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Theorem ntrneineine0 42514
Description: If (pseudo-)interior and (pseudo-)neighborhood functions are related by the operator, 𝐹, then conditions equal to claiming that for every point, at least one (pseudo-)neighborbood exists hold equally. (Contributed by RP, 29-May-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
ntrneineine0 (𝜑 → (∀𝑥𝐵𝑠 ∈ 𝒫 𝐵𝑥 ∈ (𝐼𝑠) ↔ ∀𝑥𝐵 (𝑁𝑥) ≠ ∅))
Distinct variable groups:   𝐵,𝑖,𝑗,𝑘,𝑙,𝑚,𝑠   𝑘,𝐼,𝑙,𝑚   𝑁,𝑠   𝜑,𝑖,𝑗,𝑘,𝑙,𝑠,𝑥   𝑥,𝑚
Allowed substitution hints:   𝜑(𝑚)   𝐵(𝑥)   𝐹(𝑥,𝑖,𝑗,𝑘,𝑚,𝑠,𝑙)   𝐼(𝑥,𝑖,𝑗,𝑠)   𝑁(𝑥,𝑖,𝑗,𝑘,𝑚,𝑙)   𝑂(𝑥,𝑖,𝑗,𝑘,𝑚,𝑠,𝑙)

Proof of Theorem ntrneineine0
StepHypRef Expression
1 ntrnei.o . . 3 𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗m 𝑖) ↦ (𝑙𝑗 ↦ {𝑚𝑖𝑙 ∈ (𝑘𝑚)})))
2 ntrnei.f . . 3 𝐹 = (𝒫 𝐵𝑂𝐵)
3 ntrnei.r . . . 4 (𝜑𝐼𝐹𝑁)
43adantr 481 . . 3 ((𝜑𝑥𝐵) → 𝐼𝐹𝑁)
5 simpr 485 . . 3 ((𝜑𝑥𝐵) → 𝑥𝐵)
61, 2, 4, 5ntrneineine0lem 42510 . 2 ((𝜑𝑥𝐵) → (∃𝑠 ∈ 𝒫 𝐵𝑥 ∈ (𝐼𝑠) ↔ (𝑁𝑥) ≠ ∅))
76ralbidva 3174 1 (𝜑 → (∀𝑥𝐵𝑠 ∈ 𝒫 𝐵𝑥 ∈ (𝐼𝑠) ↔ ∀𝑥𝐵 (𝑁𝑥) ≠ ∅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396   = wceq 1541  wcel 2106  wne 2939  wral 3060  wrex 3069  {crab 3418  Vcvv 3459  c0 4302  𝒫 cpw 4580   class class class wbr 5125  cmpt 5208  cfv 6516  (class class class)co 7377  cmpo 7379  m cmap 8787
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702  ax-rep 5262  ax-sep 5276  ax-nul 5283  ax-pow 5340  ax-pr 5404  ax-un 7692
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3365  df-rab 3419  df-v 3461  df-sbc 3758  df-csb 3874  df-dif 3931  df-un 3933  df-in 3935  df-ss 3945  df-nul 4303  df-if 4507  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4886  df-iun 4976  df-br 5126  df-opab 5188  df-mpt 5209  df-id 5551  df-xp 5659  df-rel 5660  df-cnv 5661  df-co 5662  df-dm 5663  df-rn 5664  df-res 5665  df-ima 5666  df-iota 6468  df-fun 6518  df-fn 6519  df-f 6520  df-f1 6521  df-fo 6522  df-f1o 6523  df-fv 6524  df-ov 7380  df-oprab 7381  df-mpo 7382  df-1st 7941  df-2nd 7942  df-map 8789
This theorem is referenced by: (None)
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