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Theorem ntrneiiso 43603
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 df-ss 3962 . . . . . . . 8 ((𝐼𝑠) ⊆ (𝐼𝑡) ↔ ∀𝑥(𝑥 ∈ (𝐼𝑠) → 𝑥 ∈ (𝐼𝑡)))
21imbi2i 335 . . . . . . 7 ((𝑠𝑡 → (𝐼𝑠) ⊆ (𝐼𝑡)) ↔ (𝑠𝑡 → ∀𝑥(𝑥 ∈ (𝐼𝑠) → 𝑥 ∈ (𝐼𝑡))))
3 19.21v 1934 . . . . . . 7 (∀𝑥(𝑠𝑡 → (𝑥 ∈ (𝐼𝑠) → 𝑥 ∈ (𝐼𝑡))) ↔ (𝑠𝑡 → ∀𝑥(𝑥 ∈ (𝐼𝑠) → 𝑥 ∈ (𝐼𝑡))))
42, 3bitr4i 277 . . . . . 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 765 . . . . . . . . . . . . . . . . . . 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 43588 . . . . . . . . . . . . . . . . . . 19 (𝜑𝐼 ∈ (𝒫 𝐵m 𝒫 𝐵))
11 elmapi 8866 . . . . . . . . . . . . . . . . . . 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 767 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) → 𝑠 ∈ 𝒫 𝐵)
1412, 13ffvelcdmd 7092 . . . . . . . . . . . . . . . . 17 (((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) → (𝐼𝑠) ∈ 𝒫 𝐵)
1514elpwid 4612 . . . . . . . . . . . . . . . 16 (((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) → (𝐼𝑠) ⊆ 𝐵)
1615sselda 3977 . . . . . . . . . . . . . . 15 ((((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) ∧ 𝑥 ∈ (𝐼𝑠)) → 𝑥𝐵)
1716pm2.24d 151 . . . . . . . . . . . . . 14 ((((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) ∧ 𝑥 ∈ (𝐼𝑠)) → (¬ 𝑥𝐵𝑥 ∈ (𝐼𝑡)))
1817ex 411 . . . . . . . . . . . . 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 187 . . . . . . . . . 10 (((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) → ((𝑥𝐵 → (𝑠𝑡 → (𝑥 ∈ (𝐼𝑠) → 𝑥 ∈ (𝐼𝑡)))) → (𝑠𝑡 → (𝑥 ∈ (𝐼𝑠) → 𝑥 ∈ (𝐼𝑡)))))
235, 22impbid2 225 . . . . . . . . 9 (((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) → ((𝑠𝑡 → (𝑥 ∈ (𝐼𝑠) → 𝑥 ∈ (𝐼𝑡))) ↔ (𝑥𝐵 → (𝑠𝑡 → (𝑥 ∈ (𝐼𝑠) → 𝑥 ∈ (𝐼𝑡))))))
2423albidv 1915 . . . . . . . 8 (((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) → (∀𝑥(𝑠𝑡 → (𝑥 ∈ (𝐼𝑠) → 𝑥 ∈ (𝐼𝑡))) ↔ ∀𝑥(𝑥𝐵 → (𝑠𝑡 → (𝑥 ∈ (𝐼𝑠) → 𝑥 ∈ (𝐼𝑡))))))
