Theorem bj-restpw 37546

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 23218 (which uses distop 23035 and restopn2 23217). (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 5334	. . . 4 ⊢ (𝑌 ∈ 𝑉 → 𝒫 𝑌 ∈ V)
2		elrest 17439	. . . 4 ⊢ ((𝒫 𝑌 ∈ V ∧ 𝐴 ∈ 𝑊) → (𝑥 ∈ (𝒫 𝑌 ↾t 𝐴) ↔ ∃𝑦 ∈ 𝒫 𝑌𝑥 = (𝑦 ∩ 𝐴)))
3	1, 2	sylan 589	. . 3 ⊢ ((𝑌 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) → (𝑥 ∈ (𝒫 𝑌 ↾t 𝐴) ↔ ∃𝑦 ∈ 𝒫 𝑌𝑥 = (𝑦 ∩ 𝐴)))
4		velpw 4559	. . . . . . 7 ⊢ (𝑦 ∈ 𝒫 𝑌 ↔ 𝑦 ⊆ 𝑌)
5	4	anbi1i 633	. . . . . 6 ⊢ ((𝑦 ∈ 𝒫 𝑌 ∧ 𝑥 = (𝑦 ∩ 𝐴)) ↔ (𝑦 ⊆ 𝑌 ∧ 𝑥 = (𝑦 ∩ 𝐴)))
6	5	exbii 1867	. . . . 5 ⊢ (∃𝑦(𝑦 ∈ 𝒫 𝑌 ∧ 𝑥 = (𝑦 ∩ 𝐴)) ↔ ∃𝑦(𝑦 ⊆ 𝑌 ∧ 𝑥 = (𝑦 ∩ 𝐴)))
7		sstr2 3943	. . . . . . . . . 10 ⊢ (𝑥 ⊆ 𝑦 → (𝑦 ⊆ 𝑌 → 𝑥 ⊆ 𝑌))
8	7	com12 32	. . . . . . . . 9 ⊢ (𝑦 ⊆ 𝑌 → (𝑥 ⊆ 𝑦 → 𝑥 ⊆ 𝑌))
9		inss1 4188	. . . . . . . . . 10 ⊢ (𝑦 ∩ 𝐴) ⊆ 𝑦
10		sseq1 3961	. . . . . . . . . 10 ⊢ (𝑥 = (𝑦 ∩ 𝐴) → (𝑥 ⊆ 𝑦 ↔ (𝑦 ∩ 𝐴) ⊆ 𝑦))
11	9, 10	mpbiri 260	. . . . . . . . 9 ⊢ (𝑥 = (𝑦 ∩ 𝐴) → 𝑥 ⊆ 𝑦)
12	8, 11	impel 513	. . . . . . . 8 ⊢ ((𝑦 ⊆ 𝑌 ∧ 𝑥 = (𝑦 ∩ 𝐴)) → 𝑥 ⊆ 𝑌)
13		inss2 4189	. . . . . . . . . 10 ⊢ (𝑦 ∩ 𝐴) ⊆ 𝐴
14		sseq1 3961	. . . . . . . . . 10 ⊢ (𝑥 = (𝑦 ∩ 𝐴) → (𝑥 ⊆ 𝐴 ↔ (𝑦 ∩ 𝐴) ⊆ 𝐴))
15	13, 14	mpbiri 260	. . . . . . . . 9 ⊢ (𝑥 = (𝑦 ∩ 𝐴) → 𝑥 ⊆ 𝐴)
16	15	adantl 485	. . . . . . . 8 ⊢ ((𝑦 ⊆ 𝑌 ∧ 𝑥 = (𝑦 ∩ 𝐴)) → 𝑥 ⊆ 𝐴)
17	12, 16	ssind 4192	. . . . . . 7 ⊢ ((𝑦 ⊆ 𝑌 ∧ 𝑥 = (𝑦 ∩ 𝐴)) → 𝑥 ⊆ (𝑌 ∩ 𝐴))
18	17	exlimiv 1949	. . . . . 6 ⊢ (∃𝑦(𝑦 ⊆ 𝑌 ∧ 𝑥 = (𝑦 ∩ 𝐴)) → 𝑥 ⊆ (𝑌 ∩ 𝐴))
19		inss1 4188	. . . . . . . 8 ⊢ (𝑌 ∩ 𝐴) ⊆ 𝑌
20		sstr2 3943	. . . . . . . 8 ⊢ (𝑥 ⊆ (𝑌 ∩ 𝐴) → ((𝑌 ∩ 𝐴) ⊆ 𝑌 → 𝑥 ⊆ 𝑌))
21	19, 20	mpi 20	. . . . . . 7 ⊢ (𝑥 ⊆ (𝑌 ∩ 𝐴) → 𝑥 ⊆ 𝑌)
22		inss2 4189	. . . . . . . 8 ⊢ (𝑌 ∩ 𝐴) ⊆ 𝐴
23		sstr2 3943	. . . . . . . 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 4192	. . . . . . . . 9 ⊢ (𝑥 ⊆ 𝐴 → 𝑥 ⊆ (𝑥 ∩ 𝐴))
28		inss1 4188	. . . . . . . . . 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 3457	. . . . . . . . 9 ⊢ 𝑥 ∈ V
32		sseq1 3961	. . . . . . . . . 10 ⊢ (𝑦 = 𝑥 → (𝑦 ⊆ 𝑌 ↔ 𝑥 ⊆ 𝑌))
