Theorem ntrk0kbimka 44280

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 44279. (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 4574	. . . . 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 5301	. . . . 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 4165	. . . . . . 7 ⊢ (𝑠 = 𝐵 → (𝑠 ∩ 𝑡) = (𝐵 ∩ 𝑡))
7	6	eqeq1d 2738	. . . . . 6 ⊢ (𝑠 = 𝐵 → ((𝑠 ∩ 𝑡) = ∅ ↔ (𝐵 ∩ 𝑡) = ∅))
8		fveq2 6834	. . . . . . . 8 ⊢ (𝑠 = 𝐵 → (𝐼‘𝑠) = (𝐼‘𝐵))
9	8	ineq1d 4171	. . . . . . 7 ⊢ (𝑠 = 𝐵 → ((𝐼‘𝑠) ∩ (𝐼‘𝑡)) = ((𝐼‘𝐵) ∩ (𝐼‘𝑡)))
10	9	eqeq1d 2738	. . . . . 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 4166	. . . . . . . 8 ⊢ (𝑡 = ∅ → (𝐵 ∩ 𝑡) = (𝐵 ∩ ∅))
13	12	eqeq1d 2738	. . . . . . 7 ⊢ (𝑡 = ∅ → ((𝐵 ∩ 𝑡) = ∅ ↔ (𝐵 ∩ ∅) = ∅))
14		fveq2 6834	. . . . . . . . 9 ⊢ (𝑡 = ∅ → (𝐼‘𝑡) = (𝐼‘∅))
15	14	ineq2d 4172	. . . . . . . 8 ⊢ (𝑡 = ∅ → ((𝐼‘𝐵) ∩ (𝐼‘𝑡)) = ((𝐼‘𝐵) ∩ (𝐼‘∅)))
16	15	eqeq1d 2738	. . . . . . 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 4347	. . . . . . 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 3588	. . . 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 8786	. . . . . 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 7030	. . . 4 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐼 ∈ (𝒫 𝐵 ↑m 𝒫 𝐵)) → (𝐼‘∅) ∈ 𝒫 𝐵)
29	28	elpwid 4563	. . 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 4165	. . . . . . . 8 ⊢ ((𝐼‘𝐵) = 𝐵 → ((𝐼‘𝐵) ∩ (𝐼‘∅)) = (𝐵 ∩ (𝐼‘∅)))
32		incom 4161	. . . . . . . 8 ⊢ (𝐵 ∩ (𝐼‘∅)) = ((𝐼‘∅) ∩ 𝐵)
33	31, 32	eqtrdi 2787	. . . . . . 7 ⊢ ((𝐼‘𝐵) = 𝐵 → ((𝐼‘𝐵) ∩ (𝐼‘∅)) = ((𝐼‘∅) ∩ 𝐵))
34	33	eqeq1d 2738	. . . . . 6 ⊢ ((𝐼‘𝐵) = 𝐵 → (((𝐼‘𝐵) ∩ (𝐼‘∅)) = ∅ ↔ ((𝐼‘∅) ∩ 𝐵) = ∅))
35	34	biimpd 229	. . . . 5 ⊢ ((𝐼‘𝐵) = 𝐵 → (((𝐼‘𝐵) ∩ (𝐼‘∅)) = ∅ → ((𝐼‘∅) ∩ 𝐵) = ∅))
36		reldisj 4405	. . . . . . 7 ⊢ ((𝐼‘∅) ⊆ 𝐵 → (((𝐼‘∅) ∩ 𝐵) = ∅ ↔ (𝐼‘∅) ⊆ (𝐵 ∖ 𝐵)))
37	36	biimpd 229	. . . . . 6 ⊢ ((𝐼‘∅) ⊆ 𝐵 → (((𝐼‘∅) ∩ 𝐵) = ∅ → (𝐼‘∅) ⊆ (𝐵 ∖ 𝐵)))
38		difid 4328	. . . . . . . 8 ⊢ (𝐵 ∖ 𝐵) = ∅
39	38	sseq2i 3963	. . . . . . 7 ⊢ ((𝐼‘∅) ⊆ (𝐵 ∖ 𝐵) ↔ (𝐼‘∅) ⊆ ∅)
40		ss0 4354	. . . . . . 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 3051   ∖ cdif 3898   ∩ cin 3900   ⊆ wss 3901  ∅c0 4285  𝒫 cpw 4554  ⟶wf 6488  ‘cfv 6492  (class class class)co 7358   ↑_m cmap 8763

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 2184  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680

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 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-fv 6500  df-ov 7361  df-oprab 7362  df-mpo 7363  df-1st 7933  df-2nd 7934  df-map 8765

This theorem is referenced by: (None)