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Theorem ntrneiiso 37409
 Description: If (pseudo-)interior and (pseudo-)neighborhood functions are related by the operator, 𝐹, then conditions equal to claiming that the interior function is isotonic hold equally. (Contributed by RP, 3-Jun-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
ntrneiiso (𝜑 → (∀𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵(𝑠𝑡 → (𝐼𝑠) ⊆ (𝐼𝑡)) ↔ ∀𝑥𝐵𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵((𝑠 ∈ (𝑁𝑥) ∧ 𝑠𝑡) → 𝑡 ∈ (𝑁𝑥))))
Distinct variable groups:   𝐵,𝑖,𝑗,𝑘,𝑙,𝑚,𝑠,𝑡,𝑥   𝑘,𝐼,𝑙,𝑚,𝑥   𝜑,𝑖,𝑗,𝑘,𝑙,𝑠,𝑡,𝑥
Allowed substitution hints:   𝜑(𝑚)   𝐹(𝑥,𝑡,𝑖,𝑗,𝑘,𝑚,𝑠,𝑙)   𝐼(𝑡,𝑖,𝑗,𝑠)   𝑁(𝑥,𝑡,𝑖,𝑗,𝑘,𝑚,𝑠,𝑙)   𝑂(𝑥,𝑡,𝑖,𝑗,𝑘,𝑚,𝑠,𝑙)

Proof of Theorem ntrneiiso
StepHypRef Expression
1 dfss2 3557 . . . . . . . 8 ((𝐼𝑠) ⊆ (𝐼𝑡) ↔ ∀𝑥(𝑥 ∈ (𝐼𝑠) → 𝑥 ∈ (𝐼𝑡)))
21imbi2i 325 . . . . . . 7 ((𝑠𝑡 → (𝐼𝑠) ⊆ (𝐼𝑡)) ↔ (𝑠𝑡 → ∀𝑥(𝑥 ∈ (𝐼𝑠) → 𝑥 ∈ (𝐼𝑡))))
3 19.21v 1855 . . . . . . 7 (∀𝑥(𝑠𝑡 → (𝑥 ∈ (𝐼𝑠) → 𝑥 ∈ (𝐼𝑡))) ↔ (𝑠𝑡 → ∀𝑥(𝑥 ∈ (𝐼𝑠) → 𝑥 ∈ (𝐼𝑡))))
42, 3bitr4i 266 . . . . . 6 ((𝑠𝑡 → (𝐼𝑠) ⊆ (𝐼𝑡)) ↔ ∀𝑥(𝑠𝑡 → (𝑥 ∈ (𝐼𝑠) → 𝑥 ∈ (𝐼𝑡))))
5 ax-1 6 . . . . . . . . . 10 ((𝑠𝑡 → (𝑥 ∈ (𝐼𝑠) → 𝑥 ∈ (𝐼𝑡))) → (𝑥𝐵 → (𝑠𝑡 → (𝑥 ∈ (𝐼𝑠) → 𝑥 ∈ (𝐼𝑡)))))
6 simpll 786 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) → 𝜑)
7 ntrnei.o . . . . . . . . . . . . . . . . . . . 20 𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗𝑚 𝑖) ↦ (𝑙𝑗 ↦ {𝑚𝑖𝑙 ∈ (𝑘𝑚)})))
8 ntrnei.f . . . . . . . . . . . . . . . . . . . 20 𝐹 = (𝒫 𝐵𝑂𝐵)
9 ntrnei.r . . . . . . . . . . . . . . . . . . . 20 (𝜑𝐼𝐹𝑁)
107, 8, 9ntrneiiex 37394 . . . . . . . . . . . . . . . . . . 19 (𝜑𝐼 ∈ (𝒫 𝐵𝑚 𝒫 𝐵))
11 elmapi 7765 . . . . . . . . . . . . . . . . . . 19 (𝐼 ∈ (𝒫 𝐵𝑚 𝒫 𝐵) → 𝐼:𝒫 𝐵⟶𝒫 𝐵)
126, 10, 113syl 18 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) → 𝐼:𝒫 𝐵⟶𝒫 𝐵)
13 simplr 788 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) → 𝑠 ∈ 𝒫 𝐵)
1412, 13ffvelrnd 6268 . . . . . . . . . . . . . . . . 17 (((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) → (𝐼𝑠) ∈ 𝒫 𝐵)
1514elpwid 4118 . . . . . . . . . . . . . . . 16 (((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) → (𝐼𝑠) ⊆ 𝐵)
1615sselda 3568 . . . . . . . . . . . . . . 15 ((((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) ∧ 𝑥 ∈ (𝐼𝑠)) → 𝑥𝐵)
