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Theorem ntrk0kbimka 38831
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 38830. (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 4366 . . . . 5 (𝐵𝑉𝐵 ∈ 𝒫 𝐵)
21ad2antrr 708 . . . 4 (((𝐵𝑉𝐼 ∈ (𝒫 𝐵𝑚 𝒫 𝐵)) ∧ ((𝐼𝐵) = 𝐵 ∧ ∀𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵((𝑠𝑡) = ∅ → ((𝐼𝑠) ∩ (𝐼𝑡)) = ∅))) → 𝐵 ∈ 𝒫 𝐵)
3 0elpw 5026 . . . . 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 780 . . . 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 4006 . . . . . . 7 (𝑠 = 𝐵 → (𝑠𝑡) = (𝐵𝑡))
76eqeq1d 2808 . . . . . 6 (𝑠 = 𝐵 → ((𝑠𝑡) = ∅ ↔ (𝐵𝑡) = ∅))
8 fveq2 6404 . . . . . . . 8 (𝑠 = 𝐵 → (𝐼𝑠) = (𝐼𝐵))
98ineq1d 4012 . . . . . . 7 (𝑠 = 𝐵 → ((𝐼𝑠) ∩ (𝐼𝑡)) = ((𝐼𝐵) ∩ (𝐼𝑡)))
109eqeq1d 2808 . . . . . 6 (𝑠 = 𝐵 → (((𝐼𝑠) ∩ (𝐼𝑡)) = ∅ ↔ ((𝐼𝐵) ∩ (𝐼𝑡)) = ∅))
117, 10imbi12d 335 . . . . 5 (𝑠 = 𝐵 → (((𝑠𝑡) = ∅ → ((𝐼𝑠) ∩ (𝐼𝑡)) = ∅) ↔ ((𝐵𝑡) = ∅ → ((𝐼𝐵) ∩ (𝐼𝑡)) = ∅)))
12 ineq2 4007 . . . . . . . 8 (𝑡 = ∅ → (𝐵𝑡) = (𝐵 ∩ ∅))
1312eqeq1d 2808 . . . . . . 7 (𝑡 = ∅ → ((𝐵𝑡) = ∅ ↔ (𝐵 ∩ ∅) = ∅))
14 fveq2 6404 . . . . . . . . 9 (𝑡 = ∅ → (𝐼𝑡) = (𝐼‘∅))
1514ineq2d 4013 . . . . . . . 8 (𝑡 = ∅ → ((𝐼𝐵) ∩ (𝐼𝑡)) = ((𝐼𝐵) ∩ (𝐼‘∅)))
1615eqeq1d 2808 . . . . . . 7 (𝑡 = ∅ → (((𝐼𝐵) ∩ (𝐼𝑡)) = ∅ ↔ ((𝐼𝐵) ∩ (𝐼‘∅)) = ∅))
1713, 16imbi12d 335 . . . . . 6 (𝑡 = ∅ → (((𝐵𝑡) = ∅ → ((𝐼𝐵) ∩ (𝐼𝑡)) = ∅) ↔ ((𝐵 ∩ ∅) = ∅ → ((𝐼𝐵) ∩ (𝐼‘∅)) = ∅)))
18 in0 4166 . . . . . . 7 (𝐵 ∩ ∅) = ∅
19 pm5.5 352 . . . . . . 7 ((𝐵 ∩ ∅) = ∅ → (((𝐵 ∩ ∅) = ∅ → ((𝐼𝐵) ∩ (𝐼‘∅)) = ∅) ↔ ((𝐼𝐵) ∩ (𝐼‘∅)) = ∅))
2018, 19mp1i 13 . . . . . 6 (𝑡 = ∅ → (((𝐵 ∩ ∅) = ∅ → ((𝐼𝐵) ∩ (𝐼‘∅)) = ∅) ↔ ((𝐼𝐵) ∩ (𝐼‘∅)) = ∅))
2117, 20bitrd 270 . . . . 5 (𝑡 = ∅ → (((𝐵𝑡) = ∅ → ((𝐼𝐵) ∩ (𝐼𝑡)) = ∅) ↔ ((𝐼𝐵) ∩ (𝐼‘∅)) = ∅))
2211, 21rspc2va 3516 . . . 4 (((𝐵 ∈ 𝒫 𝐵 ∧ ∅ ∈ 𝒫 𝐵) ∧ ∀𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵((𝑠𝑡) = ∅ → ((𝐼𝑠) ∩ (𝐼𝑡)) = ∅)) → ((𝐼𝐵) ∩ (𝐼‘∅)) = ∅)
232, 4, 5, 22syl21anc 857 . . 3 (((𝐵𝑉𝐼 ∈ (𝒫 𝐵𝑚 𝒫 𝐵)) ∧ ((𝐼𝐵) = 𝐵 ∧ ∀𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵((𝑠𝑡) = ∅ → ((𝐼𝑠) ∩ (𝐼𝑡)) = ∅))) → ((𝐼𝐵) ∩ (𝐼‘∅)) = ∅)
2423ex 399 . 2 ((𝐵𝑉𝐼 ∈ (𝒫 𝐵𝑚 𝒫 𝐵)) → (((𝐼𝐵) = 𝐵 ∧ ∀𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵((𝑠𝑡) = ∅ → ((𝐼𝑠) ∩ (𝐼𝑡)) = ∅)) → ((𝐼𝐵) ∩ (𝐼‘∅)) = ∅))
25 elmapi 8110 . . . . . 6 (𝐼 ∈ (𝒫 𝐵𝑚 𝒫 𝐵) → 𝐼:𝒫 𝐵⟶𝒫 𝐵)
