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Theorem ntrneiiso 37868
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 3572 . . . . . . . 8 ((𝐼𝑠) ⊆ (𝐼𝑡) ↔ ∀𝑥(𝑥 ∈ (𝐼𝑠) → 𝑥 ∈ (𝐼𝑡)))
21imbi2i 326 . . . . . . 7 ((𝑠𝑡 → (𝐼𝑠) ⊆ (𝐼𝑡)) ↔ (𝑠𝑡 → ∀𝑥(𝑥 ∈ (𝐼𝑠) → 𝑥 ∈ (𝐼𝑡))))
3 19.21v 1865 . . . . . . 7 (∀𝑥(𝑠𝑡 → (𝑥 ∈ (𝐼𝑠) → 𝑥 ∈ (𝐼𝑡))) ↔ (𝑠𝑡 → ∀𝑥(𝑥 ∈ (𝐼𝑠) → 𝑥 ∈ (𝐼𝑡))))
42, 3bitr4i 267 . . . . . 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 789 . . . . . . . . . . . . . . . . . . 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 37853 . . . . . . . . . . . . . . . . . . 19 (𝜑𝐼 ∈ (𝒫 𝐵𝑚 𝒫 𝐵))
11 elmapi 7823 . . . . . . . . . . . . . . . . . . 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 791 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) → 𝑠 ∈ 𝒫 𝐵)
1412, 13ffvelrnd 6316 . . . . . . . . . . . . . . . . 17 (((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) → (𝐼𝑠) ∈ 𝒫 𝐵)
1514elpwid 4141 . . . . . . . . . . . . . . . 16 (((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) → (𝐼𝑠) ⊆ 𝐵)
1615sselda 3583 . . . . . . . . . . . . . . 15 ((((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) ∧ 𝑥 ∈ (𝐼𝑠)) → 𝑥𝐵)
1716pm2.24d 147 . . . . . . . . . . . . . 14 ((((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) ∧ 𝑥 ∈ (𝐼𝑠)) → (¬ 𝑥𝐵𝑥 ∈ (𝐼𝑡)))
1817ex 450 . . . . . . . . . . . . 13 (((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) → (𝑥 ∈ (𝐼𝑠) → (¬ 𝑥𝐵𝑥 ∈ (𝐼𝑡))))
1918com23 86 . . . . . . . . . . . 12 (((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) → (¬ 𝑥𝐵 → (𝑥 ∈ (𝐼𝑠) → 𝑥 ∈ (𝐼𝑡))))
2019a1dd 50 . . . . . . . . . . 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 174 . . . . . . . . . 10 (((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) → ((𝑥𝐵 → (𝑠𝑡 → (𝑥 ∈ (𝐼𝑠) → 𝑥 ∈ (𝐼𝑡)))) → (𝑠𝑡 → (𝑥 ∈ (𝐼𝑠) → 𝑥 ∈ (𝐼𝑡)))))
