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Theorem ntrclscls00 38890
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 38879 . . . . 5 (𝜑 → (𝐷𝐼) = 𝐾)
54fveq1d 6335 . . . 4 (𝜑 → ((𝐷𝐼)‘∅) = (𝐾‘∅))
62, 3ntrclsbex 38858 . . . . 5 (𝜑𝐵 ∈ V)
71, 2, 3ntrclsiex 38877 . . . . 5 (𝜑𝐼 ∈ (𝒫 𝐵𝑚 𝒫 𝐵))
8 eqid 2771 . . . . 5 (𝐷𝐼) = (𝐷𝐼)
9 0elpw 4966 . . . . . 6 ∅ ∈ 𝒫 𝐵
109a1i 11 . . . . 5 (𝜑 → ∅ ∈ 𝒫 𝐵)
11 eqid 2771 . . . . 5 ((𝐷𝐼)‘∅) = ((𝐷𝐼)‘∅)
121, 2, 6, 7, 8, 10, 11dssmapfv3d 38839 . . . 4 (𝜑 → ((𝐷𝐼)‘∅) = (𝐵 ∖ (𝐼‘(𝐵 ∖ ∅))))
135, 12eqtr3d 2807 . . 3 (𝜑 → (𝐾‘∅) = (𝐵 ∖ (𝐼‘(𝐵 ∖ ∅))))
14 dif0 4098 . . . . . . 7 (𝐵 ∖ ∅) = 𝐵
1514fveq2i 6336 . . . . . 6 (𝐼‘(𝐵 ∖ ∅)) = (𝐼𝐵)
16 id 22 . . . . . 6 ((𝐼𝐵) = 𝐵 → (𝐼𝐵) = 𝐵)
1715, 16syl5eq 2817 . . . . 5 ((𝐼𝐵) = 𝐵 → (𝐼‘(𝐵 ∖ ∅)) = 𝐵)
1817difeq2d 3879 . . . 4 ((𝐼𝐵) = 𝐵 → (𝐵 ∖ (𝐼‘(𝐵 ∖ ∅))) = (𝐵𝐵))
19 difid 4096 . . . 4 (𝐵𝐵) = ∅
2018, 19syl6eq 2821 . . 3 ((𝐼𝐵) = 𝐵 → (𝐵 ∖ (𝐼‘(𝐵 ∖ ∅))) = ∅)
2113, 20sylan9eq 2825 . 2 ((𝜑 ∧ (𝐼𝐵) = 𝐵) → (𝐾‘∅) = ∅)
22 pwidg 4313 . . . . 5 (𝐵 ∈ V → 𝐵 ∈ 𝒫 𝐵)
236, 22syl 17 . . . 4 (𝜑𝐵 ∈ 𝒫 𝐵)
241, 2, 3, 23ntrclsfv 38883 . . 3 (𝜑 → (𝐼𝐵) = (𝐵 ∖ (𝐾‘(𝐵𝐵))))
2519fveq2i 6336 . . . . . 6 (𝐾‘(𝐵𝐵)) = (𝐾‘∅)
26 id 22 . . . . . 6 ((𝐾‘∅) = ∅ → (𝐾‘∅) = ∅)
2725, 26syl5eq 2817 . . . . 5 ((𝐾‘∅) = ∅ → (𝐾‘(𝐵𝐵)) = ∅)
2827difeq2d 3879 . . . 4 ((𝐾‘∅) = ∅ → (𝐵 ∖ (𝐾‘(𝐵𝐵))) = (𝐵 ∖ ∅))
2928, 14syl6eq 2821 . . 3 ((𝐾‘∅) = ∅ → (𝐵 ∖ (𝐾‘(𝐵𝐵))) = 𝐵)
3024, 29sylan9eq 2825 . 2 ((𝜑 ∧ (𝐾‘∅) = ∅) → (𝐼𝐵) = 𝐵)
3121, 30impbida 802 1 (𝜑 → ((𝐼𝐵) = 𝐵 ↔ (𝐾‘∅) = ∅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196   = wceq 1631  wcel 2145  Vcvv 3351  cdif 3720  c0 4063  𝒫 cpw 4298   class class class wbr 4787  cmpt 4864  cfv 6030  (class class class)co 6796  𝑚 cmap 8013
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-rep 4905  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035  ax-un 7100
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-reu 3068  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 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-iun 4657  df-br 4788  df-opab 4848  df-mpt 4865  df-id 5158  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263  df-iota 5993  df-fun 6032  df-fn 6033  df-f 6034  df-f1 6035  df-fo 6036  df-f1o 6037  df-fv 6038  df-ov 6799  df-oprab 6800  df-mpt2 6801  df-1st 7319  df-2nd 7320  df-map 8015
This theorem is referenced by: (None)
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