Mathbox for Richard Penner < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  ntrneineine0lem Structured version   Visualization version   GIF version

Theorem ntrneineine0lem 40850
 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 484 . . . 4 ((𝜑𝑠 ∈ 𝒫 𝐵) → 𝐼𝐹𝑁)
5 ntrnei.x . . . . 5 (𝜑𝑋𝐵)
65adantr 484 . . . 4 ((𝜑𝑠 ∈ 𝒫 𝐵) → 𝑋𝐵)
7 simpr 488 . . . 4 ((𝜑𝑠 ∈ 𝒫 𝐵) → 𝑠 ∈ 𝒫 𝐵)
81, 2, 4, 6, 7ntrneiel 40848 . . 3 ((𝜑𝑠 ∈ 𝒫 𝐵) → (𝑋 ∈ (𝐼𝑠) ↔ 𝑠 ∈ (𝑁𝑋)))
98rexbidva 3255 . 2 (𝜑 → (∃𝑠 ∈ 𝒫 𝐵𝑋 ∈ (𝐼𝑠) ↔ ∃𝑠 ∈ 𝒫 𝐵𝑠 ∈ (𝑁𝑋)))
101, 2, 3ntrneinex 40844 . . . . . . . . . 10 (𝜑𝑁 ∈ (𝒫 𝒫 𝐵m 𝐵))
11 elmapi 8418 . . . . . . . . . 10 (𝑁 ∈ (𝒫 𝒫 𝐵m 𝐵) → 𝑁:𝐵⟶𝒫 𝒫 𝐵)
1210, 11syl 17 . . . . . . . . 9 (𝜑𝑁:𝐵⟶𝒫 𝒫 𝐵)
1312, 5ffvelrnd 6834 . . . . . . . 8 (𝜑 → (𝑁𝑋) ∈ 𝒫 𝒫 𝐵)
1413elpwid 4508 . . . . . . 7 (𝜑 → (𝑁𝑋) ⊆ 𝒫 𝐵)
1514sseld 3914 . . . . . 6 (𝜑 → (𝑠 ∈ (𝑁𝑋) → 𝑠 ∈ 𝒫 𝐵))
1615pm4.71rd 566 . . . . 5 (𝜑 → (𝑠 ∈ (𝑁𝑋) ↔ (𝑠 ∈ 𝒫 𝐵𝑠 ∈ (𝑁𝑋))))
1716exbidv 1922 . . . 4 (𝜑 → (∃𝑠 𝑠 ∈ (𝑁𝑋) ↔ ∃𝑠(𝑠 ∈ 𝒫 𝐵𝑠 ∈ (𝑁𝑋))))
1817bicomd 226 . . 3 (𝜑 → (∃𝑠(𝑠 ∈ 𝒫 𝐵𝑠 ∈ (𝑁𝑋)) ↔ ∃𝑠 𝑠 ∈ (𝑁𝑋)))
19 df-rex 3112 . . 3 (∃𝑠 ∈ 𝒫 𝐵𝑠 ∈ (𝑁𝑋) ↔ ∃𝑠(𝑠 ∈ 𝒫 𝐵𝑠 ∈ (𝑁𝑋)))
20 n0 4260 . . 3 ((𝑁𝑋) ≠ ∅ ↔ ∃𝑠 𝑠 ∈ (𝑁𝑋))
2118, 19, 203bitr4g 317 . 2 (𝜑 → (∃𝑠 ∈ 𝒫 𝐵𝑠 ∈ (𝑁𝑋) ↔ (𝑁𝑋) ≠ ∅))
229, 21bitrd 282 1 (𝜑 → (∃𝑠 ∈ 𝒫 𝐵𝑋 ∈ (𝐼𝑠) ↔ (𝑁𝑋) ≠ ∅))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538  ∃wex 1781   ∈ wcel 2111   ≠ wne 2987  ∃wrex 3107  {crab 3110  Vcvv 3441  ∅c0 4243  𝒫 cpw 4497   class class class wbr 5031   ↦ cmpt 5111  ⟶wf 6323  ‘cfv 6327  (class class class)co 7140   ∈ cmpo 7142   ↑m cmap 8396 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 2770  ax-rep 5155  ax-sep 5168  ax-nul 5175  ax-pow 5232  ax-pr 5296  ax-un 7448 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-iun 4884  df-br 5032  df-opab 5094  df-mpt 5112  df-id 5426  df-xp 5526  df-rel 5527  df-cnv 5528  df-co 5529  df-dm 5530  df-rn 5531  df-res 5532  df-ima 5533  df-iota 6286  df-fun 6329  df-fn 6330  df-f 6331  df-f1 6332  df-fo 6333  df-f1o 6334  df-fv 6335  df-ov 7143  df-oprab 7144  df-mpo 7145  df-1st 7678  df-2nd 7679  df-map 8398 This theorem is referenced by:  ntrneineine0  40854
 Copyright terms: Public domain W3C validator