Theorem bj-restpw 33170

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 21030 (which uses distop 20847 and restopn2 21029). (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 4880	. . . 4 ⊢ (𝑌 ∈ 𝑉 → 𝒫 𝑌 ∈ V)
2		elrest 16135	. . . 4 ⊢ ((𝒫 𝑌 ∈ V ∧ 𝐴 ∈ 𝑊) → (𝑥 ∈ (𝒫 𝑌 ↾t 𝐴) ↔ ∃𝑦 ∈ 𝒫 𝑌𝑥 = (𝑦 ∩ 𝐴)))
3	1, 2	sylan 487	. . 3 ⊢ ((𝑌 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) → (𝑥 ∈ (𝒫 𝑌 ↾t 𝐴) ↔ ∃𝑦 ∈ 𝒫 𝑌𝑥 = (𝑦 ∩ 𝐴)))
4		selpw 4198	. . . . . . 7 ⊢ (𝑦 ∈ 𝒫 𝑌 ↔ 𝑦 ⊆ 𝑌)
5	4	anbi1i 731	. . . . . 6 ⊢ ((𝑦 ∈ 𝒫 𝑌 ∧ 𝑥 = (𝑦 ∩ 𝐴)) ↔ (𝑦 ⊆ 𝑌 ∧ 𝑥 = (𝑦 ∩ 𝐴)))
6	5	exbii 1814	. . . . 5 ⊢ (∃𝑦(𝑦 ∈ 𝒫 𝑌 ∧ 𝑥 = (𝑦 ∩ 𝐴)) ↔ ∃𝑦(𝑦 ⊆ 𝑌 ∧ 𝑥 = (𝑦 ∩ 𝐴)))
7		sstr2 3643	. . . . . . . . . 10 ⊢ (𝑥 ⊆ 𝑦 → (𝑦 ⊆ 𝑌 → 𝑥 ⊆ 𝑌))
8	7	com12 32	. . . . . . . . 9 ⊢ (𝑦 ⊆ 𝑌 → (𝑥 ⊆ 𝑦 → 𝑥 ⊆ 𝑌))
9		inss1 3866	. . . . . . . . . 10 ⊢ (𝑦 ∩ 𝐴) ⊆ 𝑦
10		sseq1 3659	. . . . . . . . . 10 ⊢ (𝑥 = (𝑦 ∩ 𝐴) → (𝑥 ⊆ 𝑦 ↔ (𝑦 ∩ 𝐴) ⊆ 𝑦))
11	9, 10	mpbiri 248	. . . . . . . . 9 ⊢ (𝑥 = (𝑦 ∩ 𝐴) → 𝑥 ⊆ 𝑦)
12	8, 11	impel 484	. . . . . . . 8 ⊢ ((𝑦 ⊆ 𝑌 ∧ 𝑥 = (𝑦 ∩ 𝐴)) → 𝑥 ⊆ 𝑌)
13		inss2 3867	. . . . . . . . . 10 ⊢ (𝑦 ∩ 𝐴) ⊆ 𝐴
14		sseq1 3659	. . . . . . . . . 10 ⊢ (𝑥 = (𝑦 ∩ 𝐴) → (𝑥 ⊆ 𝐴 ↔ (𝑦 ∩ 𝐴) ⊆ 𝐴))
15	13, 14	mpbiri 248	. . . . . . . . 9 ⊢ (𝑥 = (𝑦 ∩ 𝐴) → 𝑥 ⊆ 𝐴)
16	15	adantl 481	. . . . . . . 8 ⊢ ((𝑦 ⊆ 𝑌 ∧ 𝑥 = (𝑦 ∩ 𝐴)) → 𝑥 ⊆ 𝐴)
17	12, 16	ssind 3870	. . . . . . 7 ⊢ ((𝑦 ⊆ 𝑌 ∧ 𝑥 = (𝑦 ∩ 𝐴)) → 𝑥 ⊆ (𝑌 ∩ 𝐴))
18	17	exlimiv 1898	. . . . . 6 ⊢ (∃𝑦(𝑦 ⊆ 𝑌 ∧ 𝑥 = (𝑦 ∩ 𝐴)) → 𝑥 ⊆ (𝑌 ∩ 𝐴))
19		inss1 3866	. . . . . . . 8 ⊢ (𝑌 ∩ 𝐴) ⊆ 𝑌
20		sstr2 3643	. . . . . . . 8 ⊢ (𝑥 ⊆ (𝑌 ∩ 𝐴) → ((𝑌 ∩ 𝐴) ⊆ 𝑌 → 𝑥 ⊆ 𝑌))
21	19, 20	mpi 20	. . . . . . 7 ⊢ (𝑥 ⊆ (𝑌 ∩ 𝐴) → 𝑥 ⊆ 𝑌)
22		inss2 3867	. . . . . . . 8 ⊢ (𝑌 ∩ 𝐴) ⊆ 𝐴
23		sstr2 3643	. . . . . . . 8 ⊢ (𝑥 ⊆ (𝑌 ∩ 𝐴) → ((𝑌 ∩ 𝐴) ⊆ 𝐴 → 𝑥 ⊆ 𝐴))
24	22, 23	mpi 20	. . . . . . 7 ⊢ (𝑥 ⊆ (𝑌 ∩ 𝐴) → 𝑥 ⊆ 𝐴)
25		ssid 3657	. . . . . . . . . . 11 ⊢ 𝑥 ⊆ 𝑥
