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Theorem ntrneineine0lem 38901
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 472 . . . 4 ((𝜑𝑠 ∈ 𝒫 𝐵) → 𝐼𝐹𝑁)
5 ntrnei.x . . . . 5 (𝜑𝑋𝐵)
65adantr 472 . . . 4 ((𝜑𝑠 ∈ 𝒫 𝐵) → 𝑋𝐵)
7 simpr 479 . . . 4 ((𝜑𝑠 ∈ 𝒫 𝐵) → 𝑠 ∈ 𝒫 𝐵)
81, 2, 4, 6, 7ntrneiel 38899 . . 3 ((𝜑𝑠 ∈ 𝒫 𝐵) → (𝑋 ∈ (𝐼𝑠) ↔ 𝑠 ∈ (𝑁𝑋)))
98rexbidva 3187 . 2 (𝜑 → (∃𝑠 ∈ 𝒫 𝐵𝑋 ∈ (𝐼𝑠) ↔ ∃𝑠 ∈ 𝒫 𝐵𝑠 ∈ (𝑁𝑋)))
101, 2, 3ntrneinex 38895 . . . . . . . . . 10 (𝜑𝑁 ∈ (𝒫 𝒫 𝐵𝑚 𝐵))
11 elmapi 8047 . . . . . . . . . 10 (𝑁 ∈ (𝒫 𝒫 𝐵𝑚 𝐵) → 𝑁:𝐵⟶𝒫 𝒫 𝐵)
1210, 11syl 17 . . . . . . . . 9 (𝜑𝑁:𝐵⟶𝒫 𝒫 𝐵)
1312, 5ffvelrnd 6524 . . . . . . . 8 (𝜑 → (𝑁𝑋) ∈ 𝒫 𝒫 𝐵)
1413elpwid 4314 . . . . . . 7 (𝜑 → (𝑁𝑋) ⊆ 𝒫 𝐵)
1514sseld 3743 . . . . . 6 (𝜑 → (𝑠 ∈ (𝑁𝑋) → 𝑠 ∈ 𝒫 𝐵))
1615pm4.71rd 670 . . . . 5 (𝜑 → (𝑠 ∈ (𝑁𝑋) ↔ (𝑠 ∈ 𝒫 𝐵𝑠 ∈ (𝑁𝑋))))
1716exbidv 1999 . . . 4 (𝜑 → (∃𝑠 𝑠 ∈ (𝑁𝑋) ↔ ∃𝑠(𝑠 ∈ 𝒫 𝐵𝑠 ∈ (𝑁𝑋))))
1817bicomd 213 . . 3 (𝜑 → (∃𝑠(𝑠 ∈ 𝒫 𝐵𝑠 ∈ (𝑁𝑋)) ↔ ∃𝑠 𝑠 ∈ (𝑁𝑋)))
19 df-rex 3056 . . 3 (∃𝑠 ∈ 𝒫 𝐵𝑠 ∈ (𝑁𝑋) ↔ ∃𝑠(𝑠 ∈ 𝒫 𝐵𝑠 ∈ (𝑁𝑋)))
20 n0 4074 . . 3 ((𝑁𝑋) ≠ ∅ ↔ ∃𝑠 𝑠 ∈ (𝑁𝑋))
2118, 19, 203bitr4g 303 . 2 (𝜑 → (∃𝑠 ∈ 𝒫 𝐵𝑠 ∈ (𝑁𝑋) ↔ (𝑁𝑋) ≠ ∅))
229, 21bitrd 268 1 (𝜑 → (∃𝑠 ∈ 𝒫 𝐵𝑋 ∈ (𝐼𝑠) ↔ (𝑁𝑋) ≠ ∅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1632  wex 1853  wcel 2139  wne 2932  wrex 3051  {crab 3054  Vcvv 3340  c0 4058  𝒫 cpw 4302   class class class wbr 4804  cmpt 4881  wf 6045  cfv 6049  (class class class)co 6814  cmpt2 6816  𝑚 cmap 8025
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7115
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-ov 6817  df-oprab 6818  df-mpt2 6819  df-1st 7334  df-2nd 7335  df-map 8027
This theorem is referenced by:  ntrneineine0  38905
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