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Theorem ntrneicls11 41166
 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 41152 . . . . . . . 8 (𝜑𝐼 ∈ (𝒫 𝐵m 𝒫 𝐵))
5 elmapi 8438 . . . . . . . 8 (𝐼 ∈ (𝒫 𝐵m 𝒫 𝐵) → 𝐼:𝒫 𝐵⟶𝒫 𝐵)
64, 5syl 17 . . . . . . 7 (𝜑𝐼:𝒫 𝐵⟶𝒫 𝐵)
7 0elpw 5224 . . . . . . . 8 ∅ ∈ 𝒫 𝐵
87a1i 11 . . . . . . 7 (𝜑 → ∅ ∈ 𝒫 𝐵)
96, 8ffvelrnd 6843 . . . . . 6 (𝜑 → (𝐼‘∅) ∈ 𝒫 𝐵)
109elpwid 4505 . . . . 5 (𝜑 → (𝐼‘∅) ⊆ 𝐵)
11 reldisj 4348 . . . . 5 ((𝐼‘∅) ⊆ 𝐵 → (((𝐼‘∅) ∩ 𝐵) = ∅ ↔ (𝐼‘∅) ⊆ (𝐵𝐵)))
1210, 11syl 17 . . . 4 (𝜑 → (((𝐼‘∅) ∩ 𝐵) = ∅ ↔ (𝐼‘∅) ⊆ (𝐵𝐵)))
1312bicomd 226 . . 3 (𝜑 → ((𝐼‘∅) ⊆ (𝐵𝐵) ↔ ((𝐼‘∅) ∩ 𝐵) = ∅))
14 difid 4269 . . . . 5 (𝐵𝐵) = ∅
1514sseq2i 3921 . . . 4 ((𝐼‘∅) ⊆ (𝐵𝐵) ↔ (𝐼‘∅) ⊆ ∅)
16 ss0b 4293 . . . 4 ((𝐼‘∅) ⊆ ∅ ↔ (𝐼‘∅) = ∅)
1715, 16bitri 278 . . 3 ((𝐼‘∅) ⊆ (𝐵𝐵) ↔ (𝐼‘∅) = ∅)
18 disjr 4346 . . 3 (((𝐼‘∅) ∩ 𝐵) = ∅ ↔ ∀𝑥𝐵 ¬ 𝑥 ∈ (𝐼‘∅))
1913, 17, 183bitr3g 316 . 2 (𝜑 → ((𝐼‘∅) = ∅ ↔ ∀𝑥𝐵 ¬ 𝑥 ∈ (𝐼‘∅)))
203adantr 484 . . . . 5 ((𝜑𝑥𝐵) → 𝐼𝐹𝑁)
21 simpr 488 . . . . 5 ((𝜑𝑥𝐵) → 𝑥𝐵)
227a1i 11 . . . . 5 ((𝜑𝑥𝐵) → ∅ ∈ 𝒫 𝐵)
231, 2, 20, 21, 22ntrneiel 41157 . . . 4 ((𝜑𝑥𝐵) → (𝑥 ∈ (𝐼‘∅) ↔ ∅ ∈ (𝑁𝑥)))
2423notbid 321 . . 3 ((𝜑𝑥𝐵) → (¬ 𝑥 ∈ (𝐼‘∅) ↔ ¬ ∅ ∈ (𝑁𝑥)))
2524ralbidva 3125 . 2 (𝜑 → (∀𝑥𝐵 ¬ 𝑥 ∈ (𝐼‘∅) ↔ ∀𝑥𝐵 ¬ ∅ ∈ (𝑁𝑥)))
2619, 25bitrd 282 1 (𝜑 → ((𝐼‘∅) = ∅ ↔ ∀𝑥𝐵 ¬ ∅ ∈ (𝑁𝑥)))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2111  ∀wral 3070  {crab 3074  Vcvv 3409   ∖ cdif 3855   ∩ cin 3857   ⊆ wss 3858  ∅c0 4225  𝒫 cpw 4494   class class class wbr 5032   ↦ cmpt 5112  ⟶wf 6331  ‘cfv 6335  (class class class)co 7150   ∈ cmpo 7152   ↑m cmap 8416 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-rep 5156  ax-sep 5169  ax-nul 5176  ax-pow 5234  ax-pr 5298  ax-un 7459 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-reu 3077  df-rab 3079  df-v 3411  df-sbc 3697  df-csb 3806  df-dif 3861  df-un 3863  df-in 3865  df-ss 3875  df-nul 4226  df-if 4421  df-pw 4496  df-sn 4523  df-pr 4525  df-op 4529  df-uni 4799  df-iun 4885  df-br 5033  df-opab 5095  df-mpt 5113  df-id 5430  df-xp 5530  df-rel 5531  df-cnv 5532  df-co 5533  df-dm 5534  df-rn 5535  df-res 5536  df-ima 5537  df-iota 6294  df-fun 6337  df-fn 6338  df-f 6339  df-f1 6340  df-fo 6341  df-f1o 6342  df-fv 6343  df-ov 7153  df-oprab 7154  df-mpo 7155  df-1st 7693  df-2nd 7694  df-map 8418 This theorem is referenced by: (None)
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