235, 22impbid2 216 . . . . . . . . 9 (((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) → ((𝑠𝑡 → (𝑥 ∈ (𝐼𝑠) → 𝑥 ∈ (𝐼𝑡))) ↔ (𝑥𝐵 → (𝑠𝑡 → (𝑥 ∈ (𝐼𝑠) → 𝑥 ∈ (𝐼𝑡))))))
2423albidv 1846 . . . . . . . 8 (((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) → (∀𝑥(𝑠𝑡 → (𝑥 ∈ (𝐼𝑠) → 𝑥 ∈ (𝐼𝑡))) ↔ ∀𝑥(𝑥𝐵 → (𝑠𝑡 → (𝑥 ∈ (𝐼𝑠) → 𝑥 ∈ (𝐼𝑡))))))
25 df-ral 2912 . . . . . . . 8 (∀𝑥𝐵 (𝑠𝑡 → (𝑥 ∈ (𝐼𝑠) → 𝑥 ∈ (𝐼𝑡))) ↔ ∀𝑥(𝑥𝐵 → (𝑠𝑡 → (𝑥 ∈ (𝐼𝑠) → 𝑥 ∈ (𝐼𝑡)))))
2624, 25syl6bbr 278 . . . . . . 7 (((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) → (∀𝑥(𝑠𝑡 → (𝑥 ∈ (𝐼𝑠) → 𝑥 ∈ (𝐼𝑡))) ↔ ∀𝑥𝐵 (𝑠𝑡 → (𝑥 ∈ (𝐼𝑠) → 𝑥 ∈ (𝐼𝑡)))))
279ad3antrrr 765 . . . . . . . . . . . 12 ((((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) ∧ 𝑥𝐵) → 𝐼𝐹𝑁)
28 simpr 477 . . . . . . . . . . . 12 ((((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) ∧ 𝑥𝐵) → 𝑥𝐵)
29 simpllr 798 . . . . . . . . . . . 12 ((((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) ∧ 𝑥𝐵) → 𝑠 ∈ 𝒫 𝐵)
307, 8, 27, 28, 29ntrneiel 37858 . . . . . . . . . . 11 ((((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) ∧ 𝑥𝐵) → (𝑥 ∈ (𝐼𝑠) ↔ 𝑠 ∈ (𝑁𝑥)))
31 simplr 791 . . . . . . . . . . . 12 ((((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) ∧ 𝑥𝐵) → 𝑡 ∈ 𝒫 𝐵)
327, 8, 27, 28, 31ntrneiel 37858 . . . . . . . . . . 11 ((((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) ∧ 𝑥𝐵) → (𝑥 ∈ (𝐼𝑡) ↔ 𝑡 ∈ (𝑁𝑥)))
3330, 32imbi12d 334 . . . . . . . . . 10 ((((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) ∧ 𝑥𝐵) → ((𝑥 ∈ (𝐼𝑠) → 𝑥 ∈ (𝐼𝑡)) ↔ (𝑠 ∈ (𝑁𝑥) → 𝑡 ∈ (𝑁𝑥))))
3433imbi2d 330 . . . . . . . . 9 ((((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) ∧ 𝑥𝐵) → ((𝑠𝑡 → (𝑥 ∈ (𝐼𝑠) → 𝑥 ∈ (𝐼𝑡))) ↔ (𝑠𝑡 → (𝑠 ∈ (𝑁𝑥) → 𝑡 ∈ (𝑁𝑥)))))
35 impexp 462 . . . . . . . . . 10 (((𝑠𝑡𝑠 ∈ (𝑁𝑥)) → 𝑡 ∈ (𝑁𝑥)) ↔ (𝑠𝑡 → (𝑠 ∈ (𝑁𝑥) → 𝑡 ∈ (𝑁𝑥))))
36 ancomst 468 . . . . . . . . . 10 (((𝑠𝑡𝑠 ∈ (𝑁𝑥)) → 𝑡 ∈ (𝑁𝑥)) ↔ ((𝑠 ∈ (𝑁𝑥) ∧ 𝑠𝑡) → 𝑡 ∈ (𝑁𝑥)))
