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Theorem bj-restpw 33476
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 21265 (which uses distop 21082 and restopn2 21264). (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 5016 . . . 4 (𝑌𝑉 → 𝒫 𝑌 ∈ V)
2 elrest 16357 . . . 4 ((𝒫 𝑌 ∈ V ∧ 𝐴𝑊) → (𝑥 ∈ (𝒫 𝑌t 𝐴) ↔ ∃𝑦 ∈ 𝒫 𝑌𝑥 = (𝑦𝐴)))
31, 2sylan 575 . . 3 ((𝑌𝑉𝐴𝑊) → (𝑥 ∈ (𝒫 𝑌t 𝐴) ↔ ∃𝑦 ∈ 𝒫 𝑌𝑥 = (𝑦𝐴)))
4 selpw 4324 . . . . . . 7 (𝑦 ∈ 𝒫 𝑌𝑦𝑌)
54anbi1i 617 . . . . . 6 ((𝑦 ∈ 𝒫 𝑌𝑥 = (𝑦𝐴)) ↔ (𝑦𝑌𝑥 = (𝑦𝐴)))
65exbii 1943 . . . . 5 (∃𝑦(𝑦 ∈ 𝒫 𝑌𝑥 = (𝑦𝐴)) ↔ ∃𝑦(𝑦𝑌𝑥 = (𝑦𝐴)))
7 sstr2 3770 . . . . . . . . . 10 (𝑥𝑦 → (𝑦𝑌𝑥𝑌))
87com12 32 . . . . . . . . 9 (𝑦𝑌 → (𝑥𝑦𝑥𝑌))
9 inss1 3994 . . . . . . . . . 10 (𝑦𝐴) ⊆ 𝑦
10 sseq1 3788 . . . . . . . . . 10 (𝑥 = (𝑦𝐴) → (𝑥𝑦 ↔ (𝑦𝐴) ⊆ 𝑦))
119, 10mpbiri 249 . . . . . . . . 9 (𝑥 = (𝑦𝐴) → 𝑥𝑦)
128, 11impel 501 . . . . . . . 8 ((𝑦𝑌𝑥 = (𝑦𝐴)) → 𝑥𝑌)
13 inss2 3995 . . . . . . . . . 10 (𝑦𝐴) ⊆ 𝐴
14 sseq1 3788 . . . . . . . . . 10 (𝑥 = (𝑦𝐴) → (𝑥𝐴 ↔ (𝑦𝐴) ⊆ 𝐴))
1513, 14mpbiri 249 . . . . . . . . 9 (𝑥 = (𝑦𝐴) → 𝑥𝐴)
1615adantl 473 . . . . . . . 8 ((𝑦𝑌𝑥 = (𝑦𝐴)) → 𝑥𝐴)
1712, 16ssind 3998 . . . . . . 7 ((𝑦𝑌𝑥 = (𝑦𝐴)) → 𝑥 ⊆ (𝑌𝐴))
1817exlimiv 2025 . . . . . 6 (∃𝑦(𝑦𝑌𝑥 = (𝑦𝐴)) → 𝑥 ⊆ (𝑌𝐴))
19 inss1 3994 . . . . . . . 8 (𝑌𝐴) ⊆ 𝑌
20 sstr2 3770 . . . . . . . 8 (𝑥 ⊆ (𝑌𝐴) → ((𝑌𝐴) ⊆ 𝑌𝑥𝑌))
2119, 20mpi 20 . . . . . . 7 (𝑥 ⊆ (𝑌𝐴) → 𝑥𝑌)
22 inss2 3995 . . . . . . . 8 (𝑌𝐴) ⊆ 𝐴
23 sstr2 3770 . . . . . . . 8 (𝑥 ⊆ (𝑌𝐴) → ((𝑌𝐴) ⊆ 𝐴𝑥𝐴))
2422, 23mpi 20 . . . . . . 7 (𝑥 ⊆ (𝑌𝐴) → 𝑥𝐴)
25 ssidd 3786 . . . . . . . . . 10 (𝑥𝐴𝑥𝑥)
26 id 22 . . . . . . . . . 10 (𝑥𝐴𝑥𝐴)
2725, 26ssind 3998 . . . . . . . . 9 (𝑥𝐴𝑥 ⊆ (𝑥𝐴))
28 inss1 3994 . . . . . . . . . 10 (𝑥𝐴) ⊆ 𝑥
2928a1i 11 . . . . . . . . 9 (𝑥𝐴 → (𝑥𝐴) ⊆ 𝑥)
3027, 29eqssd 3780 . . . . . . . 8 (𝑥𝐴𝑥 = (𝑥𝐴))
31 vex 3353 . . . . . . . . 9 𝑥 ∈ V
