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Theorem ntrneicls11 39086
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 39072 . . . . . . . 8 (𝜑𝐼 ∈ (𝒫 𝐵𝑚 𝒫 𝐵))
5 elmapi 8086 . . . . . . . 8 (𝐼 ∈ (𝒫 𝐵𝑚 𝒫 𝐵) → 𝐼:𝒫 𝐵⟶𝒫 𝐵)
64, 5syl 17 . . . . . . 7 (𝜑𝐼:𝒫 𝐵⟶𝒫 𝐵)
7 0elpw 4994 . . . . . . . 8 ∅ ∈ 𝒫 𝐵
87a1i 11 . . . . . . 7 (𝜑 → ∅ ∈ 𝒫 𝐵)
96, 8ffvelrnd 6554 . . . . . 6 (𝜑 → (𝐼‘∅) ∈ 𝒫 𝐵)
109elpwid 4329 . . . . 5 (𝜑 → (𝐼‘∅) ⊆ 𝐵)
11 reldisj 4183 . . . . 5 ((𝐼‘∅) ⊆ 𝐵 → (((𝐼‘∅) ∩ 𝐵) = ∅ ↔ (𝐼‘∅) ⊆ (𝐵𝐵)))
1210, 11syl 17 . . . 4 (𝜑 → (((𝐼‘∅) ∩ 𝐵) = ∅ ↔ (𝐼‘∅) ⊆ (𝐵𝐵)))
1312bicomd 214 . . 3 (𝜑 → ((𝐼‘∅) ⊆ (𝐵𝐵) ↔ ((𝐼‘∅) ∩ 𝐵) = ∅))
14 difid 4115 . . . . 5 (𝐵𝐵) = ∅
1514sseq2i 3792 . . . 4 ((𝐼‘∅) ⊆ (𝐵𝐵) ↔ (𝐼‘∅) ⊆ ∅)
16 ss0b 4137 . . . 4 ((𝐼‘∅) ⊆ ∅ ↔ (𝐼‘∅) = ∅)
1715, 16bitri 266 . . 3 ((𝐼‘∅) ⊆ (𝐵𝐵) ↔ (𝐼‘∅) = ∅)
18 disjr 4181 . . 3 (((𝐼‘∅) ∩ 𝐵) = ∅ ↔ ∀𝑥𝐵 ¬ 𝑥 ∈ (𝐼‘∅))
1913, 17, 183bitr3g 304 . 2 (𝜑 → ((𝐼‘∅) = ∅ ↔ ∀𝑥𝐵 ¬ 𝑥 ∈ (𝐼‘∅)))
203adantr 472 . . . . 5 ((𝜑𝑥𝐵) → 𝐼𝐹𝑁)
21 simpr 477 . . . . 5 ((𝜑𝑥𝐵) → 𝑥𝐵)
227a1i 11 . . . . 5 ((𝜑𝑥𝐵) → ∅ ∈ 𝒫 𝐵)
231, 2, 20, 21, 22ntrneiel 39077 . . . 4 ((𝜑𝑥𝐵) → (𝑥 ∈ (𝐼‘∅) ↔ ∅ ∈ (𝑁𝑥)))
2423notbid 309 . . 3 ((𝜑𝑥𝐵) → (¬ 𝑥 ∈ (𝐼‘∅) ↔ ¬ ∅ ∈ (𝑁𝑥)))
2524ralbidva 3132 . 2 (𝜑 → (∀𝑥𝐵 ¬ 𝑥 ∈ (𝐼‘∅) ↔ ∀𝑥𝐵 ¬ ∅ ∈ (𝑁𝑥)))
2619, 25bitrd 270 1 (𝜑 → ((𝐼‘∅) = ∅ ↔ ∀𝑥𝐵 ¬ ∅ ∈ (𝑁𝑥)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 197  wa 384   = wceq 1652  wcel 2155  wral 3055  {crab 3059  Vcvv 3350  cdif 3731  cin 3733  wss 3734  c0 4081  𝒫 cpw 4317   class class class wbr 4811  cmpt 4890  wf 6066  cfv 6070  (class class class)co 6846  cmpt2 6848  𝑚 cmap 8064
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4932  ax-sep 4943  ax-nul 4951  ax-pow 5003  ax-pr 5064  ax-un 7151
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-reu 3062  df-rab 3064  df-v 3352  df-sbc 3599  df-csb 3694  df-dif 3737  df-un 3739  df-in 3741  df-ss 3748  df-nul 4082  df-if 4246  df-pw 4319  df-sn 4337  df-pr 4339  df-op 4343  df-uni 4597  df-iun 4680  df-br 4812  df-opab 4874  df-mpt 4891  df-id 5187  df-xp 5285  df-rel 5286  df-cnv 5287  df-co 5288  df-dm 5289  df-rn 5290  df-res 5291  df-ima 5292  df-iota 6033  df-fun 6072  df-fn 6073  df-f 6074  df-f1 6075  df-fo 6076  df-f1o 6077  df-fv 6078  df-ov 6849  df-oprab 6850  df-mpt2 6851  df-1st 7370  df-2nd 7371  df-map 8066
This theorem is referenced by: (None)
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