Theorem bj-restpw 36059

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 22689 (which uses distop 22505 and restopn2 22688). (Contributed by BJ, 27-Apr-2021.)

Assertion

Ref	Expression
bj-restpw	⊢ ((𝑌 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) → (𝒫 𝑌 ↾t 𝐴) = 𝒫 (𝑌 ∩ 𝐴))

Proof of Theorem bj-restpw

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		pwexg 5376	. . . 4 ⊢ (𝑌 ∈ 𝑉 → 𝒫 𝑌 ∈ V)
2		elrest 17375	. . . 4 ⊢ ((𝒫 𝑌 ∈ V ∧ 𝐴 ∈ 𝑊) → (𝑥 ∈ (𝒫 𝑌 ↾t 𝐴) ↔ ∃𝑦 ∈ 𝒫 𝑌𝑥 = (𝑦 ∩ 𝐴)))
3	1, 2	sylan 580	. . 3 ⊢ ((𝑌 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) → (𝑥 ∈ (𝒫 𝑌 ↾t 𝐴) ↔ ∃𝑦 ∈ 𝒫 𝑌𝑥 = (𝑦 ∩ 𝐴)))
4		velpw 4607	. . . . . . 7 ⊢ (𝑦 ∈ 𝒫 𝑌 ↔ 𝑦 ⊆ 𝑌)
5	4	anbi1i 624	. . . . . 6 ⊢ ((𝑦 ∈ 𝒫 𝑌 ∧ 𝑥 = (𝑦 ∩ 𝐴)) ↔ (𝑦 ⊆ 𝑌 ∧ 𝑥 = (𝑦 ∩ 𝐴)))
6	5	exbii 1850	. . . . 5 ⊢ (∃𝑦(𝑦 ∈ 𝒫 𝑌 ∧ 𝑥 = (𝑦 ∩ 𝐴)) ↔ ∃𝑦(𝑦 ⊆ 𝑌 ∧ 𝑥 = (𝑦 ∩ 𝐴)))
7		sstr2 3989	. . . . . . . . . 10 ⊢ (𝑥 ⊆ 𝑦 → (𝑦 ⊆ 𝑌 → 𝑥 ⊆ 𝑌))
8	7	com12 32	. . . . . . . . 9 ⊢ (𝑦 ⊆ 𝑌 → (𝑥 ⊆ 𝑦 → 𝑥 ⊆ 𝑌))
9		inss1 4228	. . . . . . . . . 10 ⊢ (𝑦 ∩ 𝐴) ⊆ 𝑦
10		sseq1 4007	. . . . . . . . . 10 ⊢ (𝑥 = (𝑦 ∩ 𝐴) → (𝑥 ⊆ 𝑦 ↔ (𝑦 ∩ 𝐴) ⊆ 𝑦))
11	9, 10	mpbiri 257	. . . . . . . . 9 ⊢ (𝑥 = (𝑦 ∩ 𝐴) → 𝑥 ⊆ 𝑦)
12	8, 11	impel 506	. . . . . . . 8 ⊢ ((𝑦 ⊆ 𝑌 ∧ 𝑥 = (𝑦 ∩ 𝐴)) → 𝑥 ⊆ 𝑌)
13		inss2 4229	. . . . . . . . . 10 ⊢ (𝑦 ∩ 𝐴) ⊆ 𝐴
14		sseq1 4007	. . . . . . . . . 10 ⊢ (𝑥 = (𝑦 ∩ 𝐴) → (𝑥 ⊆ 𝐴 ↔ (𝑦 ∩ 𝐴) ⊆ 𝐴))
15	13, 14	mpbiri 257	. . . . . . . . 9 ⊢ (𝑥 = (𝑦 ∩ 𝐴) → 𝑥 ⊆ 𝐴)
16	15	adantl 482	. . . . . . . 8 ⊢ ((𝑦 ⊆ 𝑌 ∧ 𝑥 = (𝑦 ∩ 𝐴)) → 𝑥 ⊆ 𝐴)
17	12, 16	ssind 4232	. . . . . . 7 ⊢ ((𝑦 ⊆ 𝑌 ∧ 𝑥 = (𝑦 ∩ 𝐴)) → 𝑥 ⊆ (𝑌 ∩ 𝐴))
18	17	exlimiv 1933	. . . . . 6 ⊢ (∃𝑦(𝑦 ⊆ 𝑌 ∧ 𝑥 = (𝑦 ∩ 𝐴)) → 𝑥 ⊆ (𝑌 ∩ 𝐴))
19		inss1 4228	. . . . . . . 8 ⊢ (𝑌 ∩ 𝐴) ⊆ 𝑌
20		sstr2 3989	. . . . . . . 8 ⊢ (𝑥 ⊆ (𝑌 ∩ 𝐴) → ((𝑌 ∩ 𝐴) ⊆ 𝑌 → 𝑥 ⊆ 𝑌))
21	19, 20	mpi 20	. . . . . . 7 ⊢ (𝑥 ⊆ (𝑌 ∩ 𝐴) → 𝑥 ⊆ 𝑌)
22		inss2 4229	. . . . . . . 8 ⊢ (𝑌 ∩ 𝐴) ⊆ 𝐴
23		sstr2 3989	. . . . . . . 8 ⊢ (𝑥 ⊆ (𝑌 ∩ 𝐴) → ((𝑌 ∩ 𝐴) ⊆ 𝐴 → 𝑥 ⊆ 𝐴))
24	22, 23	mpi 20	. . . . . . 7 ⊢ (𝑥 ⊆ (𝑌 ∩ 𝐴) → 𝑥 ⊆ 𝐴)
