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Theorem ntrk0kbimka 38856
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 38855. (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 4312 . . . . 5 (𝐵𝑉𝐵 ∈ 𝒫 𝐵)
21ad2antrr 705 . . . 4 (((𝐵𝑉𝐼 ∈ (𝒫 𝐵𝑚 𝒫 𝐵)) ∧ ((𝐼𝐵) = 𝐵 ∧ ∀𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵((𝑠𝑡) = ∅ → ((𝐼𝑠) ∩ (𝐼𝑡)) = ∅))) → 𝐵 ∈ 𝒫 𝐵)
3 0elpw 4965 . . . . 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 756 . . . 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 3958 . . . . . . 7 (𝑠 = 𝐵 → (𝑠𝑡) = (𝐵𝑡))
76eqeq1d 2773 . . . . . 6 (𝑠 = 𝐵 → ((𝑠𝑡) = ∅ ↔ (𝐵𝑡) = ∅))
8 fveq2 6330 . . . . . . . 8 (𝑠 = 𝐵 → (𝐼𝑠) = (𝐼𝐵))
98ineq1d 3964 . . . . . . 7 (𝑠 = 𝐵 → ((𝐼𝑠) ∩ (𝐼𝑡)) = ((𝐼𝐵) ∩ (𝐼𝑡)))
109eqeq1d 2773 . . . . . 6 (𝑠 = 𝐵 → (((𝐼𝑠) ∩ (𝐼𝑡)) = ∅ ↔ ((𝐼𝐵) ∩ (𝐼𝑡)) = ∅))
117, 10imbi12d 333 . . . . 5 (𝑠 = 𝐵 → (((𝑠𝑡) = ∅ → ((𝐼𝑠) ∩ (𝐼𝑡)) = ∅) ↔ ((𝐵𝑡) = ∅ → ((𝐼𝐵) ∩ (𝐼𝑡)) = ∅)))
12 ineq2 3959 . . . . . . . 8 (𝑡 = ∅ → (𝐵𝑡) = (𝐵 ∩ ∅))
1312eqeq1d 2773 . . . . . . 7 (𝑡 = ∅ → ((𝐵𝑡) = ∅ ↔ (𝐵 ∩ ∅) = ∅))
14 fveq2 6330 . . . . . . . . 9 (𝑡 = ∅ → (𝐼𝑡) = (𝐼‘∅))
1514ineq2d 3965 . . . . . . . 8 (𝑡 = ∅ → ((𝐼𝐵) ∩ (𝐼𝑡)) = ((𝐼𝐵) ∩ (𝐼‘∅)))
1615eqeq1d 2773 . . . . . . 7 (𝑡 = ∅ → (((𝐼𝐵) ∩ (𝐼𝑡)) = ∅ ↔ ((𝐼𝐵) ∩ (𝐼‘∅)) = ∅))
1713, 16imbi12d 333 . . . . . 6 (𝑡 = ∅ → (((𝐵𝑡) = ∅ → ((𝐼𝐵) ∩ (𝐼𝑡)) = ∅) ↔ ((𝐵 ∩ ∅) = ∅ → ((𝐼𝐵) ∩ (𝐼‘∅)) = ∅)))
18 in0 4112 . . . . . . 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 268 . . . . 5 (𝑡 = ∅ → (((𝐵𝑡) = ∅ → ((𝐼𝐵) ∩ (𝐼𝑡)) = ∅) ↔ ((𝐼𝐵) ∩ (𝐼‘∅)) = ∅))
2211, 21rspc2va 3473 . . . 4 (((𝐵 ∈ 𝒫 𝐵 ∧ ∅ ∈ 𝒫 𝐵) ∧ ∀𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵((𝑠𝑡) = ∅ → ((𝐼𝑠) ∩ (𝐼𝑡)) = ∅)) → ((𝐼𝐵) ∩ (𝐼‘∅)) = ∅)
232, 4, 5, 22syl21anc 1475 . . 3 (((𝐵𝑉𝐼 ∈ (𝒫 𝐵𝑚 𝒫 𝐵)) ∧ ((𝐼𝐵) = 𝐵 ∧ ∀𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵((𝑠𝑡) = ∅ → ((𝐼𝑠) ∩ (𝐼𝑡)) = ∅))) → ((𝐼𝐵) ∩ (𝐼‘∅)) = ∅)
2423ex 397 . 2 ((𝐵𝑉𝐼 ∈ (𝒫 𝐵𝑚 𝒫 𝐵)) → (((𝐼𝐵) = 𝐵 ∧ ∀𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵((𝑠𝑡) = ∅ → ((𝐼𝑠) ∩ (𝐼𝑡)) = ∅)) → ((𝐼𝐵) ∩ (𝐼‘∅)) = ∅))
25 elmapi 8029 . . . . . 6 (𝐼 ∈ (𝒫 𝐵𝑚 𝒫 𝐵) → 𝐼:𝒫 𝐵⟶𝒫 𝐵)
