Theorem ntrk0kbimka 44880

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 44879. (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 4577	. . . . 5 ⊢ (𝐵 ∈ 𝑉 → 𝐵 ∈ 𝒫 𝐵)
2	1	ad2antrr 739	. . . 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 5320	. . . . 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 785	. . . 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 4159	. . . . . . 7 ⊢ (𝑠 = 𝐵 → (𝑠 ∩ 𝑡) = (𝐵 ∩ 𝑡))
7	6	eqeq1d 2762	. . . . . 6 ⊢ (𝑠 = 𝐵 → ((𝑠 ∩ 𝑡) = ∅ ↔ (𝐵 ∩ 𝑡) = ∅))
8		fveq2 6879	. . . . . . . 8 ⊢ (𝑠 = 𝐵 → (𝐼‘𝑠) = (𝐼‘𝐵))
9	8	ineq1d 4165	. . . . . . 7 ⊢ (𝑠 = 𝐵 → ((𝐼‘𝑠) ∩ (𝐼‘𝑡)) = ((𝐼‘𝐵) ∩ (𝐼‘𝑡)))
10	9	eqeq1d 2762	. . . . . 6 ⊢ (𝑠 = 𝐵 → (((𝐼‘𝑠) ∩ (𝐼‘𝑡)) = ∅ ↔ ((𝐼‘𝐵) ∩ (𝐼‘𝑡)) = ∅))
11	7, 10	imbi12d 347	. . . . 5 ⊢ (𝑠 = 𝐵 → (((𝑠 ∩ 𝑡) = ∅ → ((𝐼‘𝑠) ∩ (𝐼‘𝑡)) = ∅) ↔ ((𝐵 ∩ 𝑡) = ∅ → ((𝐼‘𝐵) ∩ (𝐼‘𝑡)) = ∅)))
12		ineq2 4160	. . . . . . . 8 ⊢ (𝑡 = ∅ → (𝐵 ∩ 𝑡) = (𝐵 ∩ ∅))
13	12	eqeq1d 2762	. . . . . . 7 ⊢ (𝑡 = ∅ → ((𝐵 ∩ 𝑡) = ∅ ↔ (𝐵 ∩ ∅) = ∅))
14		fveq2 6879	. . . . . . . . 9 ⊢ (𝑡 = ∅ → (𝐼‘𝑡) = (𝐼‘∅))
15	14	ineq2d 4166	. . . . . . . 8 ⊢ (𝑡 = ∅ → ((𝐼‘𝐵) ∩ (𝐼‘𝑡)) = ((𝐼‘𝐵) ∩ (𝐼‘∅)))
16	15	eqeq1d 2762	. . . . . . 7 ⊢ (𝑡 = ∅ → (((𝐼‘𝐵) ∩ (𝐼‘𝑡)) = ∅ ↔ ((𝐼‘𝐵) ∩ (𝐼‘∅)) = ∅))
17	13, 16	imbi12d 347	. . . . . 6 ⊢ (𝑡 = ∅ → (((𝐵 ∩ 𝑡) = ∅ → ((𝐼‘𝐵) ∩ (𝐼‘𝑡)) = ∅) ↔ ((𝐵 ∩ ∅) = ∅ → ((𝐼‘𝐵) ∩ (𝐼‘∅)) = ∅)))
18		in0 4345	. . . . . . 7 ⊢ (𝐵 ∩ ∅) = ∅
19		pm5.5 364	. . . . . . 7 ⊢ ((𝐵 ∩ ∅) = ∅ → (((𝐵 ∩ ∅) = ∅ → ((𝐼‘𝐵) ∩ (𝐼‘∅)) = ∅) ↔ ((𝐼‘𝐵) ∩ (𝐼‘∅)) = ∅))
20	18, 19	mp1i 14	. . . . . 6 ⊢ (𝑡 = ∅ → (((𝐵 ∩ ∅) = ∅ → ((𝐼‘𝐵) ∩ (𝐼‘∅)) = ∅) ↔ ((𝐼‘𝐵) ∩ (𝐼‘∅)) = ∅))
21	17, 20	bitrd 282	. . . . 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 851	. . 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 418	. 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 8849	. . . . . 6 ⊢ (𝐼 ∈ (𝒫 𝐵 ↑m 𝒫 𝐵) → 𝐼:𝒫 𝐵⟶𝒫 𝐵)
26	25	adantl 487	. . . . 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 7079	. . . 4 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐼 ∈ (𝒫 𝐵 ↑m 𝒫 𝐵)) → (𝐼‘∅) ∈ 𝒫 𝐵)
