Mathbox for Richard Penner < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  ntrk0kbimka Structured version   Visualization version   GIF version

Theorem ntrk0kbimka 37357
 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 37356. (Contributed by RP, 12-Jun-2021.)
Assertion
Ref Expression
ntrk0kbimka ((𝐵𝑉𝐼 ∈ (𝒫 𝐵𝑚 𝒫 𝐵)) → (((𝐼𝐵) = 𝐵 ∧ ∀𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵((𝑠𝑡) = ∅ → ((𝐼𝑠) ∩ (𝐼𝑡)) = ∅)) → (𝐼‘∅) = ∅))
Distinct variable groups:   𝐵,𝑠,𝑡   𝐼,𝑠,𝑡
Allowed substitution hints:   𝑉(𝑡,𝑠)

Proof of Theorem ntrk0kbimka
StepHypRef Expression
1 pwidg 4121 . . . . 5 (𝐵𝑉𝐵 ∈ 𝒫 𝐵)
21ad2antrr 758 . . . 4 (((𝐵𝑉𝐼 ∈ (𝒫 𝐵𝑚 𝒫 𝐵)) ∧ ((𝐼𝐵) = 𝐵 ∧ ∀𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵((𝑠𝑡) = ∅ → ((𝐼𝑠) ∩ (𝐼𝑡)) = ∅))) → 𝐵 ∈ 𝒫 𝐵)
3 0elpw 4760 . . . . 5 ∅ ∈ 𝒫 𝐵
43a1i 11 . . . 4 (((𝐵𝑉𝐼 ∈ (𝒫 𝐵𝑚 𝒫 𝐵)) ∧ ((𝐼𝐵) = 𝐵 ∧ ∀𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵((𝑠𝑡) = ∅ → ((𝐼𝑠) ∩ (𝐼𝑡)) = ∅))) → ∅ ∈ 𝒫 𝐵)
5 simprr 792 . . . 4 (((𝐵𝑉𝐼 ∈ (𝒫 𝐵𝑚 𝒫 𝐵)) ∧ ((𝐼𝐵) = 𝐵 ∧ ∀𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵((𝑠𝑡) = ∅ → ((𝐼𝑠) ∩ (𝐼𝑡)) = ∅))) → ∀𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵((𝑠𝑡) = ∅ → ((𝐼𝑠) ∩ (𝐼𝑡)) = ∅))
6 ineq1 3769 . . . . . . 7 (𝑠 = 𝐵 → (𝑠𝑡) = (𝐵𝑡))
76eqeq1d 2612 . . . . . 6 (𝑠 = 𝐵 → ((𝑠𝑡) = ∅ ↔ (𝐵𝑡) = ∅))
8 fveq2 6103 . . . . . . . 8 (𝑠 = 𝐵 → (𝐼𝑠) = (𝐼𝐵))
98ineq1d 3775 . . . . . . 7 (𝑠 = 𝐵 → ((𝐼𝑠) ∩ (𝐼𝑡)) = ((𝐼𝐵) ∩ (𝐼𝑡)))
109eqeq1d 2612 . . . . . 6 (𝑠 = 𝐵 → (((𝐼𝑠) ∩ (𝐼𝑡)) = ∅ ↔ ((𝐼𝐵) ∩ (𝐼𝑡)) = ∅))
117, 10imbi12d 333 . . . . 5 (𝑠 = 𝐵 → (((𝑠𝑡) = ∅ → ((𝐼𝑠) ∩ (𝐼𝑡)) = ∅) ↔ ((𝐵𝑡) = ∅ → ((𝐼𝐵) ∩ (𝐼𝑡)) = ∅)))
12 ineq2 3770 . . . . . . . 8 (𝑡 = ∅ → (𝐵𝑡) = (𝐵 ∩ ∅))
