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Theorem ntrk0kbimka 37819
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 37818. (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 4144 . . . . 5 (𝐵𝑉𝐵 ∈ 𝒫 𝐵)
21ad2antrr 761 . . . 4 (((𝐵𝑉𝐼 ∈ (𝒫 𝐵𝑚 𝒫 𝐵)) ∧ ((𝐼𝐵) = 𝐵 ∧ ∀𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵((𝑠𝑡) = ∅ → ((𝐼𝑠) ∩ (𝐼𝑡)) = ∅))) → 𝐵 ∈ 𝒫 𝐵)
3 0elpw 4794 . . . . 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 795 . . . 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 3785 . . . . . . 7 (𝑠 = 𝐵 → (𝑠𝑡) = (𝐵𝑡))
76eqeq1d 2623 . . . . . 6 (𝑠 = 𝐵 → ((𝑠𝑡) = ∅ ↔ (𝐵𝑡) = ∅))
8 fveq2 6148 . . . . . . . 8 (𝑠 = 𝐵 → (𝐼𝑠) = (𝐼𝐵))
98ineq1d 3791 . . . . . . 7 (𝑠 = 𝐵 → ((𝐼𝑠) ∩ (𝐼𝑡)) = ((𝐼𝐵) ∩ (𝐼𝑡)))
109eqeq1d 2623 . . . . . 6 (𝑠 = 𝐵 → (((𝐼𝑠) ∩ (𝐼𝑡)) = ∅ ↔ ((𝐼𝐵) ∩ (𝐼𝑡)) = ∅))
117, 10imbi12d 334 . . . . 5 (𝑠 = 𝐵 → (((𝑠𝑡) = ∅ → ((𝐼𝑠) ∩ (𝐼𝑡)) = ∅) ↔ ((𝐵𝑡) = ∅ → ((𝐼𝐵) ∩ (𝐼𝑡)) = ∅)))
12 ineq2 3786 . . . . . . . 8 (𝑡 = ∅ → (𝐵𝑡) = (𝐵 ∩ ∅))
1312eqeq1d 2623 . . . . . . 7 (𝑡 = ∅ → ((𝐵𝑡) = ∅ ↔ (𝐵 ∩ ∅) = ∅))
14 fveq2 6148 . . . . . . . . 9 (𝑡 = ∅ → (𝐼𝑡) = (𝐼‘∅))
1514ineq2d 3792 . . . . . . . 8 (𝑡 = ∅ → ((𝐼𝐵) ∩ (𝐼𝑡)) = ((𝐼𝐵) ∩ (𝐼‘∅)))
1615eqeq1d 2623 . . . . . . 7 (𝑡 = ∅ → (((𝐼𝐵) ∩ (𝐼𝑡)) = ∅ ↔ ((𝐼𝐵) ∩ (𝐼‘∅)) = ∅))
1713, 16imbi12d 334 . . . . . 6 (𝑡 = ∅ → (((𝐵𝑡) = ∅ → ((𝐼𝐵) ∩ (𝐼𝑡)) = ∅) ↔ ((𝐵 ∩ ∅) = ∅ → ((𝐼𝐵) ∩ (𝐼‘∅)) = ∅)))
18 in0 3940 . . . . . . 7 (𝐵 ∩ ∅) = ∅
19 pm5.5 351 . . . . . . 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 3307 . . . 4 (((𝐵 ∈ 𝒫 𝐵 ∧ ∅ ∈ 𝒫 𝐵) ∧ ∀𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵((𝑠𝑡) = ∅ → ((𝐼𝑠) ∩ (𝐼𝑡)) = ∅)) → ((𝐼𝐵) ∩ (𝐼‘∅)) = ∅)
232, 4, 5, 22syl21anc 1322 . . 3 (((𝐵𝑉𝐼 ∈ (𝒫 𝐵𝑚 𝒫 𝐵)) ∧ ((𝐼𝐵) = 𝐵 ∧ ∀𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵((𝑠𝑡) = ∅ → ((𝐼𝑠) ∩ (𝐼𝑡)) = ∅))) → ((𝐼𝐵) ∩ (𝐼‘∅)) = ∅)
