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Theorem ntrneineine0lem 37401
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)
ntrnei.x (𝜑𝑋𝐵)
Assertion
Ref Expression
ntrneineine0lem (𝜑 → (∃𝑠 ∈ 𝒫 𝐵𝑋 ∈ (𝐼𝑠) ↔ (𝑁𝑋) ≠ ∅))
Distinct variable groups:   𝐵,𝑖,𝑗,𝑘,𝑙,𝑚   𝑘,𝐼,𝑙,𝑚   𝑁,𝑠   𝑋,𝑙,𝑚,𝑠   𝜑,𝑖,𝑗,𝑘,𝑙,𝑠
Allowed substitution hints:   𝜑(𝑚)   𝐵(𝑠)   𝐹(𝑖,𝑗,𝑘,𝑚,𝑠,𝑙)   𝐼(𝑖,𝑗,𝑠)   𝑁(𝑖,𝑗,𝑘,𝑚,𝑙)   𝑂(𝑖,𝑗,𝑘,𝑚,𝑠,𝑙)   𝑋(𝑖,𝑗,𝑘)

Proof of Theorem ntrneineine0lem
StepHypRef Expression
1 ntrnei.o . . . 4 𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗𝑚 𝑖) ↦ (𝑙𝑗 ↦ {𝑚𝑖𝑙 ∈ (𝑘𝑚)})))
2 ntrnei.f . . . 4 𝐹 = (𝒫 𝐵𝑂𝐵)
3 ntrnei.r . . . . 5 (𝜑𝐼𝐹𝑁)
43adantr 480 . . . 4 ((𝜑𝑠 ∈ 𝒫 𝐵) → 𝐼𝐹𝑁)
5 ntrnei.x . . . . 5 (𝜑𝑋𝐵)
65adantr 480 . . . 4 ((𝜑𝑠 ∈ 𝒫 𝐵) → 𝑋𝐵)
7 simpr 476 . . . 4 ((𝜑𝑠 ∈ 𝒫 𝐵) → 𝑠 ∈ 𝒫 𝐵)
81, 2, 4, 6, 7ntrneiel 37399 . . 3 ((𝜑𝑠 ∈ 𝒫 𝐵) → (𝑋 ∈ (𝐼𝑠) ↔ 𝑠 ∈ (𝑁𝑋)))
98rexbidva 3031 . 2 (𝜑 → (∃𝑠 ∈ 𝒫 𝐵𝑋 ∈ (𝐼𝑠) ↔ ∃𝑠 ∈ 𝒫 𝐵𝑠 ∈ (𝑁𝑋)))
101, 2, 3ntrneinex 37395 . . . . . . . . . 10 (𝜑𝑁 ∈ (𝒫 𝒫 𝐵𝑚 𝐵))
11 elmapi 7765 . . . . . . . . . 10 (𝑁 ∈ (𝒫 𝒫 𝐵𝑚 𝐵) → 𝑁:𝐵⟶𝒫 𝒫 𝐵)
1210, 11syl 17 . . . . . . . . 9 (𝜑𝑁:𝐵⟶𝒫 𝒫 𝐵)
1312, 5ffvelrnd 6268 . . . . . . . 8 (𝜑 → (𝑁𝑋) ∈ 𝒫 𝒫 𝐵)
1413elpwid 4118 . . . . . . 7 (𝜑 → (𝑁𝑋) ⊆ 𝒫 𝐵)
1514sseld 3567 . . . . . 6 (𝜑 → (𝑠 ∈ (𝑁𝑋) → 𝑠 ∈ 𝒫 𝐵))
1615pm4.71rd 665 . . . . 5 (𝜑 → (𝑠 ∈ (𝑁𝑋) ↔ (𝑠 ∈ 𝒫 𝐵𝑠 ∈ (𝑁𝑋))))
1716exbidv 1837 . . . 4 (𝜑 → (∃𝑠 𝑠 ∈ (𝑁𝑋) ↔ ∃𝑠(𝑠 ∈ 𝒫 𝐵𝑠 ∈ (𝑁𝑋))))
1817bicomd 212 . . 3 (𝜑 → (∃𝑠(𝑠 ∈ 𝒫 𝐵𝑠 ∈ (𝑁𝑋)) ↔ ∃𝑠 𝑠 ∈ (𝑁𝑋)))
19 df-rex 2902 . . 3 (∃𝑠 ∈ 𝒫 𝐵𝑠 ∈ (𝑁𝑋) ↔ ∃𝑠(𝑠 ∈ 𝒫 𝐵𝑠 ∈ (𝑁𝑋)))
20 n0 3890 . . 3 ((𝑁𝑋) ≠ ∅ ↔ ∃𝑠 𝑠 ∈ (𝑁𝑋))
2118, 19, 203bitr4g 302 . 2 (𝜑 → (∃𝑠 ∈ 𝒫 𝐵𝑠 ∈ (𝑁𝑋) ↔ (𝑁𝑋) ≠ ∅))
229, 21bitrd 267 1 (𝜑 → (∃𝑠 ∈ 𝒫 𝐵𝑋 ∈ (𝐼𝑠) ↔ (𝑁𝑋) ≠ ∅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 195  wa 383   = wceq 1475  wex 1695  wcel 1977  wne 2780  wrex 2897  {crab 2900  Vcvv 3173  c0 3874  𝒫 cpw 4108   class class class wbr 4583  cmpt 4643  wf 5800  cfv 5804  (class class class)co 6549  cmpt2 6551  𝑚 cmap 7744
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-1st 7059  df-2nd 7060  df-map 7746
This theorem is referenced by:  ntrneineine0  37405
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