25 df-ral 3052 . . . . . . . 8 (∀𝑥𝐵 (𝑠𝑡 → (𝑥 ∈ (𝐼𝑠) → 𝑥 ∈ (𝐼𝑡))) ↔ ∀𝑥(𝑥𝐵 → (𝑠𝑡 → (𝑥 ∈ (𝐼𝑠) → 𝑥 ∈ (𝐼𝑡)))))
2624, 25bitr4di 288 . . . . . . 7 (((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) → (∀𝑥(𝑠𝑡 → (𝑥 ∈ (𝐼𝑠) → 𝑥 ∈ (𝐼𝑡))) ↔ ∀𝑥𝐵 (𝑠𝑡 → (𝑥 ∈ (𝐼𝑠) → 𝑥 ∈ (𝐼𝑡)))))
279ad3antrrr 728 . . . . . . . . . . . 12 ((((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) ∧ 𝑥𝐵) → 𝐼𝐹𝑁)
28 simpr 483 . . . . . . . . . . . 12 ((((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) ∧ 𝑥𝐵) → 𝑥𝐵)
29 simpllr 774 . . . . . . . . . . . 12 ((((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) ∧ 𝑥𝐵) → 𝑠 ∈ 𝒫 𝐵)
307, 8, 27, 28, 29ntrneiel 43593 . . . . . . . . . . 11 ((((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) ∧ 𝑥𝐵) → (𝑥 ∈ (𝐼𝑠) ↔ 𝑠 ∈ (𝑁𝑥)))
31 simplr 767 . . . . . . . . . . . 12 ((((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) ∧ 𝑥𝐵) → 𝑡 ∈ 𝒫 𝐵)
327, 8, 27, 28, 31ntrneiel 43593 . . . . . . . . . . 11 ((((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) ∧ 𝑥𝐵) → (𝑥 ∈ (𝐼𝑡) ↔ 𝑡 ∈ (𝑁𝑥)))
3330, 32imbi12d 343 . . . . . . . . . 10 ((((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) ∧ 𝑥𝐵) → ((𝑥 ∈ (𝐼𝑠) → 𝑥 ∈ (𝐼𝑡)) ↔ (𝑠 ∈ (𝑁𝑥) → 𝑡 ∈ (𝑁𝑥))))
3433imbi2d 339 . . . . . . . . 9 ((((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) ∧ 𝑥𝐵) → ((𝑠𝑡 → (𝑥 ∈ (𝐼𝑠) → 𝑥 ∈ (𝐼𝑡))) ↔ (𝑠𝑡 → (𝑠 ∈ (𝑁𝑥) → 𝑡 ∈ (𝑁𝑥)))))
35 impexp 449 . . . . . . . . . 10 (((𝑠𝑡𝑠 ∈ (𝑁𝑥)) → 𝑡 ∈ (𝑁𝑥)) ↔ (𝑠𝑡 → (𝑠 ∈ (𝑁𝑥) → 𝑡 ∈ (𝑁𝑥))))
36 ancomst 463 . . . . . . . . . 10 (((𝑠𝑡𝑠 ∈ (𝑁𝑥)) → 𝑡 ∈ (𝑁𝑥)) ↔ ((𝑠 ∈ (𝑁𝑥) ∧ 𝑠𝑡) → 𝑡 ∈ (𝑁𝑥)))
3735, 36bitr3i 276 . . . . . . . . 9 ((𝑠𝑡 → (𝑠 ∈ (𝑁𝑥) → 𝑡 ∈ (𝑁𝑥))) ↔ ((𝑠 ∈ (𝑁𝑥) ∧ 𝑠𝑡) → 𝑡 ∈ (𝑁𝑥)))
3834, 37bitrdi 286 . . . . . . . 8 ((((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) ∧ 𝑥𝐵) → ((𝑠𝑡 → (𝑥 ∈ (𝐼𝑠) → 𝑥 ∈ (𝐼𝑡))) ↔ ((𝑠 ∈ (𝑁𝑥) ∧ 𝑠𝑡) → 𝑡 ∈ (𝑁𝑥))))
