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Theorem ntrneicls11 40308
Description: If (pseudo-)interior and (pseudo-)neighborhood functions are related by the operator, 𝐹, then conditions equal to claiming that the interior of the empty set is the empty set hold equally. (Contributed by RP, 2-Jun-2021.)
Hypotheses
Ref Expression
ntrnei.o 𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗m 𝑖) ↦ (𝑙𝑗 ↦ {𝑚𝑖𝑙 ∈ (𝑘𝑚)})))
ntrnei.f 𝐹 = (𝒫 𝐵𝑂𝐵)
ntrnei.r (𝜑𝐼𝐹𝑁)
Assertion
Ref Expression
ntrneicls11 (𝜑 → ((𝐼‘∅) = ∅ ↔ ∀𝑥𝐵 ¬ ∅ ∈ (𝑁𝑥)))
Distinct variable groups:   𝐵,𝑖,𝑗,𝑘,𝑙,𝑚,𝑥   𝑘,𝐼,𝑙,𝑚,𝑥   𝜑,𝑖,𝑗,𝑘,𝑙,𝑥
Allowed substitution hints:   𝜑(𝑚)   𝐹(𝑥,𝑖,𝑗,𝑘,𝑚,𝑙)   𝐼(𝑖,𝑗)   𝑁(𝑥,𝑖,𝑗,𝑘,𝑚,𝑙)   𝑂(𝑥,𝑖,𝑗,𝑘,𝑚,𝑙)

Proof of Theorem ntrneicls11
StepHypRef Expression
1 ntrnei.o . . . . . . . . 9 𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗m 𝑖) ↦ (𝑙𝑗 ↦ {𝑚𝑖𝑙 ∈ (𝑘𝑚)})))
2 ntrnei.f . . . . . . . . 9 𝐹 = (𝒫 𝐵𝑂𝐵)
3 ntrnei.r . . . . . . . . 9 (𝜑𝐼𝐹𝑁)
41, 2, 3ntrneiiex 40294 . . . . . . . 8 (𝜑𝐼 ∈ (𝒫 𝐵m 𝒫 𝐵))
5 elmapi 8418 . . . . . . . 8 (𝐼 ∈ (𝒫 𝐵m 𝒫 𝐵) → 𝐼:𝒫 𝐵⟶𝒫 𝐵)
64, 5syl 17 . . . . . . 7 (𝜑𝐼:𝒫 𝐵⟶𝒫 𝐵)
7 0elpw 5253 . . . . . . . 8 ∅ ∈ 𝒫 𝐵
87a1i 11 . . . . . . 7 (𝜑 → ∅ ∈ 𝒫 𝐵)
96, 8ffvelrnd 6848 . . . . . 6 (𝜑 → (𝐼‘∅) ∈ 𝒫 𝐵)
109elpwid 4556 . . . . 5 (𝜑 → (𝐼‘∅) ⊆ 𝐵)
11 reldisj 4405 . . . . 5 ((𝐼‘∅) ⊆ 𝐵 → (((𝐼‘∅) ∩ 𝐵) = ∅ ↔ (𝐼‘∅) ⊆ (𝐵𝐵)))
1210, 11syl 17 . . . 4 (𝜑 → (((𝐼‘∅) ∩ 𝐵) = ∅ ↔ (𝐼‘∅) ⊆ (𝐵𝐵)))
1312bicomd 224 . . 3 (𝜑 → ((𝐼‘∅) ⊆ (𝐵𝐵) ↔ ((𝐼‘∅) ∩ 𝐵) = ∅))
14 difid 4334 . . . . 5 (𝐵𝐵) = ∅
1514sseq2i 4000 . . . 4 ((𝐼‘∅) ⊆ (𝐵𝐵) ↔ (𝐼‘∅) ⊆ ∅)
16 ss0b 4355 . . . 4 ((𝐼‘∅) ⊆ ∅ ↔ (𝐼‘∅) = ∅)
1715, 16bitri 276 . . 3 ((𝐼‘∅) ⊆ (𝐵𝐵) ↔ (𝐼‘∅) = ∅)
18 disjr 4403 . . 3 (((𝐼‘∅) ∩ 𝐵) = ∅ ↔ ∀𝑥𝐵 ¬ 𝑥 ∈ (𝐼‘∅))
1913, 17, 183bitr3g 314 . 2 (𝜑 → ((𝐼‘∅) = ∅ ↔ ∀𝑥𝐵 ¬ 𝑥 ∈ (𝐼‘∅)))
203adantr 481 . . . . 5 ((𝜑𝑥𝐵) → 𝐼𝐹𝑁)
21 simpr 485 . . . . 5 ((𝜑𝑥𝐵) → 𝑥𝐵)
227a1i 11 . . . . 5 ((𝜑𝑥𝐵) → ∅ ∈ 𝒫 𝐵)
231, 2, 20, 21, 22ntrneiel 40299 . . . 4 ((𝜑𝑥𝐵) → (𝑥 ∈ (𝐼‘∅) ↔ ∅ ∈ (𝑁𝑥)))
2423notbid 319 . . 3 ((𝜑𝑥𝐵) → (¬ 𝑥 ∈ (𝐼‘∅) ↔ ¬ ∅ ∈ (𝑁𝑥)))
2524ralbidva 3201 . 2 (𝜑 → (∀𝑥𝐵 ¬ 𝑥 ∈ (𝐼‘∅) ↔ ∀𝑥𝐵 ¬ ∅ ∈ (𝑁𝑥)))
2619, 25bitrd 280 1 (𝜑 → ((𝐼‘∅) = ∅ ↔ ∀𝑥𝐵 ¬ ∅ ∈ (𝑁𝑥)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 207  wa 396   = wceq 1530  wcel 2107  wral 3143  {crab 3147  Vcvv 3500  cdif 3937  cin 3939  wss 3940  c0 4295  𝒫 cpw 4542   class class class wbr 5063  cmpt 5143  wf 6348  cfv 6352  (class class class)co 7148  cmpo 7150  m cmap 8396
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-rep 5187  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326  ax-un 7451
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-ral 3148  df-rex 3149  df-reu 3150  df-rab 3152  df-v 3502  df-sbc 3777  df-csb 3888  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-nul 4296  df-if 4471  df-pw 4544  df-sn 4565  df-pr 4567  df-op 4571  df-uni 4838  df-iun 4919  df-br 5064  df-opab 5126  df-mpt 5144  df-id 5459  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-iota 6312  df-fun 6354  df-fn 6355  df-f 6356  df-f1 6357  df-fo 6358  df-f1o 6359  df-fv 6360  df-ov 7151  df-oprab 7152  df-mpo 7153  df-1st 7680  df-2nd 7681  df-map 8398
This theorem is referenced by: (None)
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