32 sseq1 3788 . . . . . . . . . 10 (𝑦 = 𝑥 → (𝑦𝑌𝑥𝑌))
33 ineq1 3971 . . . . . . . . . . 11 (𝑦 = 𝑥 → (𝑦𝐴) = (𝑥𝐴))
3433eqeq2d 2775 . . . . . . . . . 10 (𝑦 = 𝑥 → (𝑥 = (𝑦𝐴) ↔ 𝑥 = (𝑥𝐴)))
3532, 34anbi12d 624 . . . . . . . . 9 (𝑦 = 𝑥 → ((𝑦𝑌𝑥 = (𝑦𝐴)) ↔ (𝑥𝑌𝑥 = (𝑥𝐴))))
3631, 35spcev 3453 . . . . . . . 8 ((𝑥𝑌𝑥 = (𝑥𝐴)) → ∃𝑦(𝑦𝑌𝑥 = (𝑦𝐴)))
3730, 36sylan2 586 . . . . . . 7 ((𝑥𝑌𝑥𝐴) → ∃𝑦(𝑦𝑌𝑥 = (𝑦𝐴)))
3821, 24, 37syl2anc 579 . . . . . 6 (𝑥 ⊆ (𝑌𝐴) → ∃𝑦(𝑦𝑌𝑥 = (𝑦𝐴)))
3918, 38impbii 200 . . . . 5 (∃𝑦(𝑦𝑌𝑥 = (𝑦𝐴)) ↔ 𝑥 ⊆ (𝑌𝐴))
406, 39bitri 266 . . . 4 (∃𝑦(𝑦 ∈ 𝒫 𝑌𝑥 = (𝑦𝐴)) ↔ 𝑥 ⊆ (𝑌𝐴))
41 df-rex 3061 . . . 4 (∃𝑦 ∈ 𝒫 𝑌𝑥 = (𝑦𝐴) ↔ ∃𝑦(𝑦 ∈ 𝒫 𝑌𝑥 = (𝑦𝐴)))
42 selpw 4324 . . . 4 (𝑥 ∈ 𝒫 (𝑌𝐴) ↔ 𝑥 ⊆ (𝑌𝐴))
4340, 41, 423bitr4i 294 . . 3 (∃𝑦 ∈ 𝒫 𝑌𝑥 = (𝑦𝐴) ↔ 𝑥 ∈ 𝒫 (𝑌𝐴))
443, 43syl6bb 278 . 2 ((𝑌𝑉𝐴𝑊) → (𝑥 ∈ (𝒫 𝑌t 𝐴) ↔ 𝑥 ∈ 𝒫 (𝑌𝐴)))
4544eqrdv 2763 1 ((𝑌𝑉𝐴𝑊) → (𝒫 𝑌t 𝐴) = 𝒫 (𝑌𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384   = wceq 1652  wex 1874  wcel 2155  wrex 3056  Vcvv 3350  cin 3733  wss 3734  𝒫 cpw 4317  (class class class)co 6844  t crest 16350
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4932  ax-sep 4943  ax-nul 4951  ax-pow 5003  ax-pr 5064  ax-un 7149
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-reu 3062  df-rab 3064  df-v 3352  df-sbc 3599  df-csb 3694  df-dif 3737  df-un 3739  df-in 3741  df-ss 3748  df-nul 4082  df-if 4246  df-pw 4319  df-sn 4337  df-pr 4339  df-op 4343  df-uni 4597  df-iun 4680  df-br 4812  df-opab 4874  df-mpt 4891  df-id 5187  df-xp 5285  df-rel 5286  df-cnv 5287  df-co 5288  df-dm 5289  df-rn 5290  df-res 5291  df-ima 5292  df-iota 6033  df-fun 6072  df-fn 6073  df-f 6074  df-f1 6075  df-fo 6076  df-f1o 6077  df-fv 6078  df-ov 6847  df-oprab 6848  df-mpt2 6849  df-rest 16352
This theorem is referenced by: (None)
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