Theorem ntrk0kbimka 41538

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 41537. (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 4552	. . . . 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 5273	. . . . 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 4136	. . . . . . 7 ⊢ (𝑠 = 𝐵 → (𝑠 ∩ 𝑡) = (𝐵 ∩ 𝑡))
7	6	eqeq1d 2740	. . . . . 6 ⊢ (𝑠 = 𝐵 → ((𝑠 ∩ 𝑡) = ∅ ↔ (𝐵 ∩ 𝑡) = ∅))
8		fveq2 6756	. . . . . . . 8 ⊢ (𝑠 = 𝐵 → (𝐼‘𝑠) = (𝐼‘𝐵))
9	8	ineq1d 4142	. . . . . . 7 ⊢ (𝑠 = 𝐵 → ((𝐼‘𝑠) ∩ (𝐼‘𝑡)) = ((𝐼‘𝐵) ∩ (𝐼‘𝑡)))
10	9	eqeq1d 2740	. . . . . 6 ⊢ (𝑠 = 𝐵 → (((𝐼‘𝑠) ∩ (𝐼‘𝑡)) = ∅ ↔ ((𝐼‘𝐵) ∩ (𝐼‘𝑡)) = ∅))
11	7, 10	imbi12d 344	. . . . 5 ⊢ (𝑠 = 𝐵 → (((𝑠 ∩ 𝑡) = ∅ → ((𝐼‘𝑠) ∩ (𝐼‘𝑡)) = ∅) ↔ ((𝐵 ∩ 𝑡) = ∅ → ((𝐼‘𝐵) ∩ (𝐼‘𝑡)) = ∅)))
12		ineq2 4137	. . . . . . . 8 ⊢ (𝑡 = ∅ → (𝐵 ∩ 𝑡) = (𝐵 ∩ ∅))
13	12	eqeq1d 2740	. . . . . . 7 ⊢ (𝑡 = ∅ → ((𝐵 ∩ 𝑡) = ∅ ↔ (𝐵 ∩ ∅) = ∅))
14		fveq2 6756	. . . . . . . . 9 ⊢ (𝑡 = ∅ → (𝐼‘𝑡) = (𝐼‘∅))
15	14	ineq2d 4143	. . . . . . . 8 ⊢ (𝑡 = ∅ → ((𝐼‘𝐵) ∩ (𝐼‘𝑡)) = ((𝐼‘𝐵) ∩ (𝐼‘∅)))
16	15	eqeq1d 2740	. . . . . . 7 ⊢ (𝑡 = ∅ → (((𝐼‘𝐵) ∩ (𝐼‘𝑡)) = ∅ ↔ ((𝐼‘𝐵) ∩ (𝐼‘∅)) = ∅))
17	13, 16	imbi12d 344	. . . . . 6 ⊢ (𝑡 = ∅ → (((𝐵 ∩ 𝑡) = ∅ → ((𝐼‘𝐵) ∩ (𝐼‘𝑡)) = ∅) ↔ ((𝐵 ∩ ∅) = ∅ → ((𝐼‘𝐵) ∩ (𝐼‘∅)) = ∅)))
18		in0 4322	. . . . . . 7 ⊢ (𝐵 ∩ ∅) = ∅
19		pm5.5 361	. . . . . . 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 278	. . . . 5 ⊢ (𝑡 = ∅ → (((𝐵 ∩ 𝑡) = ∅ → ((𝐼‘𝐵) ∩ (𝐼‘𝑡)) = ∅) ↔ ((𝐼‘𝐵) ∩ (𝐼‘∅)) = ∅))
22	11, 21	rspc2va 3563	. . . 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 412	. 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 8595	. . . . . 6 ⊢ (𝐼 ∈ (𝒫 𝐵 ↑m 𝒫 𝐵) → 𝐼:𝒫 𝐵⟶𝒫 𝐵)
26	25	adantl 481	. . . . 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 6944	. . . 4 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐼 ∈ (𝒫 𝐵 ↑m 𝒫 𝐵)) → (𝐼‘∅) ∈ 𝒫 𝐵)
29	28	elpwid 4541	. . 3 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐼 ∈ (𝒫 𝐵 ↑m 𝒫 𝐵)) → (𝐼‘∅) ⊆ 𝐵)
30		simpl 482	. . 3 ⊢ (((𝐼‘𝐵) = 𝐵 ∧ ∀𝑠 ∈ 𝒫 𝐵∀𝑡 ∈ 𝒫 𝐵((𝑠 ∩ 𝑡) = ∅ → ((𝐼‘𝑠) ∩ (𝐼‘𝑡)) = ∅)) → (𝐼‘𝐵) = 𝐵)
31		ineq1 4136	. . . . . . . 8 ⊢ ((𝐼‘𝐵) = 𝐵 → ((𝐼‘𝐵) ∩ (𝐼‘∅)) = (𝐵 ∩ (𝐼‘∅)))
32		incom 4131	. . . . . . . 8 ⊢ (𝐵 ∩ (𝐼‘∅)) = ((𝐼‘∅) ∩ 𝐵)
33	31, 32	eqtrdi 2795	. . . . . . 7 ⊢ ((𝐼‘𝐵) = 𝐵 → ((𝐼‘𝐵) ∩ (𝐼‘∅)) = ((𝐼‘∅) ∩ 𝐵))
34	33	eqeq1d 2740	. . . . . 6 ⊢ ((𝐼‘𝐵) = 𝐵 → (((𝐼‘𝐵) ∩ (𝐼‘∅)) = ∅ ↔ ((𝐼‘∅) ∩ 𝐵) = ∅))
35	34	biimpd 228	. . . . 5 ⊢ ((𝐼‘𝐵) = 𝐵 → (((𝐼‘𝐵) ∩ (𝐼‘∅)) = ∅ → ((𝐼‘∅) ∩ 𝐵) = ∅))
36		reldisj 4382	. . . . . . 7 ⊢ ((𝐼‘∅) ⊆ 𝐵 → (((𝐼‘∅) ∩ 𝐵) = ∅ ↔ (𝐼‘∅) ⊆ (𝐵 ∖ 𝐵)))
37	36	biimpd 228	. . . . . 6 ⊢ ((𝐼‘∅) ⊆ 𝐵 → (((𝐼‘∅) ∩ 𝐵) = ∅ → (𝐼‘∅) ⊆ (𝐵 ∖ 𝐵)))
38		difid 4301	. . . . . . . 8 ⊢ (𝐵 ∖ 𝐵) = ∅
39	38	sseq2i 3946	. . . . . . 7 ⊢ ((𝐼‘∅) ⊆ (𝐵 ∖ 𝐵) ↔ (𝐼‘∅) ⊆ ∅)
40		ss0 4329	. . . . . . 7 ⊢ ((𝐼‘∅) ⊆ ∅ → (𝐼‘∅) = ∅)
41	39, 40	sylbi 216	. . . . . 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 205   ∧ wa 395   = wceq 1539   ∈ wcel 2108  ∀wral 3063   ∖ cdif 3880   ∩ cin 3882   ⊆ wss 3883  ∅c0 4253  𝒫 cpw 4530  ⟶wf 6414  ‘cfv 6418  (class class class)co 7255   ↑_m cmap 8573

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-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-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-fv 6426  df-ov 7258  df-oprab 7259  df-mpo 7260  df-1st 7804  df-2nd 7805  df-map 8575

This theorem is referenced by: (None)