Theorem ntrk0kbimka 39887

Description: If the interiors of disjoint sets are disjoint and the interior of the base set is the base set, then the interior of the empty set is the empty set. Obsolete version of ntrkbimka 39886. (Contributed by RP, 12-Jun-2021.)

Assertion

Ref	Expression
ntrk0kbimka	⊢ ((𝐵 ∈ 𝑉 ∧ 𝐼 ∈ (𝒫 𝐵 ↑𝑚 𝒫 𝐵)) → (((𝐼‘𝐵) = 𝐵 ∧ ∀𝑠 ∈ 𝒫 𝐵∀𝑡 ∈ 𝒫 𝐵((𝑠 ∩ 𝑡) = ∅ → ((𝐼‘𝑠) ∩ (𝐼‘𝑡)) = ∅)) → (𝐼‘∅) = ∅))

Distinct variable groups:   𝐵,𝑠,𝑡   𝐼,𝑠,𝑡

Allowed substitution hints:   𝑉(𝑡,𝑠)

Proof of Theorem ntrk0kbimka

Step	Hyp	Ref	Expression
1		pwidg 4470	. . . . 5 ⊢ (𝐵 ∈ 𝑉 → 𝐵 ∈ 𝒫 𝐵)
2	1	ad2antrr 722	. . . 4 ⊢ (((𝐵 ∈ 𝑉 ∧ 𝐼 ∈ (𝒫 𝐵 ↑𝑚 𝒫 𝐵)) ∧ ((𝐼‘𝐵) = 𝐵 ∧ ∀𝑠 ∈ 𝒫 𝐵∀𝑡 ∈ 𝒫 𝐵((𝑠 ∩ 𝑡) = ∅ → ((𝐼‘𝑠) ∩ (𝐼‘𝑡)) = ∅))) → 𝐵 ∈ 𝒫 𝐵)
3		0elpw 5150	. . . . 5 ⊢ ∅ ∈ 𝒫 𝐵
4	3	a1i 11	. . . 4 ⊢ (((𝐵 ∈ 𝑉 ∧ 𝐼 ∈ (𝒫 𝐵 ↑𝑚 𝒫 𝐵)) ∧ ((𝐼‘𝐵) = 𝐵 ∧ ∀𝑠 ∈ 𝒫 𝐵∀𝑡 ∈ 𝒫 𝐵((𝑠 ∩ 𝑡) = ∅ → ((𝐼‘𝑠) ∩ (𝐼‘𝑡)) = ∅))) → ∅ ∈ 𝒫 𝐵)
5		simprr 769	. . . 4 ⊢ (((𝐵 ∈ 𝑉 ∧ 𝐼 ∈ (𝒫 𝐵 ↑𝑚 𝒫 𝐵)) ∧ ((𝐼‘𝐵) = 𝐵 ∧ ∀𝑠 ∈ 𝒫 𝐵∀𝑡 ∈ 𝒫 𝐵((𝑠 ∩ 𝑡) = ∅ → ((𝐼‘𝑠) ∩ (𝐼‘𝑡)) = ∅))) → ∀𝑠 ∈ 𝒫 𝐵∀𝑡 ∈ 𝒫 𝐵((𝑠 ∩ 𝑡) = ∅ → ((𝐼‘𝑠) ∩ (𝐼‘𝑡)) = ∅))
6		ineq1 4103	. . . . . . 7 ⊢ (𝑠 = 𝐵 → (𝑠 ∩ 𝑡) = (𝐵 ∩ 𝑡))
7	6	eqeq1d 2796	. . . . . 6 ⊢ (𝑠 = 𝐵 → ((𝑠 ∩ 𝑡) = ∅ ↔ (𝐵 ∩ 𝑡) = ∅))
8		fveq2 6541	. . . . . . . 8 ⊢ (𝑠 = 𝐵 → (𝐼‘𝑠) = (𝐼‘𝐵))
9	8	ineq1d 4110	. . . . . . 7 ⊢ (𝑠 = 𝐵 → ((𝐼‘𝑠) ∩ (𝐼‘𝑡)) = ((𝐼‘𝐵) ∩ (𝐼‘𝑡)))
10	9	eqeq1d 2796	. . . . . 6 ⊢ (𝑠 = 𝐵 → (((𝐼‘𝑠) ∩ (𝐼‘𝑡)) = ∅ ↔ ((𝐼‘𝐵) ∩ (𝐼‘𝑡)) = ∅))
11	7, 10	imbi12d 346	. . . . 5 ⊢ (𝑠 = 𝐵 → (((𝑠 ∩ 𝑡) = ∅ → ((𝐼‘𝑠) ∩ (𝐼‘𝑡)) = ∅) ↔ ((𝐵 ∩ 𝑡) = ∅ → ((𝐼‘𝐵) ∩ (𝐼‘𝑡)) = ∅)))
12		ineq2 4105	. . . . . . . 8 ⊢ (𝑡 = ∅ → (𝐵 ∩ 𝑡) = (𝐵 ∩ ∅))