29	28	elpwid 4566	. . 3 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐼 ∈ (𝒫 𝐵 ↑m 𝒫 𝐵)) → (𝐼‘∅) ⊆ 𝐵)
30		simpl 488	. . 3 ⊢ (((𝐼‘𝐵) = 𝐵 ∧ ∀𝑠 ∈ 𝒫 𝐵∀𝑡 ∈ 𝒫 𝐵((𝑠 ∩ 𝑡) = ∅ → ((𝐼‘𝑠) ∩ (𝐼‘𝑡)) = ∅)) → (𝐼‘𝐵) = 𝐵)
31		ineq1 4159	. . . . . . . 8 ⊢ ((𝐼‘𝐵) = 𝐵 → ((𝐼‘𝐵) ∩ (𝐼‘∅)) = (𝐵 ∩ (𝐼‘∅)))
32		incom 4155	. . . . . . . 8 ⊢ (𝐵 ∩ (𝐼‘∅)) = ((𝐼‘∅) ∩ 𝐵)
33	31, 32	eqtrdi 2811	. . . . . . 7 ⊢ ((𝐼‘𝐵) = 𝐵 → ((𝐼‘𝐵) ∩ (𝐼‘∅)) = ((𝐼‘∅) ∩ 𝐵))
34	33	eqeq1d 2762	. . . . . 6 ⊢ ((𝐼‘𝐵) = 𝐵 → (((𝐼‘𝐵) ∩ (𝐼‘∅)) = ∅ ↔ ((𝐼‘∅) ∩ 𝐵) = ∅))
35	34	biimpd 232	. . . . 5 ⊢ ((𝐼‘𝐵) = 𝐵 → (((𝐼‘𝐵) ∩ (𝐼‘∅)) = ∅ → ((𝐼‘∅) ∩ 𝐵) = ∅))
36		reldisj 4406	. . . . . . 7 ⊢ ((𝐼‘∅) ⊆ 𝐵 → (((𝐼‘∅) ∩ 𝐵) = ∅ ↔ (𝐼‘∅) ⊆ (𝐵 ∖ 𝐵)))
37	36	biimpd 232	. . . . . 6 ⊢ ((𝐼‘∅) ⊆ 𝐵 → (((𝐼‘∅) ∩ 𝐵) = ∅ → (𝐼‘∅) ⊆ (𝐵 ∖ 𝐵)))
38		difid 4325	. . . . . . . 8 ⊢ (𝐵 ∖ 𝐵) = ∅
39	38	sseq2i 3960	. . . . . . 7 ⊢ ((𝐼‘∅) ⊆ (𝐵 ∖ 𝐵) ↔ (𝐼‘∅) ⊆ ∅)
40		ss0 4352	. . . . . . 7 ⊢ ((𝐼‘∅) ⊆ ∅ → (𝐼‘∅) = ∅)
41	39, 40	sylbi 220	. . . . . 6 ⊢ ((𝐼‘∅) ⊆ (𝐵 ∖ 𝐵) → (𝐼‘∅) = ∅)
42	37, 41	syl6com 38	. . . . 5 ⊢ (((𝐼‘∅) ∩ 𝐵) = ∅ → ((𝐼‘∅) ⊆ 𝐵 → (𝐼‘∅) = ∅))
43	35, 42	syl6com 38	. . . 4 ⊢ (((𝐼‘𝐵) ∩ (𝐼‘∅)) = ∅ → ((𝐼‘𝐵) = 𝐵 → ((𝐼‘∅) ⊆ 𝐵 → (𝐼‘∅) = ∅)))
44	43	com13 89	. . 3 ⊢ ((𝐼‘∅) ⊆ 𝐵 → ((𝐼‘𝐵) = 𝐵 → (((𝐼‘𝐵) ∩ (𝐼‘∅)) = ∅ → (𝐼‘∅) = ∅)))
45	29, 30, 44	syl2im 41	. 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 44	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‘∅) = ∅))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ∀wral 3076   ∖ cdif 3896   ∩ cin 3898   ⊆ wss 3899  ∅c0 4279  𝒫 cpw 4557  ⟶wf 6529  ‘cfv 6533  (class class class)co 7414   ↑_m cmap 8827

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-sep 5251  ax-nul 5263  ax-pow 5330  ax-pr 5398  ax-un 7737

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-fv 6541  df-ov 7417  df-oprab 7418  df-mpo 7419  df-1st 7987  df-2nd 7988  df-map 8829

This theorem is used by: (None)