25		ssidd 4005	. . . . . . . . . 10 ⊢ (𝑥 ⊆ 𝐴 → 𝑥 ⊆ 𝑥)
26		id 22	. . . . . . . . . 10 ⊢ (𝑥 ⊆ 𝐴 → 𝑥 ⊆ 𝐴)
27	25, 26	ssind 4232	. . . . . . . . 9 ⊢ (𝑥 ⊆ 𝐴 → 𝑥 ⊆ (𝑥 ∩ 𝐴))
28		inss1 4228	. . . . . . . . . 10 ⊢ (𝑥 ∩ 𝐴) ⊆ 𝑥
29	28	a1i 11	. . . . . . . . 9 ⊢ (𝑥 ⊆ 𝐴 → (𝑥 ∩ 𝐴) ⊆ 𝑥)
30	27, 29	eqssd 3999	. . . . . . . 8 ⊢ (𝑥 ⊆ 𝐴 → 𝑥 = (𝑥 ∩ 𝐴))
31		vex 3478	. . . . . . . . 9 ⊢ 𝑥 ∈ V
32		sseq1 4007	. . . . . . . . . 10 ⊢ (𝑦 = 𝑥 → (𝑦 ⊆ 𝑌 ↔ 𝑥 ⊆ 𝑌))
33		ineq1 4205	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑥 → (𝑦 ∩ 𝐴) = (𝑥 ∩ 𝐴))
34	33	eqeq2d 2743	. . . . . . . . . 10 ⊢ (𝑦 = 𝑥 → (𝑥 = (𝑦 ∩ 𝐴) ↔ 𝑥 = (𝑥 ∩ 𝐴)))
35	32, 34	anbi12d 631	. . . . . . . . 9 ⊢ (𝑦 = 𝑥 → ((𝑦 ⊆ 𝑌 ∧ 𝑥 = (𝑦 ∩ 𝐴)) ↔ (𝑥 ⊆ 𝑌 ∧ 𝑥 = (𝑥 ∩ 𝐴))))
36	31, 35	spcev 3596	. . . . . . . 8 ⊢ ((𝑥 ⊆ 𝑌 ∧ 𝑥 = (𝑥 ∩ 𝐴)) → ∃𝑦(𝑦 ⊆ 𝑌 ∧ 𝑥 = (𝑦 ∩ 𝐴)))
37	30, 36	sylan2 593	. . . . . . 7 ⊢ ((𝑥 ⊆ 𝑌 ∧ 𝑥 ⊆ 𝐴) → ∃𝑦(𝑦 ⊆ 𝑌 ∧ 𝑥 = (𝑦 ∩ 𝐴)))
38	21, 24, 37	syl2anc 584	. . . . . 6 ⊢ (𝑥 ⊆ (𝑌 ∩ 𝐴) → ∃𝑦(𝑦 ⊆ 𝑌 ∧ 𝑥 = (𝑦 ∩ 𝐴)))
39	18, 38	impbii 208	. . . . 5 ⊢ (∃𝑦(𝑦 ⊆ 𝑌 ∧ 𝑥 = (𝑦 ∩ 𝐴)) ↔ 𝑥 ⊆ (𝑌 ∩ 𝐴))
40	6, 39	bitri 274	. . . 4 ⊢ (∃𝑦(𝑦 ∈ 𝒫 𝑌 ∧ 𝑥 = (𝑦 ∩ 𝐴)) ↔ 𝑥 ⊆ (𝑌 ∩ 𝐴))
41		df-rex 3071	. . . 4 ⊢ (∃𝑦 ∈ 𝒫 𝑌𝑥 = (𝑦 ∩ 𝐴) ↔ ∃𝑦(𝑦 ∈ 𝒫 𝑌 ∧ 𝑥 = (𝑦 ∩ 𝐴)))
42		velpw 4607	. . . 4 ⊢ (𝑥 ∈ 𝒫 (𝑌 ∩ 𝐴) ↔ 𝑥 ⊆ (𝑌 ∩ 𝐴))
43	40, 41, 42	3bitr4i 302	. . 3 ⊢ (∃𝑦 ∈ 𝒫 𝑌𝑥 = (𝑦 ∩ 𝐴) ↔ 𝑥 ∈ 𝒫 (𝑌 ∩ 𝐴))
44	3, 43	bitrdi 286	. 2 ⊢ ((𝑌 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) → (𝑥 ∈ (𝒫 𝑌 ↾t 𝐴) ↔ 𝑥 ∈ 𝒫 (𝑌 ∩ 𝐴)))
45	44	eqrdv 2730	1 ⊢ ((𝑌 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) → (𝒫 𝑌 ↾t 𝐴) = 𝒫 (𝑌 ∩ 𝐴))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1541  ∃wex 1781   ∈ wcel 2106  ∃wrex 3070  Vcvv 3474   ∩ cin 3947   ⊆ wss 3948  𝒫 cpw 4602  (class class class)co 7411   ↾_t crest 17368

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7727

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7414  df-oprab 7415  df-mpo 7416  df-rest 17370

This theorem is referenced by: (None)