33		ineq1 4165	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑥 → (𝑦 ∩ 𝐴) = (𝑥 ∩ 𝐴))
34	33	eqeq2d 2772	. . . . . . . . . 10 ⊢ (𝑦 = 𝑥 → (𝑥 = (𝑦 ∩ 𝐴) ↔ 𝑥 = (𝑥 ∩ 𝐴)))
35	32, 34	anbi12d 641	. . . . . . . . 9 ⊢ (𝑦 = 𝑥 → ((𝑦 ⊆ 𝑌 ∧ 𝑥 = (𝑦 ∩ 𝐴)) ↔ (𝑥 ⊆ 𝑌 ∧ 𝑥 = (𝑥 ∩ 𝐴))))
36	31, 35	spcev 3565	. . . . . . . 8 ⊢ ((𝑥 ⊆ 𝑌 ∧ 𝑥 = (𝑥 ∩ 𝐴)) → ∃𝑦(𝑦 ⊆ 𝑌 ∧ 𝑥 = (𝑦 ∩ 𝐴)))
37	30, 36	sylan2 602	. . . . . . 7 ⊢ ((𝑥 ⊆ 𝑌 ∧ 𝑥 ⊆ 𝐴) → ∃𝑦(𝑦 ⊆ 𝑌 ∧ 𝑥 = (𝑦 ∩ 𝐴)))
38	21, 24, 37	syl2anc 593	. . . . . 6 ⊢ (𝑥 ⊆ (𝑌 ∩ 𝐴) → ∃𝑦(𝑦 ⊆ 𝑌 ∧ 𝑥 = (𝑦 ∩ 𝐴)))
39	18, 38	impbii 211	. . . . 5 ⊢ (∃𝑦(𝑦 ⊆ 𝑌 ∧ 𝑥 = (𝑦 ∩ 𝐴)) ↔ 𝑥 ⊆ (𝑌 ∩ 𝐴))
40	6, 39	bitri 277	. . . 4 ⊢ (∃𝑦(𝑦 ∈ 𝒫 𝑌 ∧ 𝑥 = (𝑦 ∩ 𝐴)) ↔ 𝑥 ⊆ (𝑌 ∩ 𝐴))
41		df-rex 3086	. . . 4 ⊢ (∃𝑦 ∈ 𝒫 𝑌𝑥 = (𝑦 ∩ 𝐴) ↔ ∃𝑦(𝑦 ∈ 𝒫 𝑌 ∧ 𝑥 = (𝑦 ∩ 𝐴)))
42		velpw 4559	. . . 4 ⊢ (𝑥 ∈ 𝒫 (𝑌 ∩ 𝐴) ↔ 𝑥 ⊆ (𝑌 ∩ 𝐴))
43	40, 41, 42	3bitr4i 305	. . 3 ⊢ (∃𝑦 ∈ 𝒫 𝑌𝑥 = (𝑦 ∩ 𝐴) ↔ 𝑥 ∈ 𝒫 (𝑌 ∩ 𝐴))
44	3, 43	bitrdi 289	. 2 ⊢ ((𝑌 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) → (𝑥 ∈ (𝒫 𝑌 ↾t 𝐴) ↔ 𝑥 ∈ 𝒫 (𝑌 ∩ 𝐴)))
45	44	eqrdv 2759	1 ⊢ ((𝑌 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) → (𝒫 𝑌 ↾t 𝐴) = 𝒫 (𝑌 ∩ 𝐴))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399   = wceq 1559  ∃wex 1798   ∈ wcel 2141  ∃wrex 3085  Vcvv 3453   ∩ cin 3903   ⊆ wss 3904  𝒫 cpw 4554  (class class class)co 7392   ↾_t crest 17432

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-rep 5226  ax-sep 5245  ax-nul 5255  ax-pow 5321  ax-pr 5389  ax-un 7714

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3076  df-rex 3086  df-reu 3367  df-rab 3414  df-v 3455  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-iun 4950  df-br 5100  df-opab 5162  df-mpt 5181  df-id 5540  df-xp 5651  df-rel 5652  df-cnv 5653  df-co 5654  df-dm 5655  df-rn 5656  df-res 5657  df-ima 5658  df-iota 6473  df-fun 6519  df-fn 6520  df-f 6521  df-f1 6522  df-fo 6523  df-f1o 6524  df-fv 6525  df-ov 7395  df-oprab 7396  df-mpo 7397  df-rest 17434

This theorem is referenced by: (None)