1312eqeq1d 2612 . . . . . . 7 (𝑡 = ∅ → ((𝐵𝑡) = ∅ ↔ (𝐵 ∩ ∅) = ∅))
14 fveq2 6103 . . . . . . . . 9 (𝑡 = ∅ → (𝐼𝑡) = (𝐼‘∅))
1514ineq2d 3776 . . . . . . . 8 (𝑡 = ∅ → ((𝐼𝐵) ∩ (𝐼𝑡)) = ((𝐼𝐵) ∩ (𝐼‘∅)))
1615eqeq1d 2612 . . . . . . 7 (𝑡 = ∅ → (((𝐼𝐵) ∩ (𝐼𝑡)) = ∅ ↔ ((𝐼𝐵) ∩ (𝐼‘∅)) = ∅))
1713, 16imbi12d 333 . . . . . 6 (𝑡 = ∅ → (((𝐵𝑡) = ∅ → ((𝐼𝐵) ∩ (𝐼𝑡)) = ∅) ↔ ((𝐵 ∩ ∅) = ∅ → ((𝐼𝐵) ∩ (𝐼‘∅)) = ∅)))
18 in0 3920 . . . . . . 7 (𝐵 ∩ ∅) = ∅
19 pm5.5 350 . . . . . . 7 ((𝐵 ∩ ∅) = ∅ → (((𝐵 ∩ ∅) = ∅ → ((𝐼𝐵) ∩ (𝐼‘∅)) = ∅) ↔ ((𝐼𝐵) ∩ (𝐼‘∅)) = ∅))
2018, 19mp1i 13 . . . . . 6 (𝑡 = ∅ → (((𝐵 ∩ ∅) = ∅ → ((𝐼𝐵) ∩ (𝐼‘∅)) = ∅) ↔ ((𝐼𝐵) ∩ (𝐼‘∅)) = ∅))
2117, 20bitrd 267 . . . . 5 (𝑡 = ∅ → (((𝐵𝑡) = ∅ → ((𝐼𝐵) ∩ (𝐼𝑡)) = ∅) ↔ ((𝐼𝐵) ∩ (𝐼‘∅)) = ∅))
2211, 21rspc2va 3294 . . . 4 (((𝐵 ∈ 𝒫 𝐵 ∧ ∅ ∈ 𝒫 𝐵) ∧ ∀𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵((𝑠𝑡) = ∅ → ((𝐼𝑠) ∩ (𝐼𝑡)) = ∅)) → ((𝐼𝐵) ∩ (𝐼‘∅)) = ∅)
232, 4, 5, 22syl21anc 1317 . . 3 (((𝐵𝑉𝐼 ∈ (𝒫 𝐵𝑚 𝒫 𝐵)) ∧ ((𝐼𝐵) = 𝐵 ∧ ∀𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵((𝑠𝑡) = ∅ → ((𝐼𝑠) ∩ (𝐼𝑡)) = ∅))) → ((𝐼𝐵) ∩ (𝐼‘∅)) = ∅)
2423ex 449 . 2 ((𝐵𝑉𝐼 ∈ (𝒫 𝐵𝑚 𝒫 𝐵)) → (((𝐼𝐵) = 𝐵 ∧ ∀𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵((𝑠𝑡) = ∅ → ((𝐼𝑠) ∩ (𝐼𝑡)) = ∅)) → ((𝐼𝐵) ∩ (𝐼‘∅)) = ∅))
25 elmapi 7765 . . . . . 6 (𝐼 ∈ (𝒫 𝐵𝑚 𝒫 𝐵) → 𝐼:𝒫 𝐵⟶𝒫 𝐵)
2625adantl 481 . . . . 5 ((𝐵𝑉𝐼 ∈ (𝒫 𝐵𝑚 𝒫 𝐵)) → 𝐼:𝒫 𝐵⟶𝒫 𝐵)
273a1i 11 . . . . 5 ((𝐵𝑉𝐼 ∈ (𝒫 𝐵𝑚 𝒫 𝐵)) → ∅ ∈ 𝒫 𝐵)
2826, 27ffvelrnd 6268 . . . 4 ((𝐵𝑉𝐼 ∈ (𝒫 𝐵𝑚 𝒫 𝐵)) → (𝐼‘∅) ∈ 𝒫 𝐵)
2928elpwid 4118 . . 3 ((𝐵𝑉𝐼 ∈ (𝒫 𝐵𝑚 𝒫 𝐵)) → (𝐼‘∅) ⊆ 𝐵)
30 simpl 472 . . 3 (((𝐼𝐵) = 𝐵 ∧ ∀𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵((𝑠𝑡) = ∅ → ((𝐼𝑠) ∩ (𝐼𝑡)) = ∅)) → (𝐼𝐵) = 𝐵)
