 ∈ (𝒫 𝑌t 𝐴) ↔ 𝑥 ∈ 𝒫 (𝑌𝐴)))
4544eqrdv 2796 1 ((𝑌𝑉𝐴𝑊) → (𝒫 𝑌t 𝐴) = 𝒫 (𝑌𝐴))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538  ∃wex 1781   ∈ wcel 2111  ∃wrex 3107  Vcvv 3441   ∩ cin 3880   ⊆ wss 3881  𝒫 cpw 4497  (class class class)co 7136   ↾t crest 16689 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5155  ax-sep 5168  ax-nul 5175  ax-pow 5232  ax-pr 5296  ax-un 7444 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-iun 4884  df-br 5032  df-opab 5094  df-mpt 5112  df-id 5426  df-xp 5526  df-rel 5527  df-cnv 5528  df-co 5529  df-dm 5530  df-rn 5531  df-res 5532  df-ima 5533  df-iota 6284  df-fun 6327  df-fn 6328  df-f 6329  df-f1 6330  df-fo 6331  df-f1o 6332  df-fv 6333  df-ov 7139  df-oprab 7140  df-mpo 7141  df-rest 16691 This theorem is referenced by: (None)
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