Theorem bj-restpw 35190

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 22237 (which uses distop 22053 and restopn2 22236). (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 5296	. . . 4 ⊢ (𝑌 ∈ 𝑉 → 𝒫 𝑌 ∈ V)
2		elrest 17055	. . . 4 ⊢ ((𝒫 𝑌 ∈ V ∧ 𝐴 ∈ 𝑊) → (𝑥 ∈ (𝒫 𝑌 ↾t 𝐴) ↔ ∃𝑦 ∈ 𝒫 𝑌𝑥 = (𝑦 ∩ 𝐴)))
3	1, 2	sylan 579	. . 3 ⊢ ((𝑌 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) → (𝑥 ∈ (𝒫 𝑌 ↾t 𝐴) ↔ ∃𝑦 ∈ 𝒫 𝑌𝑥 = (𝑦 ∩ 𝐴)))
4		velpw 4535	. . . . . . 7 ⊢ (𝑦 ∈ 𝒫 𝑌 ↔ 𝑦 ⊆ 𝑌)
5	4	anbi1i 623	. . . . . 6 ⊢ ((𝑦 ∈ 𝒫 𝑌 ∧ 𝑥 = (𝑦 ∩ 𝐴)) ↔ (𝑦 ⊆ 𝑌 ∧ 𝑥 = (𝑦 ∩ 𝐴)))
6	5	exbii 1851	. . . . 5 ⊢ (∃𝑦(𝑦 ∈ 𝒫 𝑌 ∧ 𝑥 = (𝑦 ∩ 𝐴)) ↔ ∃𝑦(𝑦 ⊆ 𝑌 ∧ 𝑥 = (𝑦 ∩ 𝐴)))
7		sstr2 3924	. . . . . . . . . 10 ⊢ (𝑥 ⊆ 𝑦 → (𝑦 ⊆ 𝑌 → 𝑥 ⊆ 𝑌))
8	7	com12 32	. . . . . . . . 9 ⊢ (𝑦 ⊆ 𝑌 → (𝑥 ⊆ 𝑦 → 𝑥 ⊆ 𝑌))
9		inss1 4159	. . . . . . . . . 10 ⊢ (𝑦 ∩ 𝐴) ⊆ 𝑦
10		sseq1 3942	. . . . . . . . . 10 ⊢ (𝑥 = (𝑦 ∩ 𝐴) → (𝑥 ⊆ 𝑦 ↔ (𝑦 ∩ 𝐴) ⊆ 𝑦))
11	9, 10	mpbiri 257	. . . . . . . . 9 ⊢ (𝑥 = (𝑦 ∩ 𝐴) → 𝑥 ⊆ 𝑦)
12	8, 11	impel 505	. . . . . . . 8 ⊢ ((𝑦 ⊆ 𝑌 ∧ 𝑥 = (𝑦 ∩ 𝐴)) → 𝑥 ⊆ 𝑌)
13		inss2 4160	. . . . . . . . . 10 ⊢ (𝑦 ∩ 𝐴) ⊆ 𝐴
14		sseq1 3942	. . . . . . . . . 10 ⊢ (𝑥 = (𝑦 ∩ 𝐴) → (𝑥 ⊆ 𝐴 ↔ (𝑦 ∩ 𝐴) ⊆ 𝐴))
15	13, 14	mpbiri 257	. . . . . . . . 9 ⊢ (𝑥 = (𝑦 ∩ 𝐴) → 𝑥 ⊆ 𝐴)
16	15	adantl 481	. . . . . . . 8 ⊢ ((𝑦 ⊆ 𝑌 ∧ 𝑥 = (𝑦 ∩ 𝐴)) → 𝑥 ⊆ 𝐴)
17	12, 16	ssind 4163	. . . . . . 7 ⊢ ((𝑦 ⊆ 𝑌 ∧ 𝑥 = (𝑦 ∩ 𝐴)) → 𝑥 ⊆ (𝑌 ∩ 𝐴))
18	17	exlimiv 1934	. . . . . 6 ⊢ (∃𝑦(𝑦 ⊆ 𝑌 ∧ 𝑥 = (𝑦 ∩ 𝐴)) → 𝑥 ⊆ (𝑌 ∩ 𝐴))
19		inss1 4159	. . . . . . . 8 ⊢ (𝑌 ∩ 𝐴) ⊆ 𝑌
20		sstr2 3924	. . . . . . . 8 ⊢ (𝑥 ⊆ (𝑌 ∩ 𝐴) → ((𝑌 ∩ 𝐴) ⊆ 𝑌 → 𝑥 ⊆ 𝑌))
21	19, 20	mpi 20	. . . . . . 7 ⊢ (𝑥 ⊆ (𝑌 ∩ 𝐴) → 𝑥 ⊆ 𝑌)
22		inss2 4160	. . . . . . . 8 ⊢ (𝑌 ∩ 𝐴) ⊆ 𝐴
23		sstr2 3924	. . . . . . . 8 ⊢ (𝑥 ⊆ (𝑌 ∩ 𝐴) → ((𝑌 ∩ 𝐴) ⊆ 𝐴 → 𝑥 ⊆ 𝐴))