31 ineq1 3769 . . . . . . . 8 ((𝐼𝐵) = 𝐵 → ((𝐼𝐵) ∩ (𝐼‘∅)) = (𝐵 ∩ (𝐼‘∅)))
32 incom 3767 . . . . . . . 8 (𝐵 ∩ (𝐼‘∅)) = ((𝐼‘∅) ∩ 𝐵)
3331, 32syl6eq 2660 . . . . . . 7 ((𝐼𝐵) = 𝐵 → ((𝐼𝐵) ∩ (𝐼‘∅)) = ((𝐼‘∅) ∩ 𝐵))
3433eqeq1d 2612 . . . . . 6 ((𝐼𝐵) = 𝐵 → (((𝐼𝐵) ∩ (𝐼‘∅)) = ∅ ↔ ((𝐼‘∅) ∩ 𝐵) = ∅))
3534biimpd 218 . . . . 5 ((𝐼𝐵) = 𝐵 → (((𝐼𝐵) ∩ (𝐼‘∅)) = ∅ → ((𝐼‘∅) ∩ 𝐵) = ∅))
36 reldisj 3972 . . . . . . 7 ((𝐼‘∅) ⊆ 𝐵 → (((𝐼‘∅) ∩ 𝐵) = ∅ ↔ (𝐼‘∅) ⊆ (𝐵𝐵)))
3736biimpd 218 . . . . . 6 ((𝐼‘∅) ⊆ 𝐵 → (((𝐼‘∅) ∩ 𝐵) = ∅ → (𝐼‘∅) ⊆ (𝐵𝐵)))
38 difid 3902 . . . . . . . 8 (𝐵𝐵) = ∅
3938sseq2i 3593 . . . . . . 7 ((𝐼‘∅) ⊆ (𝐵𝐵) ↔ (𝐼‘∅) ⊆ ∅)
40 ss0 3926 . . . . . . 7 ((𝐼‘∅) ⊆ ∅ → (𝐼‘∅) = ∅)
4139, 40sylbi 206 . . . . . 6 ((𝐼‘∅) ⊆ (𝐵𝐵) → (𝐼‘∅) = ∅)
4237, 41syl6com 36 . . . . 5 (((𝐼‘∅) ∩ 𝐵) = ∅ → ((𝐼‘∅) ⊆ 𝐵 → (𝐼‘∅) = ∅))
4335, 42syl6com 36 . . . 4 (((𝐼𝐵) ∩ (𝐼‘∅)) = ∅ → ((𝐼𝐵) = 𝐵 → ((𝐼‘∅) ⊆ 𝐵 → (𝐼‘∅) = ∅)))
4443com13 86 . . 3 ((𝐼‘∅) ⊆ 𝐵 → ((𝐼𝐵) = 𝐵 → (((𝐼𝐵) ∩ (𝐼‘∅)) = ∅ → (𝐼‘∅) = ∅)))
4529, 30, 44syl2im 39 . 2 ((𝐵𝑉𝐼 ∈ (𝒫 𝐵𝑚 𝒫 𝐵)) → (((𝐼𝐵) = 𝐵 ∧ ∀𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵((𝑠𝑡) = ∅ → ((𝐼𝑠) ∩ (𝐼𝑡)) = ∅)) → (((𝐼𝐵) ∩ (𝐼‘∅)) = ∅ → (𝐼‘∅) = ∅)))
4624, 45mpdd 42 1 ((𝐵𝑉𝐼 ∈ (𝒫 𝐵𝑚 𝒫 𝐵)) → (((𝐼𝐵) = 𝐵 ∧ ∀𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵((𝑠𝑡) = ∅ → ((𝐼𝑠) ∩ (𝐼𝑡)) = ∅)) → (𝐼‘∅) = ∅))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896   ∖ cdif 3537   ∩ cin 3539   ⊆ wss 3540  ∅c0 3874  𝒫 cpw 4108  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549   ↑𝑚 cmap 7744 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-1st 7059  df-2nd 7060  df-map 7746 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator