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Theorem ntrclscls00 38866
Description: If (pseudo-)interior and (pseudo-)closure functions are related by the duality operator then conditions equal to claiming that the closure of the empty set is the empty set hold equally. (Contributed by RP, 1-Jun-2021.)
Hypotheses
Ref Expression
ntrcls.o 𝑂 = (𝑖 ∈ V ↦ (𝑘 ∈ (𝒫 𝑖𝑚 𝒫 𝑖) ↦ (𝑗 ∈ 𝒫 𝑖 ↦ (𝑖 ∖ (𝑘‘(𝑖𝑗))))))
ntrcls.d 𝐷 = (𝑂𝐵)
ntrcls.r (𝜑𝐼𝐷𝐾)
Assertion
Ref Expression
ntrclscls00 (𝜑 → ((𝐼𝐵) = 𝐵 ↔ (𝐾‘∅) = ∅))
Distinct variable groups:   𝐵,𝑖,𝑗,𝑘   𝑗,𝐼,𝑘   𝑗,𝐾,𝑘   𝜑,𝑖,𝑗,𝑘
Allowed substitution hints:   𝐷(𝑖,𝑗,𝑘)   𝐼(𝑖)   𝐾(𝑖)   𝑂(𝑖,𝑗,𝑘)

Proof of Theorem ntrclscls00
StepHypRef Expression
1 ntrcls.o . . . . . 6 𝑂 = (𝑖 ∈ V ↦ (𝑘 ∈ (𝒫 𝑖𝑚 𝒫 𝑖) ↦ (𝑗 ∈ 𝒫 𝑖 ↦ (𝑖 ∖ (𝑘‘(𝑖𝑗))))))
2 ntrcls.d . . . . . 6 𝐷 = (𝑂𝐵)
3 ntrcls.r . . . . . 6 (𝜑𝐼𝐷𝐾)
41, 2, 3ntrclsfv1 38855 . . . . 5 (𝜑 → (𝐷𝐼) = 𝐾)
54fveq1d 6354 . . . 4 (𝜑 → ((𝐷𝐼)‘∅) = (𝐾‘∅))
62, 3ntrclsbex 38834 . . . . 5 (𝜑𝐵 ∈ V)
71, 2, 3ntrclsiex 38853 . . . . 5 (𝜑𝐼 ∈ (𝒫 𝐵𝑚 𝒫 𝐵))
8 eqid 2760 . . . . 5 (𝐷𝐼) = (𝐷𝐼)
9 0elpw 4983 . . . . . 6 ∅ ∈ 𝒫 𝐵
109a1i 11 . . . . 5 (𝜑 → ∅ ∈ 𝒫 𝐵)
11 eqid 2760 . . . . 5 ((𝐷𝐼)‘∅) = ((𝐷𝐼)‘∅)
121, 2, 6, 7, 8, 10, 11dssmapfv3d 38815 . . . 4 (𝜑 → ((𝐷𝐼)‘∅) = (𝐵 ∖ (𝐼‘(𝐵 ∖ ∅))))
135, 12eqtr3d 2796 . . 3 (𝜑 → (𝐾‘∅) = (𝐵 ∖ (𝐼‘(𝐵 ∖ ∅))))
14 dif0 4093 . . . . . . 7 (𝐵 ∖ ∅) = 𝐵
1514fveq2i 6355 . . . . . 6 (𝐼‘(𝐵 ∖ ∅)) = (𝐼𝐵)
16 id 22 . . . . . 6 ((𝐼𝐵) = 𝐵 → (𝐼𝐵) = 𝐵)
1715, 16syl5eq 2806 . . . . 5 ((𝐼𝐵) = 𝐵 → (𝐼‘(𝐵 ∖ ∅)) = 𝐵)
1817difeq2d 3871 . . . 4 ((𝐼𝐵) = 𝐵 → (𝐵 ∖ (𝐼‘(𝐵 ∖ ∅))) = (𝐵𝐵))
19 difid 4091 . . . 4 (𝐵𝐵) = ∅
2018, 19syl6eq 2810 . . 3 ((𝐼𝐵) = 𝐵 → (𝐵 ∖ (𝐼‘(𝐵 ∖ ∅))) = ∅)
2113, 20sylan9eq 2814 . 2 ((𝜑 ∧ (𝐼𝐵) = 𝐵) → (𝐾‘∅) = ∅)
22 pwidg 4317 . . . . 5 (𝐵 ∈ V → 𝐵 ∈ 𝒫 𝐵)
236, 22syl 17 . . . 4 (𝜑𝐵 ∈ 𝒫 𝐵)
241, 2, 3, 23ntrclsfv 38859 . . 3 (𝜑 → (𝐼𝐵) = (𝐵 ∖ (𝐾‘(𝐵𝐵))))
2519fveq2i 6355 . . . . . 6 (𝐾‘(𝐵𝐵)) = (𝐾‘∅)
26 id 22 . . . . . 6 ((𝐾‘∅) = ∅ → (𝐾‘∅) = ∅)
2725, 26syl5eq 2806 . . . . 5 ((𝐾‘∅) = ∅ → (𝐾‘(𝐵𝐵)) = ∅)
2827difeq2d 3871 . . . 4 ((𝐾‘∅) = ∅ → (𝐵 ∖ (𝐾‘(𝐵𝐵))) = (𝐵 ∖ ∅))
2928, 14syl6eq 2810 . . 3 ((𝐾‘∅) = ∅ → (𝐵 ∖ (𝐾‘(𝐵𝐵))) = 𝐵)
3024, 29sylan9eq 2814 . 2 ((𝜑 ∧ (𝐾‘∅) = ∅) → (𝐼𝐵) = 𝐵)
3121, 30impbida 913 1 (𝜑 → ((𝐼𝐵) = 𝐵 ↔ (𝐾‘∅) = ∅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196   = wceq 1632  wcel 2139  Vcvv 3340  cdif 3712  c0 4058  𝒫 cpw 4302   class class class wbr 4804  cmpt 4881  cfv 6049  (class class class)co 6813  𝑚 cmap 8023
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-ov 6816  df-oprab 6817  df-mpt2 6818  df-1st 7333  df-2nd 7334  df-map 8025
This theorem is referenced by: (None)
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