13	12	eqeq1d 2796	. . . . . . 7 ⊢ (𝑡 = ∅ → ((𝐵 ∩ 𝑡) = ∅ ↔ (𝐵 ∩ ∅) = ∅))
14		fveq2 6541	. . . . . . . . 9 ⊢ (𝑡 = ∅ → (𝐼‘𝑡) = (𝐼‘∅))
15	14	ineq2d 4111	. . . . . . . 8 ⊢ (𝑡 = ∅ → ((𝐼‘𝐵) ∩ (𝐼‘𝑡)) = ((𝐼‘𝐵) ∩ (𝐼‘∅)))
16	15	eqeq1d 2796	. . . . . . 7 ⊢ (𝑡 = ∅ → (((𝐼‘𝐵) ∩ (𝐼‘𝑡)) = ∅ ↔ ((𝐼‘𝐵) ∩ (𝐼‘∅)) = ∅))
17	13, 16	imbi12d 346	. . . . . 6 ⊢ (𝑡 = ∅ → (((𝐵 ∩ 𝑡) = ∅ → ((𝐼‘𝐵) ∩ (𝐼‘𝑡)) = ∅) ↔ ((𝐵 ∩ ∅) = ∅ → ((𝐼‘𝐵) ∩ (𝐼‘∅)) = ∅)))
18		in0 4267	. . . . . . 7 ⊢ (𝐵 ∩ ∅) = ∅
19		pm5.5 363	. . . . . . 7 ⊢ ((𝐵 ∩ ∅) = ∅ → (((𝐵 ∩ ∅) = ∅ → ((𝐼‘𝐵) ∩ (𝐼‘∅)) = ∅) ↔ ((𝐼‘𝐵) ∩ (𝐼‘∅)) = ∅))
20	18, 19	mp1i 13	. . . . . 6 ⊢ (𝑡 = ∅ → (((𝐵 ∩ ∅) = ∅ → ((𝐼‘𝐵) ∩ (𝐼‘∅)) = ∅) ↔ ((𝐼‘𝐵) ∩ (𝐼‘∅)) = ∅))
21	17, 20	bitrd 280	. . . . 5 ⊢ (𝑡 = ∅ → (((𝐵 ∩ 𝑡) = ∅ → ((𝐼‘𝐵) ∩ (𝐼‘𝑡)) = ∅) ↔ ((𝐼‘𝐵) ∩ (𝐼‘∅)) = ∅))
22	11, 21	rspc2va 3571	. . . 4 ⊢ (((𝐵 ∈ 𝒫 𝐵 ∧ ∅ ∈ 𝒫 𝐵) ∧ ∀𝑠 ∈ 𝒫 𝐵∀𝑡 ∈ 𝒫 𝐵((𝑠 ∩ 𝑡) = ∅ → ((𝐼‘𝑠) ∩ (𝐼‘𝑡)) = ∅)) → ((𝐼‘𝐵) ∩ (𝐼‘∅)) = ∅)
23	2, 4, 5, 22	syl21anc 834	. . 3 ⊢ (((𝐵 ∈ 𝑉 ∧ 𝐼 ∈ (𝒫 𝐵 ↑𝑚 𝒫 𝐵)) ∧ ((𝐼‘𝐵) = 𝐵 ∧ ∀𝑠 ∈ 𝒫 𝐵∀𝑡 ∈ 𝒫 𝐵((𝑠 ∩ 𝑡) = ∅ → ((𝐼‘𝑠) ∩ (𝐼‘𝑡)) = ∅))) → ((𝐼‘𝐵) ∩ (𝐼‘∅)) = ∅)
24	23	ex 413	. 2 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐼 ∈ (𝒫 𝐵 ↑𝑚 𝒫 𝐵)) → (((𝐼‘𝐵) = 𝐵 ∧ ∀𝑠 ∈ 𝒫 𝐵∀𝑡 ∈ 𝒫 𝐵((𝑠 ∩ 𝑡) = ∅ → ((𝐼‘𝑠) ∩ (𝐼‘𝑡)) = ∅)) → ((𝐼‘𝐵) ∩ (𝐼‘∅)) = ∅))
25		elmapi 8281	. . . . . 6 ⊢ (𝐼 ∈ (𝒫 𝐵 ↑𝑚 𝒫 𝐵) → 𝐼:𝒫 𝐵⟶𝒫 𝐵)
26	25	adantl 482	. . . . 5 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐼 ∈ (𝒫 𝐵 ↑𝑚 𝒫 𝐵)) → 𝐼:𝒫 𝐵⟶𝒫 𝐵)
27	3	a1i 11	. . . . 5 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐼 ∈ (𝒫 𝐵 ↑𝑚 𝒫 𝐵)) → ∅ ∈ 𝒫 𝐵)
28	26, 27	ffvelrnd 6720	. . . 4 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐼 ∈ (𝒫 𝐵 ↑𝑚 𝒫 𝐵)) → (𝐼‘∅) ∈ 𝒫 𝐵)