3735, 36bitr3i 266 . . . . . . . . 9 ((𝑠𝑡 → (𝑠 ∈ (𝑁𝑥) → 𝑡 ∈ (𝑁𝑥))) ↔ ((𝑠 ∈ (𝑁𝑥) ∧ 𝑠𝑡) → 𝑡 ∈ (𝑁𝑥)))
3834, 37syl6bb 276 . . . . . . . 8 ((((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) ∧ 𝑥𝐵) → ((𝑠𝑡 → (𝑥 ∈ (𝐼𝑠) → 𝑥 ∈ (𝐼𝑡))) ↔ ((𝑠 ∈ (𝑁𝑥) ∧ 𝑠𝑡) → 𝑡 ∈ (𝑁𝑥))))
3938ralbidva 2979 . . . . . . 7 (((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) → (∀𝑥𝐵 (𝑠𝑡 → (𝑥 ∈ (𝐼𝑠) → 𝑥 ∈ (𝐼𝑡))) ↔ ∀𝑥𝐵 ((𝑠 ∈ (𝑁𝑥) ∧ 𝑠𝑡) → 𝑡 ∈ (𝑁𝑥))))
4026, 39bitrd 268 . . . . . 6 (((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) → (∀𝑥(𝑠𝑡 → (𝑥 ∈ (𝐼𝑠) → 𝑥 ∈ (𝐼𝑡))) ↔ ∀𝑥𝐵 ((𝑠 ∈ (𝑁𝑥) ∧ 𝑠𝑡) → 𝑡 ∈ (𝑁𝑥))))
414, 40syl5bb 272 . . . . 5 (((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) → ((𝑠𝑡 → (𝐼𝑠) ⊆ (𝐼𝑡)) ↔ ∀𝑥𝐵 ((𝑠 ∈ (𝑁𝑥) ∧ 𝑠𝑡) → 𝑡 ∈ (𝑁𝑥))))
4241ralbidva 2979 . . . 4 ((𝜑𝑠 ∈ 𝒫 𝐵) → (∀𝑡 ∈ 𝒫 𝐵(𝑠𝑡 → (𝐼𝑠) ⊆ (𝐼𝑡)) ↔ ∀𝑡 ∈ 𝒫 𝐵𝑥𝐵 ((𝑠 ∈ (𝑁𝑥) ∧ 𝑠𝑡) → 𝑡 ∈ (𝑁𝑥))))
43 ralcom 3090 . . . 4 (∀𝑡 ∈ 𝒫 𝐵𝑥𝐵 ((𝑠 ∈ (𝑁𝑥) ∧ 𝑠𝑡) → 𝑡 ∈ (𝑁𝑥)) ↔ ∀𝑥𝐵𝑡 ∈ 𝒫 𝐵((𝑠 ∈ (𝑁𝑥) ∧ 𝑠𝑡) → 𝑡 ∈ (𝑁𝑥)))
4442, 43syl6bb 276 . . 3 ((𝜑𝑠 ∈ 𝒫 𝐵) → (∀𝑡 ∈ 𝒫 𝐵(𝑠𝑡 → (𝐼𝑠) ⊆ (𝐼𝑡)) ↔ ∀𝑥𝐵𝑡 ∈ 𝒫 𝐵((𝑠 ∈ (𝑁𝑥) ∧ 𝑠𝑡) → 𝑡 ∈ (𝑁𝑥))))
4544ralbidva 2979 . 2 (𝜑 → (∀𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵(𝑠𝑡 → (𝐼𝑠) ⊆ (𝐼𝑡)) ↔ ∀𝑠 ∈ 𝒫 𝐵𝑥𝐵𝑡 ∈ 𝒫 𝐵((𝑠 ∈ (𝑁𝑥) ∧ 𝑠𝑡) → 𝑡 ∈ (𝑁𝑥))))
46 ralcom 3090 . 2 (∀𝑠 ∈ 𝒫 𝐵𝑥𝐵𝑡 ∈ 𝒫 𝐵((𝑠 ∈ (𝑁𝑥) ∧ 𝑠𝑡) → 𝑡 ∈ (𝑁𝑥)) ↔ ∀𝑥𝐵𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵((𝑠 ∈ (𝑁𝑥) ∧ 𝑠𝑡) → 𝑡 ∈ (𝑁𝑥)))
4745, 46syl6bb 276 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 196  wa 384  wal 1478   = wceq 1480  wcel 1987  wral 2907  {crab 2911  Vcvv 3186  wss 3555  𝒫 cpw 4130   class class class wbr 4613  cmpt 4673  wf 5843  cfv 5847  (class class class)co 6604  cmpt2 6606  𝑚 cmap 7802
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-1st 7113  df-2nd 7114  df-map 7804
This theorem is referenced by: (None)
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