Theorem bj-restpw 37342

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 23134 (which uses distop 22951 and restopn2 23133). (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 5325	. . . 4 ⊢ (𝑌 ∈ 𝑉 → 𝒫 𝑌 ∈ V)
2		elrest 17359	. . . 4 ⊢ ((𝒫 𝑌 ∈ V ∧ 𝐴 ∈ 𝑊) → (𝑥 ∈ (𝒫 𝑌 ↾t 𝐴) ↔ ∃𝑦 ∈ 𝒫 𝑌𝑥 = (𝑦 ∩ 𝐴)))
3	1, 2	sylan 581	. . 3 ⊢ ((𝑌 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) → (𝑥 ∈ (𝒫 𝑌 ↾t 𝐴) ↔ ∃𝑦 ∈ 𝒫 𝑌𝑥 = (𝑦 ∩ 𝐴)))
4		velpw 4561	. . . . . . 7 ⊢ (𝑦 ∈ 𝒫 𝑌 ↔ 𝑦 ⊆ 𝑌)
5	4	anbi1i 625	. . . . . 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 3942	. . . . . . . . . 10 ⊢ (𝑥 ⊆ 𝑦 → (𝑦 ⊆ 𝑌 → 𝑥 ⊆ 𝑌))
8	7	com12 32	. . . . . . . . 9 ⊢ (𝑦 ⊆ 𝑌 → (𝑥 ⊆ 𝑦 → 𝑥 ⊆ 𝑌))
9		inss1 4191	. . . . . . . . . 10 ⊢ (𝑦 ∩ 𝐴) ⊆ 𝑦
10		sseq1 3961	. . . . . . . . . 10 ⊢ (𝑥 = (𝑦 ∩ 𝐴) → (𝑥 ⊆ 𝑦 ↔ (𝑦 ∩ 𝐴) ⊆ 𝑦))
11	9, 10	mpbiri 258	. . . . . . . . 9 ⊢ (𝑥 = (𝑦 ∩ 𝐴) → 𝑥 ⊆ 𝑦)
12	8, 11	impel 505	. . . . . . . 8 ⊢ ((𝑦 ⊆ 𝑌 ∧ 𝑥 = (𝑦 ∩ 𝐴)) → 𝑥 ⊆ 𝑌)
13		inss2 4192	. . . . . . . . . 10 ⊢ (𝑦 ∩ 𝐴) ⊆ 𝐴
14		sseq1 3961	. . . . . . . . . 10 ⊢ (𝑥 = (𝑦 ∩ 𝐴) → (𝑥 ⊆ 𝐴 ↔ (𝑦 ∩ 𝐴) ⊆ 𝐴))
15	13, 14	mpbiri 258	. . . . . . . . 9 ⊢ (𝑥 = (𝑦 ∩ 𝐴) → 𝑥 ⊆ 𝐴)
16	15	adantl 481	. . . . . . . 8 ⊢ ((𝑦 ⊆ 𝑌 ∧ 𝑥 = (𝑦 ∩ 𝐴)) → 𝑥 ⊆ 𝐴)
17	12, 16	ssind 4195	. . . . . . 7 ⊢ ((𝑦 ⊆ 𝑌 ∧ 𝑥 = (𝑦 ∩ 𝐴)) → 𝑥 ⊆ (𝑌 ∩ 𝐴))
18	17	exlimiv 1932	. . . . . 6 ⊢ (∃𝑦(𝑦 ⊆ 𝑌 ∧ 𝑥 = (𝑦 ∩ 𝐴)) → 𝑥 ⊆ (𝑌 ∩ 𝐴))
19		inss1 4191	. . . . . . . 8 ⊢ (𝑌 ∩ 𝐴) ⊆ 𝑌
20		sstr2 3942	. . . . . . . 8 ⊢ (𝑥 ⊆ (𝑌 ∩ 𝐴) → ((𝑌 ∩ 𝐴) ⊆ 𝑌 → 𝑥 ⊆ 𝑌))
21	19, 20	mpi 20	. . . . . . 7 ⊢ (𝑥 ⊆ (𝑌 ∩ 𝐴) → 𝑥 ⊆ 𝑌)
22		inss2 4192	. . . . . . . 8 ⊢ (𝑌 ∩ 𝐴) ⊆ 𝐴
23		sstr2 3942	. . . . . . . 8 ⊢ (𝑥 ⊆ (𝑌 ∩ 𝐴) → ((𝑌 ∩ 𝐴) ⊆ 𝐴 → 𝑥 ⊆ 𝐴))
24	22, 23	mpi 20	. . . . . . 7 ⊢ (𝑥 ⊆ (𝑌 ∩ 𝐴) → 𝑥 ⊆ 𝐴)
25		ssidd 3959	. . . . . . . . . 10 ⊢ (𝑥 ⊆ 𝐴 → 𝑥 ⊆ 𝑥)
26		id 22	. . . . . . . . . 10 ⊢ (𝑥 ⊆ 𝐴 → 𝑥 ⊆ 𝐴)
27	25, 26	ssind 4195	. . . . . . . . 9 ⊢ (𝑥 ⊆ 𝐴 → 𝑥 ⊆ (𝑥 ∩ 𝐴))
28		inss1 4191	. . . . . . . . . 10 ⊢ (𝑥 ∩ 𝐴) ⊆ 𝑥
29	28	a1i 11	. . . . . . . . 9 ⊢ (𝑥 ⊆ 𝐴 → (𝑥 ∩ 𝐴) ⊆ 𝑥)
30	27, 29	eqssd 3953	. . . . . . . 8 ⊢ (𝑥 ⊆ 𝐴 → 𝑥 = (𝑥 ∩ 𝐴))
31		vex 3446	. . . . . . . . 9 ⊢ 𝑥 ∈ V
32		sseq1 3961	. . . . . . . . . 10 ⊢ (𝑦 = 𝑥 → (𝑦 ⊆ 𝑌 ↔ 𝑥 ⊆ 𝑌))
33		ineq1 4167	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑥 → (𝑦 ∩ 𝐴) = (𝑥 ∩ 𝐴))
34	33	eqeq2d 2748	. . . . . . . . . 10 ⊢ (𝑦 = 𝑥 → (𝑥 = (𝑦 ∩ 𝐴) ↔ 𝑥 = (𝑥 ∩ 𝐴)))
35	32, 34	anbi12d 633	. . . . . . . . 9 ⊢ (𝑦 = 𝑥 → ((𝑦 ⊆ 𝑌 ∧ 𝑥 = (𝑦 ∩ 𝐴)) ↔ (𝑥 ⊆ 𝑌 ∧ 𝑥 = (𝑥 ∩ 𝐴))))
36	31, 35	spcev 3562	. . . . . . . 8 ⊢ ((𝑥 ⊆ 𝑌 ∧ 𝑥 = (𝑥 ∩ 𝐴)) → ∃𝑦(𝑦 ⊆ 𝑌 ∧ 𝑥 = (𝑦 ∩ 𝐴)))
37	30, 36	sylan2 594	. . . . . . 7 ⊢ ((𝑥 ⊆ 𝑌 ∧ 𝑥 ⊆ 𝐴) → ∃𝑦(𝑦 ⊆ 𝑌 ∧ 𝑥 = (𝑦 ∩ 𝐴)))
38	21, 24, 37	syl2anc 585	. . . . . 6 ⊢ (𝑥 ⊆ (𝑌 ∩ 𝐴) → ∃𝑦(𝑦 ⊆ 𝑌 ∧ 𝑥 = (𝑦 ∩ 𝐴)))
39	18, 38	impbii 209	. . . . 5 ⊢ (∃𝑦(𝑦 ⊆ 𝑌 ∧ 𝑥 = (𝑦 ∩ 𝐴)) ↔ 𝑥 ⊆ (𝑌 ∩ 𝐴))
40	6, 39	bitri 275	. . . 4 ⊢ (∃𝑦(𝑦 ∈ 𝒫 𝑌 ∧ 𝑥 = (𝑦 ∩ 𝐴)) ↔ 𝑥 ⊆ (𝑌 ∩ 𝐴))
41		df-rex 3063	. . . 4 ⊢ (∃𝑦 ∈ 𝒫 𝑌𝑥 = (𝑦 ∩ 𝐴) ↔ ∃𝑦(𝑦 ∈ 𝒫 𝑌 ∧ 𝑥 = (𝑦 ∩ 𝐴)))
42		velpw 4561	. . . 4 ⊢ (𝑥 ∈ 𝒫 (𝑌 ∩ 𝐴) ↔ 𝑥 ⊆ (𝑌 ∩ 𝐴))
43	40, 41, 42	3bitr4i 303	. . 3 ⊢ (∃𝑦 ∈ 𝒫 𝑌𝑥 = (𝑦 ∩ 𝐴) ↔ 𝑥 ∈ 𝒫 (𝑌 ∩ 𝐴))
44	3, 43	bitrdi 287	. 2 ⊢ ((𝑌 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) → (𝑥 ∈ (𝒫 𝑌 ↾t 𝐴) ↔ 𝑥 ∈ 𝒫 (𝑌 ∩ 𝐴)))
45	44	eqrdv 2735	1 ⊢ ((𝑌 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) → (𝒫 𝑌 ↾t 𝐴) = 𝒫 (𝑌 ∩ 𝐴))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542  ∃wex 1781   ∈ wcel 2114  ∃wrex 3062  Vcvv 3442   ∩ cin 3902   ⊆ wss 3903  𝒫 cpw 4556  (class class class)co 7368   ↾_t crest 17352

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5226  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5527  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-ov 7371  df-oprab 7372  df-mpo 7373  df-rest 17354

This theorem is referenced by: (None)