2423ex 450 . 2 ((𝐵𝑉𝐼 ∈ (𝒫 𝐵𝑚 𝒫 𝐵)) → (((𝐼𝐵) = 𝐵 ∧ ∀𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵((𝑠𝑡) = ∅ → ((𝐼𝑠) ∩ (𝐼𝑡)) = ∅)) → ((𝐼𝐵) ∩ (𝐼‘∅)) = ∅))
25 elmapi 7823 . . . . . 6 (𝐼 ∈ (𝒫 𝐵𝑚 𝒫 𝐵) → 𝐼:𝒫 𝐵⟶𝒫 𝐵)
2625adantl 482 . . . . 5 ((𝐵𝑉𝐼 ∈ (𝒫 𝐵𝑚 𝒫 𝐵)) → 𝐼:𝒫 𝐵⟶𝒫 𝐵)
273a1i 11 . . . . 5 ((𝐵𝑉𝐼 ∈ (𝒫 𝐵𝑚 𝒫 𝐵)) → ∅ ∈ 𝒫 𝐵)
2826, 27ffvelrnd 6316 . . . 4 ((𝐵𝑉𝐼 ∈ (𝒫 𝐵𝑚 𝒫 𝐵)) → (𝐼‘∅) ∈ 𝒫 𝐵)
2928elpwid 4141 . . 3 ((𝐵𝑉𝐼 ∈ (𝒫 𝐵𝑚 𝒫 𝐵)) → (𝐼‘∅) ⊆ 𝐵)
30 simpl 473 . . 3 (((𝐼𝐵) = 𝐵 ∧ ∀𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵((𝑠𝑡) = ∅ → ((𝐼𝑠) ∩ (𝐼𝑡)) = ∅)) → (𝐼𝐵) = 𝐵)
31 ineq1 3785 . . . . . . . 8 ((𝐼𝐵) = 𝐵 → ((𝐼𝐵) ∩ (𝐼‘∅)) = (𝐵 ∩ (𝐼‘∅)))
32 incom 3783 . . . . . . . 8 (𝐵 ∩ (𝐼‘∅)) = ((𝐼‘∅) ∩ 𝐵)
3331, 32syl6eq 2671 . . . . . . 7 ((𝐼𝐵) = 𝐵 → ((𝐼𝐵) ∩ (𝐼‘∅)) = ((𝐼‘∅) ∩ 𝐵))
3433eqeq1d 2623 . . . . . 6 ((𝐼𝐵) = 𝐵 → (((𝐼𝐵) ∩ (𝐼‘∅)) = ∅ ↔ ((𝐼‘∅) ∩ 𝐵) = ∅))
3534biimpd 219 . . . . 5 ((𝐼𝐵) = 𝐵 → (((𝐼𝐵) ∩ (𝐼‘∅)) = ∅ → ((𝐼‘∅) ∩ 𝐵) = ∅))
36 reldisj 3992 . . . . . . 7 ((𝐼‘∅) ⊆ 𝐵 → (((𝐼‘∅) ∩ 𝐵) = ∅ ↔ (𝐼‘∅) ⊆ (𝐵𝐵)))
3736biimpd 219 . . . . . 6 ((𝐼‘∅) ⊆ 𝐵 → (((𝐼‘∅) ∩ 𝐵) = ∅ → (𝐼‘∅) ⊆ (𝐵𝐵)))
38 difid 3922 . . . . . . . 8 (𝐵𝐵) = ∅
3938sseq2i 3609 . . . . . . 7 ((𝐼‘∅) ⊆ (𝐵𝐵) ↔ (𝐼‘∅) ⊆ ∅)
40 ss0 3946 . . . . . . 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 384   = wceq 1480  wcel 1987  wral 2907  cdif 3552  cin 3554  wss 3555  c0 3891  𝒫 cpw 4130  wf 5843  cfv 5847  (class class class)co 6604  𝑚 cmap 7802
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-fv 5855  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-1st 7113  df-2nd 7114  df-map 7804
This theorem is referenced by: (None)
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