3938ralbidva 3166 . . . . . . 7 (((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) → (∀𝑥𝐵 (𝑠𝑡 → (𝑥 ∈ (𝐼𝑠) → 𝑥 ∈ (𝐼𝑡))) ↔ ∀𝑥𝐵 ((𝑠 ∈ (𝑁𝑥) ∧ 𝑠𝑡) → 𝑡 ∈ (𝑁𝑥))))
4026, 39bitrd 278 . . . . . 6 (((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) → (∀𝑥(𝑠𝑡 → (𝑥 ∈ (𝐼𝑠) → 𝑥 ∈ (𝐼𝑡))) ↔ ∀𝑥𝐵 ((𝑠 ∈ (𝑁𝑥) ∧ 𝑠𝑡) → 𝑡 ∈ (𝑁𝑥))))
414, 40bitrid 282 . . . . 5 (((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) → ((𝑠𝑡 → (𝐼𝑠) ⊆ (𝐼𝑡)) ↔ ∀𝑥𝐵 ((𝑠 ∈ (𝑁𝑥) ∧ 𝑠𝑡) → 𝑡 ∈ (𝑁𝑥))))
4241ralbidva 3166 . . . 4 ((𝜑𝑠 ∈ 𝒫 𝐵) → (∀𝑡 ∈ 𝒫 𝐵(𝑠𝑡 → (𝐼𝑠) ⊆ (𝐼𝑡)) ↔ ∀𝑡 ∈ 𝒫 𝐵𝑥𝐵 ((𝑠 ∈ (𝑁𝑥) ∧ 𝑠𝑡) → 𝑡 ∈ (𝑁𝑥))))
43 ralcom 3277 . . . 4 (∀𝑡 ∈ 𝒫 𝐵𝑥𝐵 ((𝑠 ∈ (𝑁𝑥) ∧ 𝑠𝑡) → 𝑡 ∈ (𝑁𝑥)) ↔ ∀𝑥𝐵𝑡 ∈ 𝒫 𝐵((𝑠 ∈ (𝑁𝑥) ∧ 𝑠𝑡) → 𝑡 ∈ (𝑁𝑥)))
4442, 43bitrdi 286 . . 3 ((𝜑𝑠 ∈ 𝒫 𝐵) → (∀𝑡 ∈ 𝒫 𝐵(𝑠𝑡 → (𝐼𝑠) ⊆ (𝐼𝑡)) ↔ ∀𝑥𝐵𝑡 ∈ 𝒫 𝐵((𝑠 ∈ (𝑁𝑥) ∧ 𝑠𝑡) → 𝑡 ∈ (𝑁𝑥))))
4544ralbidva 3166 . 2 (𝜑 → (∀𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵(𝑠𝑡 → (𝐼𝑠) ⊆ (𝐼𝑡)) ↔ ∀𝑠 ∈ 𝒫 𝐵𝑥𝐵𝑡 ∈ 𝒫 𝐵((𝑠 ∈ (𝑁𝑥) ∧ 𝑠𝑡) → 𝑡 ∈ (𝑁𝑥))))
46 ralcom 3277 . 2 (∀𝑠 ∈ 𝒫 𝐵𝑥𝐵𝑡 ∈ 𝒫 𝐵((𝑠 ∈ (𝑁𝑥) ∧ 𝑠𝑡) → 𝑡 ∈ (𝑁𝑥)) ↔ ∀𝑥𝐵𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵((𝑠 ∈ (𝑁𝑥) ∧ 𝑠𝑡) → 𝑡 ∈ (𝑁𝑥)))
4745, 46bitrdi 286 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 205  wa 394  wal 1531   = wceq 1533  wcel 2098  wral 3051  {crab 3419  Vcvv 3463  wss 3945  𝒫 cpw 4603   class class class wbr 5148  cmpt 5231  wf 6543  cfv 6547  (class class class)co 7417  cmpo 7419  m cmap 8843
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5364  ax-pr 5428  ax-un 7739
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-ral 3052  df-rex 3061  df-reu 3365  df-rab 3420  df-v 3465  df-sbc 3775  df-csb 3891  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6499  df-fun 6549  df-fn 6550  df-f 6551  df-f1 6552  df-fo 6553  df-f1o 6554  df-fv 6555  df-ov 7420  df-oprab 7421  df-mpo 7422  df-1st 7992  df-2nd 7993  df-map 8845
This theorem is referenced by: (None)
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