26	25	a1i 11	. . . . . . . . . 10 ⊢ (𝑥 ⊆ 𝐴 → 𝑥 ⊆ 𝑥)
27		id 22	. . . . . . . . . 10 ⊢ (𝑥 ⊆ 𝐴 → 𝑥 ⊆ 𝐴)
28	26, 27	ssind 3870	. . . . . . . . 9 ⊢ (𝑥 ⊆ 𝐴 → 𝑥 ⊆ (𝑥 ∩ 𝐴))
29		inss1 3866	. . . . . . . . . 10 ⊢ (𝑥 ∩ 𝐴) ⊆ 𝑥
30	29	a1i 11	. . . . . . . . 9 ⊢ (𝑥 ⊆ 𝐴 → (𝑥 ∩ 𝐴) ⊆ 𝑥)
31	28, 30	eqssd 3653	. . . . . . . 8 ⊢ (𝑥 ⊆ 𝐴 → 𝑥 = (𝑥 ∩ 𝐴))
32		vex 3234	. . . . . . . . 9 ⊢ 𝑥 ∈ V
33		sseq1 3659	. . . . . . . . . 10 ⊢ (𝑦 = 𝑥 → (𝑦 ⊆ 𝑌 ↔ 𝑥 ⊆ 𝑌))
34		ineq1 3840	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑥 → (𝑦 ∩ 𝐴) = (𝑥 ∩ 𝐴))
35	34	eqeq2d 2661	. . . . . . . . . 10 ⊢ (𝑦 = 𝑥 → (𝑥 = (𝑦 ∩ 𝐴) ↔ 𝑥 = (𝑥 ∩ 𝐴)))
36	33, 35	anbi12d 747	. . . . . . . . 9 ⊢ (𝑦 = 𝑥 → ((𝑦 ⊆ 𝑌 ∧ 𝑥 = (𝑦 ∩ 𝐴)) ↔ (𝑥 ⊆ 𝑌 ∧ 𝑥 = (𝑥 ∩ 𝐴))))
37	32, 36	spcev 3331	. . . . . . . 8 ⊢ ((𝑥 ⊆ 𝑌 ∧ 𝑥 = (𝑥 ∩ 𝐴)) → ∃𝑦(𝑦 ⊆ 𝑌 ∧ 𝑥 = (𝑦 ∩ 𝐴)))
38	31, 37	sylan2 490	. . . . . . 7 ⊢ ((𝑥 ⊆ 𝑌 ∧ 𝑥 ⊆ 𝐴) → ∃𝑦(𝑦 ⊆ 𝑌 ∧ 𝑥 = (𝑦 ∩ 𝐴)))
39	21, 24, 38	syl2anc 694	. . . . . 6 ⊢ (𝑥 ⊆ (𝑌 ∩ 𝐴) → ∃𝑦(𝑦 ⊆ 𝑌 ∧ 𝑥 = (𝑦 ∩ 𝐴)))
40	18, 39	impbii 199	. . . . 5 ⊢ (∃𝑦(𝑦 ⊆ 𝑌 ∧ 𝑥 = (𝑦 ∩ 𝐴)) ↔ 𝑥 ⊆ (𝑌 ∩ 𝐴))
41	6, 40	bitri 264	. . . 4 ⊢ (∃𝑦(𝑦 ∈ 𝒫 𝑌 ∧ 𝑥 = (𝑦 ∩ 𝐴)) ↔ 𝑥 ⊆ (𝑌 ∩ 𝐴))
42		df-rex 2947	. . . 4 ⊢ (∃𝑦 ∈ 𝒫 𝑌𝑥 = (𝑦 ∩ 𝐴) ↔ ∃𝑦(𝑦 ∈ 𝒫 𝑌 ∧ 𝑥 = (𝑦 ∩ 𝐴)))
43		selpw 4198	. . . 4 ⊢ (𝑥 ∈ 𝒫 (𝑌 ∩ 𝐴) ↔ 𝑥 ⊆ (𝑌 ∩ 𝐴))
44	41, 42, 43	3bitr4i 292	. . 3 ⊢ (∃𝑦 ∈ 𝒫 𝑌𝑥 = (𝑦 ∩ 𝐴) ↔ 𝑥 ∈ 𝒫 (𝑌 ∩ 𝐴))
45	3, 44	syl6bb 276	. 2 ⊢ ((𝑌 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) → (𝑥 ∈ (𝒫 𝑌 ↾t 𝐴) ↔ 𝑥 ∈ 𝒫 (𝑌 ∩ 𝐴)))
46	45	eqrdv 2649	1 ⊢ ((𝑌 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) → (𝒫 𝑌 ↾t 𝐴) = 𝒫 (𝑌 ∩ 𝐴))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 383   = wceq 1523  ∃wex 1744   ∈ wcel 2030  ∃wrex 2942  Vcvv 3231   ∩ cin 3606   ⊆ wss 3607  𝒫 cpw 4191  (class class class)co 6690   ↾_t crest 16128

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-rest 16130

This theorem is referenced by: (None)