1716pm2.24d 146 . . . . . . . . . . . . . 14 ((((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) ∧ 𝑥 ∈ (𝐼𝑠)) → (¬ 𝑥𝐵𝑥 ∈ (𝐼𝑡)))
1817ex 449 . . . . . . . . . . . . 13 (((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) → (𝑥 ∈ (𝐼𝑠) → (¬ 𝑥𝐵𝑥 ∈ (𝐼𝑡))))
1918com23 84 . . . . . . . . . . . 12 (((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) → (¬ 𝑥𝐵 → (𝑥 ∈ (𝐼𝑠) → 𝑥 ∈ (𝐼𝑡))))
2019a1dd 48 . . . . . . . . . . 11 (((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) → (¬ 𝑥𝐵 → (𝑠𝑡 → (𝑥 ∈ (𝐼𝑠) → 𝑥 ∈ (𝐼𝑡)))))
21 idd 24 . . . . . . . . . . 11 (((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) → ((𝑠𝑡 → (𝑥 ∈ (𝐼𝑠) → 𝑥 ∈ (𝐼𝑡))) → (𝑠𝑡 → (𝑥 ∈ (𝐼𝑠) → 𝑥 ∈ (𝐼𝑡)))))
2220, 21jad 173 . . . . . . . . . 10 (((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) → ((𝑥𝐵 → (𝑠𝑡 → (𝑥 ∈ (𝐼𝑠) → 𝑥 ∈ (𝐼𝑡)))) → (𝑠𝑡 → (𝑥 ∈ (𝐼𝑠) → 𝑥 ∈ (𝐼𝑡)))))
235, 22impbid2 215 . . . . . . . . 9 (((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) → ((𝑠𝑡 → (𝑥 ∈ (𝐼𝑠) → 𝑥 ∈ (𝐼𝑡))) ↔ (𝑥𝐵 → (𝑠𝑡 → (𝑥 ∈ (𝐼𝑠) → 𝑥 ∈ (𝐼𝑡))))))
2423albidv 1836 . . . . . . . 8 (((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) → (∀𝑥(𝑠𝑡 → (𝑥 ∈ (𝐼𝑠) → 𝑥 ∈ (𝐼𝑡))) ↔ ∀𝑥(𝑥𝐵 → (𝑠𝑡 → (𝑥 ∈ (𝐼𝑠) → 𝑥 ∈ (𝐼𝑡))))))
25 df-ral 2901 . . . . . . . 8 (∀𝑥𝐵 (𝑠𝑡 → (𝑥 ∈ (𝐼𝑠) → 𝑥 ∈ (𝐼𝑡))) ↔ ∀𝑥(𝑥𝐵 → (𝑠𝑡 → (𝑥 ∈ (𝐼𝑠) → 𝑥 ∈ (𝐼𝑡)))))
2624, 25syl6bbr 277 . . . . . . 7 (((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) → (∀𝑥(𝑠𝑡 → (𝑥 ∈ (𝐼𝑠) → 𝑥 ∈ (𝐼𝑡))) ↔ ∀𝑥𝐵 (𝑠𝑡 → (𝑥 ∈ (𝐼𝑠) → 𝑥 ∈ (𝐼𝑡)))))
279ad3antrrr 762 . . . . . . . . . . . 12 ((((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) ∧ 𝑥𝐵) → 𝐼𝐹𝑁)
28 simpr 476 . . . . . . . . . . . 12 ((((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) ∧ 𝑥𝐵) → 𝑥𝐵)
29 simpllr 795 . . . . . . . . . . . 12 ((((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) ∧ 𝑥𝐵) → 𝑠 ∈ 𝒫 𝐵)
307, 8, 27, 28, 29ntrneiel 37399 . . . . . . . . . . 11 ((((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) ∧ 𝑥𝐵) → (𝑥 ∈ (𝐼𝑠) ↔ 𝑠 ∈ (𝑁𝑥)))
31 simplr 788 . . . . . . . . . . . 12 ((((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) ∧ 𝑥𝐵) → 𝑡 ∈ 𝒫 𝐵)
327, 8, 27, 28, 31ntrneiel 37399 . . . . . . . . . . 11 ((((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) ∧ 𝑥𝐵) → (𝑥 ∈ (𝐼𝑡) ↔ 𝑡 ∈ (𝑁𝑥)))
3330, 32imbi12d 333 . . . . . . . . . 10 ((((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) ∧ 𝑥𝐵) → ((𝑥 ∈ (𝐼𝑠) → 𝑥 ∈ (𝐼𝑡)) ↔ (𝑠 ∈ (𝑁𝑥) → 𝑡 ∈ (𝑁𝑥))))