2625adantl 467 . . . . 5 ((𝐵𝑉𝐼 ∈ (𝒫 𝐵𝑚 𝒫 𝐵)) → 𝐼:𝒫 𝐵⟶𝒫 𝐵)
273a1i 11 . . . . 5 ((𝐵𝑉𝐼 ∈ (𝒫 𝐵𝑚 𝒫 𝐵)) → ∅ ∈ 𝒫 𝐵)
2826, 27ffvelrnd 6501 . . . 4 ((𝐵𝑉𝐼 ∈ (𝒫 𝐵𝑚 𝒫 𝐵)) → (𝐼‘∅) ∈ 𝒫 𝐵)
2928elpwid 4309 . . 3 ((𝐵𝑉𝐼 ∈ (𝒫 𝐵𝑚 𝒫 𝐵)) → (𝐼‘∅) ⊆ 𝐵)
30 simpl 468 . . 3 (((𝐼𝐵) = 𝐵 ∧ ∀𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵((𝑠𝑡) = ∅ → ((𝐼𝑠) ∩ (𝐼𝑡)) = ∅)) → (𝐼𝐵) = 𝐵)
31 ineq1 3958 . . . . . . . 8 ((𝐼𝐵) = 𝐵 → ((𝐼𝐵) ∩ (𝐼‘∅)) = (𝐵 ∩ (𝐼‘∅)))
32 incom 3956 . . . . . . . 8 (𝐵 ∩ (𝐼‘∅)) = ((𝐼‘∅) ∩ 𝐵)
3331, 32syl6eq 2821 . . . . . . 7 ((𝐼𝐵) = 𝐵 → ((𝐼𝐵) ∩ (𝐼‘∅)) = ((𝐼‘∅) ∩ 𝐵))
3433eqeq1d 2773 . . . . . 6 ((𝐼𝐵) = 𝐵 → (((𝐼𝐵) ∩ (𝐼‘∅)) = ∅ ↔ ((𝐼‘∅) ∩ 𝐵) = ∅))
3534biimpd 219 . . . . 5 ((𝐼𝐵) = 𝐵 → (((𝐼𝐵) ∩ (𝐼‘∅)) = ∅ → ((𝐼‘∅) ∩ 𝐵) = ∅))
36 reldisj 4163 . . . . . . 7 ((𝐼‘∅) ⊆ 𝐵 → (((𝐼‘∅) ∩ 𝐵) = ∅ ↔ (𝐼‘∅) ⊆ (𝐵𝐵)))
3736biimpd 219 . . . . . 6 ((𝐼‘∅) ⊆ 𝐵 → (((𝐼‘∅) ∩ 𝐵) = ∅ → (𝐼‘∅) ⊆ (𝐵𝐵)))
38 difid 4095 . . . . . . . 8 (𝐵𝐵) = ∅
3938sseq2i 3779 . . . . . . 7 ((𝐼‘∅) ⊆ (𝐵𝐵) ↔ (𝐼‘∅) ⊆ ∅)
40 ss0 4118 . . . . . . 7 ((𝐼‘∅) ⊆ ∅ → (𝐼‘∅) = ∅)
4139, 40sylbi 207 . . . . . 6 ((𝐼‘∅) ⊆ (𝐵𝐵) → (𝐼‘∅) = ∅)
4237, 41syl6com 37 . . . . 5 (((𝐼‘∅) ∩ 𝐵) = ∅ → ((𝐼‘∅) ⊆ 𝐵 → (𝐼‘∅) = ∅))
4335, 42syl6com 37 . . . 4 (((𝐼𝐵) ∩ (𝐼‘∅)) = ∅ → ((𝐼𝐵) = 𝐵 → ((𝐼‘∅) ⊆ 𝐵 → (𝐼‘∅) = ∅)))
4443com13 88 . . 3 ((𝐼‘∅) ⊆ 𝐵 → ((𝐼𝐵) = 𝐵 → (((𝐼𝐵) ∩ (𝐼‘∅)) = ∅ → (𝐼‘∅) = ∅)))
4529, 30, 44syl2im 40 . 2 ((𝐵𝑉𝐼 ∈ (𝒫 𝐵𝑚 𝒫 𝐵)) → (((𝐼𝐵) = 𝐵 ∧ ∀𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵((𝑠𝑡) = ∅ → ((𝐼𝑠) ∩ (𝐼𝑡)) = ∅)) → (((𝐼𝐵) ∩ (𝐼‘∅)) = ∅ → (𝐼‘∅) = ∅)))
4624, 45mpdd 43 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 196  wa 382   = wceq 1631  wcel 2145  wral 3061  cdif 3720  cin 3722  wss 3723  c0 4063  𝒫 cpw 4297  wf 6025  cfv 6029  (class class class)co 6791  𝑚 cmap 8007
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7094
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5992  df-fun 6031  df-fn 6032  df-f 6033  df-fv 6037  df-ov 6794  df-oprab 6795  df-mpt2 6796  df-1st 7313  df-2nd 7314  df-map 8009
This theorem is referenced by: (None)
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