24	22, 23	mpi 20	. . . . . . 7 ⊢ (𝑥 ⊆ (𝑌 ∩ 𝐴) → 𝑥 ⊆ 𝐴)
25		ssidd 3940	. . . . . . . . . 10 ⊢ (𝑥 ⊆ 𝐴 → 𝑥 ⊆ 𝑥)
26		id 22	. . . . . . . . . 10 ⊢ (𝑥 ⊆ 𝐴 → 𝑥 ⊆ 𝐴)
27	25, 26	ssind 4163	. . . . . . . . 9 ⊢ (𝑥 ⊆ 𝐴 → 𝑥 ⊆ (𝑥 ∩ 𝐴))
28		inss1 4159	. . . . . . . . . 10 ⊢ (𝑥 ∩ 𝐴) ⊆ 𝑥
29	28	a1i 11	. . . . . . . . 9 ⊢ (𝑥 ⊆ 𝐴 → (𝑥 ∩ 𝐴) ⊆ 𝑥)
30	27, 29	eqssd 3934	. . . . . . . 8 ⊢ (𝑥 ⊆ 𝐴 → 𝑥 = (𝑥 ∩ 𝐴))
31		vex 3426	. . . . . . . . 9 ⊢ 𝑥 ∈ V
32		sseq1 3942	. . . . . . . . . 10 ⊢ (𝑦 = 𝑥 → (𝑦 ⊆ 𝑌 ↔ 𝑥 ⊆ 𝑌))
33		ineq1 4136	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑥 → (𝑦 ∩ 𝐴) = (𝑥 ∩ 𝐴))
34	33	eqeq2d 2749	. . . . . . . . . 10 ⊢ (𝑦 = 𝑥 → (𝑥 = (𝑦 ∩ 𝐴) ↔ 𝑥 = (𝑥 ∩ 𝐴)))
35	32, 34	anbi12d 630	. . . . . . . . 9 ⊢ (𝑦 = 𝑥 → ((𝑦 ⊆ 𝑌 ∧ 𝑥 = (𝑦 ∩ 𝐴)) ↔ (𝑥 ⊆ 𝑌 ∧ 𝑥 = (𝑥 ∩ 𝐴))))
36	31, 35	spcev 3535	. . . . . . . 8 ⊢ ((𝑥 ⊆ 𝑌 ∧ 𝑥 = (𝑥 ∩ 𝐴)) → ∃𝑦(𝑦 ⊆ 𝑌 ∧ 𝑥 = (𝑦 ∩ 𝐴)))
37	30, 36	sylan2 592	. . . . . . 7 ⊢ ((𝑥 ⊆ 𝑌 ∧ 𝑥 ⊆ 𝐴) → ∃𝑦(𝑦 ⊆ 𝑌 ∧ 𝑥 = (𝑦 ∩ 𝐴)))
38	21, 24, 37	syl2anc 583	. . . . . 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 3069	. . . 4 ⊢ (∃𝑦 ∈ 𝒫 𝑌𝑥 = (𝑦 ∩ 𝐴) ↔ ∃𝑦(𝑦 ∈ 𝒫 𝑌 ∧ 𝑥 = (𝑦 ∩ 𝐴)))
42		velpw 4535	. . . 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 2736	1 ⊢ ((𝑌 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) → (𝒫 𝑌 ↾t 𝐴) = 𝒫 (𝑌 ∩ 𝐴))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1539  ∃wex 1783   ∈ wcel 2108  ∃wrex 3064  Vcvv 3422   ∩ cin 3882   ⊆ wss 3883  𝒫 cpw 4530  (class class class)co 7255   ↾_t crest 17048

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-reu 3070  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-ov 7258  df-oprab 7259  df-mpo 7260  df-rest 17050

This theorem is referenced by: (None)