29	28	elpwid 4467	. . 3 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐼 ∈ (𝒫 𝐵 ↑𝑚 𝒫 𝐵)) → (𝐼‘∅) ⊆ 𝐵)
30		simpl 483	. . 3 ⊢ (((𝐼‘𝐵) = 𝐵 ∧ ∀𝑠 ∈ 𝒫 𝐵∀𝑡 ∈ 𝒫 𝐵((𝑠 ∩ 𝑡) = ∅ → ((𝐼‘𝑠) ∩ (𝐼‘𝑡)) = ∅)) → (𝐼‘𝐵) = 𝐵)
31		ineq1 4103	. . . . . . . 8 ⊢ ((𝐼‘𝐵) = 𝐵 → ((𝐼‘𝐵) ∩ (𝐼‘∅)) = (𝐵 ∩ (𝐼‘∅)))
32		incom 4101	. . . . . . . 8 ⊢ (𝐵 ∩ (𝐼‘∅)) = ((𝐼‘∅) ∩ 𝐵)
33	31, 32	syl6eq 2846	. . . . . . 7 ⊢ ((𝐼‘𝐵) = 𝐵 → ((𝐼‘𝐵) ∩ (𝐼‘∅)) = ((𝐼‘∅) ∩ 𝐵))
34	33	eqeq1d 2796	. . . . . 6 ⊢ ((𝐼‘𝐵) = 𝐵 → (((𝐼‘𝐵) ∩ (𝐼‘∅)) = ∅ ↔ ((𝐼‘∅) ∩ 𝐵) = ∅))
35	34	biimpd 230	. . . . 5 ⊢ ((𝐼‘𝐵) = 𝐵 → (((𝐼‘𝐵) ∩ (𝐼‘∅)) = ∅ → ((𝐼‘∅) ∩ 𝐵) = ∅))
36		reldisj 4318	. . . . . . 7 ⊢ ((𝐼‘∅) ⊆ 𝐵 → (((𝐼‘∅) ∩ 𝐵) = ∅ ↔ (𝐼‘∅) ⊆ (𝐵 ∖ 𝐵)))
37	36	biimpd 230	. . . . . 6 ⊢ ((𝐼‘∅) ⊆ 𝐵 → (((𝐼‘∅) ∩ 𝐵) = ∅ → (𝐼‘∅) ⊆ (𝐵 ∖ 𝐵)))
38		difid 4252	. . . . . . . 8 ⊢ (𝐵 ∖ 𝐵) = ∅
39	38	sseq2i 3919	. . . . . . 7 ⊢ ((𝐼‘∅) ⊆ (𝐵 ∖ 𝐵) ↔ (𝐼‘∅) ⊆ ∅)
40		ss0 4274	. . . . . . 7 ⊢ ((𝐼‘∅) ⊆ ∅ → (𝐼‘∅) = ∅)
41	39, 40	sylbi 218	. . . . . 6 ⊢ ((𝐼‘∅) ⊆ (𝐵 ∖ 𝐵) → (𝐼‘∅) = ∅)
42	37, 41	syl6com 37	. . . . 5 ⊢ (((𝐼‘∅) ∩ 𝐵) = ∅ → ((𝐼‘∅) ⊆ 𝐵 → (𝐼‘∅) = ∅))
43	35, 42	syl6com 37	. . . 4 ⊢ (((𝐼‘𝐵) ∩ (𝐼‘∅)) = ∅ → ((𝐼‘𝐵) = 𝐵 → ((𝐼‘∅) ⊆ 𝐵 → (𝐼‘∅) = ∅)))
44	43	com13 88	. . 3 ⊢ ((𝐼‘∅) ⊆ 𝐵 → ((𝐼‘𝐵) = 𝐵 → (((𝐼‘𝐵) ∩ (𝐼‘∅)) = ∅ → (𝐼‘∅) = ∅)))
45	29, 30, 44	syl2im 40	. 2 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐼 ∈ (𝒫 𝐵 ↑𝑚 𝒫 𝐵)) → (((𝐼‘𝐵) = 𝐵 ∧ ∀𝑠 ∈ 𝒫 𝐵∀𝑡 ∈ 𝒫 𝐵((𝑠 ∩ 𝑡) = ∅ → ((𝐼‘𝑠) ∩ (𝐼‘𝑡)) = ∅)) → (((𝐼‘𝐵) ∩ (𝐼‘∅)) = ∅ → (𝐼‘∅) = ∅)))
46	24, 45	mpdd 43	1 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐼 ∈ (𝒫 𝐵 ↑𝑚 𝒫 𝐵)) → (((𝐼‘𝐵) = 𝐵 ∧ ∀𝑠 ∈ 𝒫 𝐵∀𝑡 ∈ 𝒫 𝐵((𝑠 ∩ 𝑡) = ∅ → ((𝐼‘𝑠) ∩ (𝐼‘𝑡)) = ∅)) → (𝐼‘∅) = ∅))

