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Theorem ntrneineine0lem 42297
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 481 . . . 4 ((𝜑𝑠 ∈ 𝒫 𝐵) → 𝐼𝐹𝑁)
5 ntrnei.x . . . . 5 (𝜑𝑋𝐵)
65adantr 481 . . . 4 ((𝜑𝑠 ∈ 𝒫 𝐵) → 𝑋𝐵)
7 simpr 485 . . . 4 ((𝜑𝑠 ∈ 𝒫 𝐵) → 𝑠 ∈ 𝒫 𝐵)
81, 2, 4, 6, 7ntrneiel 42295 . . 3 ((𝜑𝑠 ∈ 𝒫 𝐵) → (𝑋 ∈ (𝐼𝑠) ↔ 𝑠 ∈ (𝑁𝑋)))
98rexbidva 3171 . 2 (𝜑 → (∃𝑠 ∈ 𝒫 𝐵𝑋 ∈ (𝐼𝑠) ↔ ∃𝑠 ∈ 𝒫 𝐵𝑠 ∈ (𝑁𝑋)))
101, 2, 3ntrneinex 42291 . . . . . . . . . 10 (𝜑𝑁 ∈ (𝒫 𝒫 𝐵m 𝐵))
11 elmapi 8783 . . . . . . . . . 10 (𝑁 ∈ (𝒫 𝒫 𝐵m 𝐵) → 𝑁:𝐵⟶𝒫 𝒫 𝐵)
1210, 11syl 17 . . . . . . . . 9 (𝜑𝑁:𝐵⟶𝒫 𝒫 𝐵)
1312, 5ffvelcdmd 7032 . . . . . . . 8 (𝜑 → (𝑁𝑋) ∈ 𝒫 𝒫 𝐵)
1413elpwid 4567 . . . . . . 7 (𝜑 → (𝑁𝑋) ⊆ 𝒫 𝐵)
1514sseld 3941 . . . . . 6 (𝜑 → (𝑠 ∈ (𝑁𝑋) → 𝑠 ∈ 𝒫 𝐵))
1615pm4.71rd 563 . . . . 5 (𝜑 → (𝑠 ∈ (𝑁𝑋) ↔ (𝑠 ∈ 𝒫 𝐵𝑠 ∈ (𝑁𝑋))))
1716exbidv 1924 . . . 4 (𝜑 → (∃𝑠 𝑠 ∈ (𝑁𝑋) ↔ ∃𝑠(𝑠 ∈ 𝒫 𝐵𝑠 ∈ (𝑁𝑋))))
1817bicomd 222 . . 3 (𝜑 → (∃𝑠(𝑠 ∈ 𝒫 𝐵𝑠 ∈ (𝑁𝑋)) ↔ ∃𝑠 𝑠 ∈ (𝑁𝑋)))
19 df-rex 3072 . . 3 (∃𝑠 ∈ 𝒫 𝐵𝑠 ∈ (𝑁𝑋) ↔ ∃𝑠(𝑠 ∈ 𝒫 𝐵𝑠 ∈ (𝑁𝑋)))
20 n0 4304 . . 3 ((𝑁𝑋) ≠ ∅ ↔ ∃𝑠 𝑠 ∈ (𝑁𝑋))
2118, 19, 203bitr4g 313 . 2 (𝜑 → (∃𝑠 ∈ 𝒫 𝐵𝑠 ∈ (𝑁𝑋) ↔ (𝑁𝑋) ≠ ∅))
229, 21bitrd 278 1 (𝜑 → (∃𝑠 ∈ 𝒫 𝐵𝑋 ∈ (𝐼𝑠) ↔ (𝑁𝑋) ≠ ∅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396   = wceq 1541  wex 1781  wcel 2106  wne 2941  wrex 3071  {crab 3405  Vcvv 3443  c0 4280  𝒫 cpw 4558   class class class wbr 5103  cmpt 5186  wf 6489  cfv 6493  (class class class)co 7353  cmpo 7355  m cmap 8761
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 2707  ax-rep 5240  ax-sep 5254  ax-nul 5261  ax-pow 5318  ax-pr 5382  ax-un 7668
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 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3352  df-rab 3406  df-v 3445  df-sbc 3738  df-csb 3854  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4281  df-if 4485  df-pw 4560  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4864  df-iun 4954  df-br 5104  df-opab 5166  df-mpt 5187  df-id 5529  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6445  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-ov 7356  df-oprab 7357  df-mpo 7358  df-1st 7917  df-2nd 7918  df-map 8763
This theorem is referenced by:  ntrneineine0  42301
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