3433imbi2d 329 . . . . . . . . 9 ((((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) ∧ 𝑥𝐵) → ((𝑠𝑡 → (𝑥 ∈ (𝐼𝑠) → 𝑥 ∈ (𝐼𝑡))) ↔ (𝑠𝑡 → (𝑠 ∈ (𝑁𝑥) → 𝑡 ∈ (𝑁𝑥)))))
35 impexp 461 . . . . . . . . . 10 (((𝑠𝑡𝑠 ∈ (𝑁𝑥)) → 𝑡 ∈ (𝑁𝑥)) ↔ (𝑠𝑡 → (𝑠 ∈ (𝑁𝑥) → 𝑡 ∈ (𝑁𝑥))))
36 ancomst 467 . . . . . . . . . 10 (((𝑠𝑡𝑠 ∈ (𝑁𝑥)) → 𝑡 ∈ (𝑁𝑥)) ↔ ((𝑠 ∈ (𝑁𝑥) ∧ 𝑠𝑡) → 𝑡 ∈ (𝑁𝑥)))
3735, 36bitr3i 265 . . . . . . . . 9 ((𝑠𝑡 → (𝑠 ∈ (𝑁𝑥) → 𝑡 ∈ (𝑁𝑥))) ↔ ((𝑠 ∈ (𝑁𝑥) ∧ 𝑠𝑡) → 𝑡 ∈ (𝑁𝑥)))
3834, 37syl6bb 275 . . . . . . . 8 ((((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) ∧ 𝑥𝐵) → ((𝑠𝑡 → (𝑥 ∈ (𝐼𝑠) → 𝑥 ∈ (𝐼𝑡))) ↔ ((𝑠 ∈ (𝑁𝑥) ∧ 𝑠𝑡) → 𝑡 ∈ (𝑁𝑥))))
3938ralbidva 2968 . . . . . . 7 (((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) → (∀𝑥𝐵 (𝑠𝑡 → (𝑥 ∈ (𝐼𝑠) → 𝑥 ∈ (𝐼𝑡))) ↔ ∀𝑥𝐵 ((𝑠 ∈ (𝑁𝑥) ∧ 𝑠𝑡) → 𝑡 ∈ (𝑁𝑥))))
4026, 39bitrd 267 . . . . . 6 (((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) → (∀𝑥(𝑠𝑡 → (𝑥 ∈ (𝐼𝑠) → 𝑥 ∈ (𝐼𝑡))) ↔ ∀𝑥𝐵 ((𝑠 ∈ (𝑁𝑥) ∧ 𝑠𝑡) → 𝑡 ∈ (𝑁𝑥))))
414, 40syl5bb 271 . . . . 5 (((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) → ((𝑠𝑡 → (𝐼𝑠) ⊆ (𝐼𝑡)) ↔ ∀𝑥𝐵 ((𝑠 ∈ (𝑁𝑥) ∧ 𝑠𝑡) → 𝑡 ∈ (𝑁𝑥))))
4241ralbidva 2968 . . . 4 ((𝜑𝑠 ∈ 𝒫 𝐵) → (∀𝑡 ∈ 𝒫 𝐵(𝑠𝑡 → (𝐼𝑠) ⊆ (𝐼𝑡)) ↔ ∀𝑡 ∈ 𝒫 𝐵𝑥𝐵 ((𝑠 ∈ (𝑁𝑥) ∧ 𝑠𝑡) → 𝑡 ∈ (𝑁𝑥))))
43 ralcom 3079 . . . 4 (∀𝑡 ∈ 𝒫 𝐵𝑥𝐵 ((𝑠 ∈ (𝑁𝑥) ∧ 𝑠𝑡) → 𝑡 ∈ (𝑁𝑥)) ↔ ∀𝑥𝐵𝑡 ∈ 𝒫 𝐵((𝑠 ∈ (𝑁𝑥) ∧ 𝑠𝑡) → 𝑡 ∈ (𝑁𝑥)))
4442, 43syl6bb 275 . . 3 ((𝜑𝑠 ∈ 𝒫 𝐵) → (∀𝑡 ∈ 𝒫 𝐵(𝑠𝑡 → (𝐼𝑠) ⊆ (𝐼𝑡)) ↔ ∀𝑥𝐵𝑡 ∈ 𝒫 𝐵((𝑠 ∈ (𝑁𝑥) ∧ 𝑠𝑡) → 𝑡 ∈ (𝑁𝑥))))
4544ralbidva 2968 . 2 (𝜑 → (∀𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵(𝑠𝑡 → (𝐼𝑠) ⊆ (𝐼𝑡)) ↔ ∀𝑠 ∈ 𝒫 𝐵𝑥𝐵𝑡 ∈ 𝒫 𝐵((𝑠 ∈ (𝑁𝑥) ∧ 𝑠𝑡) → 𝑡 ∈ (𝑁𝑥))))
46 ralcom 3079 . 2 (∀𝑠 ∈ 𝒫 𝐵𝑥𝐵𝑡 ∈ 𝒫 𝐵((𝑠 ∈ (𝑁𝑥) ∧ 𝑠𝑡) → 𝑡 ∈ (𝑁𝑥)) ↔ ∀𝑥𝐵𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵((𝑠 ∈ (𝑁𝑥) ∧ 𝑠𝑡) → 𝑡 ∈ (𝑁𝑥)))
4745, 46syl6bb 275 1 (𝜑 → (∀𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵(𝑠𝑡 → (𝐼𝑠) ⊆ (𝐼𝑡)) ↔ ∀𝑥𝐵𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵((𝑠 ∈ (𝑁𝑥) ∧ 𝑠𝑡) → 𝑡 ∈ (𝑁𝑥))))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∧ wa 383  ∀wal 1473   = wceq 1475   ∈ wcel 1977  ∀wral 2896  {crab 2900  Vcvv 3173   ⊆ wss 3540  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: (None)
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