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   = wceq 1522   ∈ wcel 2080  ∀wral 3104   ∖ cdif 3858   ∩ cin 3860   ⊆ wss 3861  ∅c0 4213  𝒫 cpw 4455  ⟶wf 6224  ‘cfv 6228  (class class class)co 7019   ↑_𝑚 cmap 8259

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1778  ax-4 1792  ax-5 1889  ax-6 1948  ax-7 1993  ax-8 2082  ax-9 2090  ax-10 2111  ax-11 2125  ax-12 2140  ax-13 2343  ax-ext 2768  ax-sep 5097  ax-nul 5104  ax-pow 5160  ax-pr 5224  ax-un 7322

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1525  df-ex 1763  df-nf 1767  df-sb 2042  df-mo 2575  df-eu 2611  df-clab 2775  df-cleq 2787  df-clel 2862  df-nfc 2934  df-ne 2984  df-ral 3109  df-rex 3110  df-rab 3113  df-v 3438  df-sbc 3708  df-csb 3814  df-dif 3864  df-un 3866  df-in 3868  df-ss 3876  df-nul 4214  df-if 4384  df-pw 4457  df-sn 4475  df-pr 4477  df-op 4481  df-uni 4748  df-iun 4829  df-br 4965  df-opab 5027  df-mpt 5044  df-id 5351  df-xp 5452  df-rel 5453  df-cnv 5454  df-co 5455  df-dm 5456  df-rn 5457  df-res 5458  df-ima 5459  df-iota 6192  df-fun 6230  df-fn 6231  df-f 6232  df-fv 6236  df-ov 7022  df-oprab 7023  df-mpo 7024  df-1st 7548  df-2nd 7549  df-map 8261

This theorem is referenced by: (None)