2625adantl 469 . . . . 5 ((𝐵𝑉𝐼 ∈ (𝒫 𝐵𝑚 𝒫 𝐵)) → 𝐼:𝒫 𝐵⟶𝒫 𝐵)
273a1i 11 . . . . 5 ((𝐵𝑉𝐼 ∈ (𝒫 𝐵𝑚 𝒫 𝐵)) → ∅ ∈ 𝒫 𝐵)
2826, 27ffvelrnd 6578 . . . 4 ((𝐵𝑉𝐼 ∈ (𝒫 𝐵𝑚 𝒫 𝐵)) → (𝐼‘∅) ∈ 𝒫 𝐵)
2928elpwid 4363 . . 3 ((𝐵𝑉𝐼 ∈ (𝒫 𝐵𝑚 𝒫 𝐵)) → (𝐼‘∅) ⊆ 𝐵)
30 simpl 470 . . 3 (((𝐼𝐵) = 𝐵 ∧ ∀𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵((𝑠𝑡) = ∅ → ((𝐼𝑠) ∩ (𝐼𝑡)) = ∅)) → (𝐼𝐵) = 𝐵)
31 ineq1 4006 . . . . . . . 8 ((𝐼𝐵) = 𝐵 → ((𝐼𝐵) ∩ (𝐼‘∅)) = (𝐵 ∩ (𝐼‘∅)))
32 incom 4004 . . . . . . . 8 (𝐵 ∩ (𝐼‘∅)) = ((𝐼‘∅) ∩ 𝐵)
3331, 32syl6eq 2856 . . . . . . 7 ((𝐼𝐵) = 𝐵 → ((𝐼𝐵) ∩ (𝐼‘∅)) = ((𝐼‘∅) ∩ 𝐵))
3433eqeq1d 2808 . . . . . 6 ((𝐼𝐵) = 𝐵 → (((𝐼𝐵) ∩ (𝐼‘∅)) = ∅ ↔ ((𝐼‘∅) ∩ 𝐵) = ∅))
3534biimpd 220 . . . . 5 ((𝐼𝐵) = 𝐵 → (((𝐼𝐵) ∩ (𝐼‘∅)) = ∅ → ((𝐼‘∅) ∩ 𝐵) = ∅))
36 reldisj 4217 . . . . . . 7 ((𝐼‘∅) ⊆ 𝐵 → (((𝐼‘∅) ∩ 𝐵) = ∅ ↔ (𝐼‘∅) ⊆ (𝐵𝐵)))
3736biimpd 220 . . . . . 6 ((𝐼‘∅) ⊆ 𝐵 → (((𝐼‘∅) ∩ 𝐵) = ∅ → (𝐼‘∅) ⊆ (𝐵𝐵)))
38 difid 4149 . . . . . . . 8 (𝐵𝐵) = ∅
3938sseq2i 3827 . . . . . . 7 ((𝐼‘∅) ⊆ (𝐵𝐵) ↔ (𝐼‘∅) ⊆ ∅)
40 ss0 4172 . . . . . . 7 ((𝐼‘∅) ⊆ ∅ → (𝐼‘∅) = ∅)
4139, 40sylbi 208 . . . . . 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 197  wa 384   = wceq 1637  wcel 2156  wral 3096  cdif 3766  cin 3768  wss 3769  c0 4116  𝒫 cpw 4351  wf 6093  cfv 6097  (class class class)co 6870  𝑚 cmap 8088
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2068  ax-7 2104  ax-8 2158  ax-9 2165  ax-10 2185  ax-11 2201  ax-12 2214  ax-13 2420  ax-ext 2784  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5096  ax-un 7175
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2061  df-eu 2634  df-mo 2635  df-clab 2793  df-cleq 2799  df-clel 2802  df-nfc 2937  df-ne 2979  df-ral 3101  df-rex 3102  df-rab 3105  df-v 3393  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4117  df-if 4280  df-pw 4353  df-sn 4371  df-pr 4373  df-op 4377  df-uni 4631  df-iun 4714  df-br 4845  df-opab 4907  df-mpt 4924  df-id 5219  df-xp 5317  df-rel 5318  df-cnv 5319  df-co 5320  df-dm 5321  df-rn 5322  df-res 5323  df-ima 5324  df-iota 6060  df-fun 6099  df-fn 6100  df-f 6101  df-fv 6105  df-ov 6873  df-oprab 6874  df-mpt2 6875  df-1st 7394  df-2nd 7395  df-map 8090
This theorem is referenced by: (None)
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