Theorem ntrk0kbimka 44222

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 44221. (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 4572	. . . . 5 ⊢ (𝐵 ∈ 𝑉 → 𝐵 ∈ 𝒫 𝐵)
2	1	ad2antrr 726	. . . 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 5299	. . . . 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 772	. . . 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 4163	. . . . . . 7 ⊢ (𝑠 = 𝐵 → (𝑠 ∩ 𝑡) = (𝐵 ∩ 𝑡))
7	6	eqeq1d 2736	. . . . . 6 ⊢ (𝑠 = 𝐵 → ((𝑠 ∩ 𝑡) = ∅ ↔ (𝐵 ∩ 𝑡) = ∅))
8		fveq2 6832	. . . . . . . 8 ⊢ (𝑠 = 𝐵 → (𝐼‘𝑠) = (𝐼‘𝐵))
9	8	ineq1d 4169	. . . . . . 7 ⊢ (𝑠 = 𝐵 → ((𝐼‘𝑠) ∩ (𝐼‘𝑡)) = ((𝐼‘𝐵) ∩ (𝐼‘𝑡)))
10	9	eqeq1d 2736	. . . . . 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 4164	. . . . . . . 8 ⊢ (𝑡 = ∅ → (𝐵 ∩ 𝑡) = (𝐵 ∩ ∅))
13	12	eqeq1d 2736	. . . . . . 7 ⊢ (𝑡 = ∅ → ((𝐵 ∩ 𝑡) = ∅ ↔ (𝐵 ∩ ∅) = ∅))
14		fveq2 6832	. . . . . . . . 9 ⊢ (𝑡 = ∅ → (𝐼‘𝑡) = (𝐼‘∅))
15	14	ineq2d 4170	. . . . . . . 8 ⊢ (𝑡 = ∅ → ((𝐼‘𝐵) ∩ (𝐼‘𝑡)) = ((𝐼‘𝐵) ∩ (𝐼‘∅)))
16	15	eqeq1d 2736	. . . . . . 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 4345	. . . . . . 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 279	. . . . 5 ⊢ (𝑡 = ∅ → (((𝐵 ∩ 𝑡) = ∅ → ((𝐼‘𝐵) ∩ (𝐼‘𝑡)) = ∅) ↔ ((𝐼‘𝐵) ∩ (𝐼‘∅)) = ∅))
22	11, 21	rspc2va 3586	. . . 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 837	. . 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 8784	. . . . . 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	ffvelcdmd 7028	. . . 4 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐼 ∈ (𝒫 𝐵 ↑m 𝒫 𝐵)) → (𝐼‘∅) ∈ 𝒫 𝐵)
29	28	elpwid 4561	. . 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 4163	. . . . . . . 8 ⊢ ((𝐼‘𝐵) = 𝐵 → ((𝐼‘𝐵) ∩ (𝐼‘∅)) = (𝐵 ∩ (𝐼‘∅)))
32		incom 4159	. . . . . . . 8 ⊢ (𝐵 ∩ (𝐼‘∅)) = ((𝐼‘∅) ∩ 𝐵)
33	31, 32	eqtrdi 2785	. . . . . . 7 ⊢ ((𝐼‘𝐵) = 𝐵 → ((𝐼‘𝐵) ∩ (𝐼‘∅)) = ((𝐼‘∅) ∩ 𝐵))
34	33	eqeq1d 2736	. . . . . 6 ⊢ ((𝐼‘𝐵) = 𝐵 → (((𝐼‘𝐵) ∩ (𝐼‘∅)) = ∅ ↔ ((𝐼‘∅) ∩ 𝐵) = ∅))
35	34	biimpd 229	. . . . 5 ⊢ ((𝐼‘𝐵) = 𝐵 → (((𝐼‘𝐵) ∩ (𝐼‘∅)) = ∅ → ((𝐼‘∅) ∩ 𝐵) = ∅))
36		reldisj 4403	. . . . . . 7 ⊢ ((𝐼‘∅) ⊆ 𝐵 → (((𝐼‘∅) ∩ 𝐵) = ∅ ↔ (𝐼‘∅) ⊆ (𝐵 ∖ 𝐵)))
37	36	biimpd 229	. . . . . 6 ⊢ ((𝐼‘∅) ⊆ 𝐵 → (((𝐼‘∅) ∩ 𝐵) = ∅ → (𝐼‘∅) ⊆ (𝐵 ∖ 𝐵)))
38		difid 4326	. . . . . . . 8 ⊢ (𝐵 ∖ 𝐵) = ∅
39	38	sseq2i 3961	. . . . . . 7 ⊢ ((𝐼‘∅) ⊆ (𝐵 ∖ 𝐵) ↔ (𝐼‘∅) ⊆ ∅)
40		ss0 4352	. . . . . . 7 ⊢ ((𝐼‘∅) ⊆ ∅ → (𝐼‘∅) = ∅)
41	39, 40	sylbi 217	. . . . . 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 206   ∧ wa 395   = wceq 1541   ∈ wcel 2113  ∀wral 3049   ∖ cdif 3896   ∩ cin 3898   ⊆ wss 3899  ∅c0 4283  𝒫 cpw 4552  ⟶wf 6486  ‘cfv 6490  (class class class)co 7356   ↑_m cmap 8761

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2706  ax-sep 5239  ax-nul 5249  ax-pow 5308  ax-pr 5375  ax-un 7678

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2809  df-nfc 2883  df-ne 2931  df-ral 3050  df-rex 3059  df-rab 3398  df-v 3440  df-sbc 3739  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4579  df-pr 4581  df-op 4585  df-uni 4862  df-iun 4946  df-br 5097  df-opab 5159  df-mpt 5178  df-id 5517  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635  df-iota 6446  df-fun 6492  df-fn 6493  df-f 6494  df-fv 6498  df-ov 7359  df-oprab 7360  df-mpo 7361  df-1st 7931  df-2nd 7932  df-map 8763

This theorem is referenced by: (None)