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Theorem ntrneiiso 40741
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 3939 . . . . . . . 8 ((𝐼𝑠) ⊆ (𝐼𝑡) ↔ ∀𝑥(𝑥 ∈ (𝐼𝑠) → 𝑥 ∈ (𝐼𝑡)))
21imbi2i 339 . . . . . . 7 ((𝑠𝑡 → (𝐼𝑠) ⊆ (𝐼𝑡)) ↔ (𝑠𝑡 → ∀𝑥(𝑥 ∈ (𝐼𝑠) → 𝑥 ∈ (𝐼𝑡))))
3 19.21v 1941 . . . . . . 7 (∀𝑥(𝑠𝑡 → (𝑥 ∈ (𝐼𝑠) → 𝑥 ∈ (𝐼𝑡))) ↔ (𝑠𝑡 → ∀𝑥(𝑥 ∈ (𝐼𝑠) → 𝑥 ∈ (𝐼𝑡))))
42, 3bitr4i 281 . . . . . 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 766 . . . . . . . . . . . . . . . . . . 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 40726 . . . . . . . . . . . . . . . . . . 19 (𝜑𝐼 ∈ (𝒫 𝐵m 𝒫 𝐵))
11 elmapi 8426 . . . . . . . . . . . . . . . . . . 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 768 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) → 𝑠 ∈ 𝒫 𝐵)
1412, 13ffvelrnd 6845 . . . . . . . . . . . . . . . . 17 (((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) → (𝐼𝑠) ∈ 𝒫 𝐵)
1514elpwid 4533 . . . . . . . . . . . . . . . 16 (((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) → (𝐼𝑠) ⊆ 𝐵)
1615sselda 3953 . . . . . . . . . . . . . . 15 ((((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) ∧ 𝑥 ∈ (𝐼𝑠)) → 𝑥𝐵)
1716pm2.24d 154 . . . . . . . . . . . . . 14 ((((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) ∧ 𝑥 ∈ (𝐼𝑠)) → (¬ 𝑥𝐵𝑥 ∈ (𝐼𝑡)))
1817ex 416 . . . . . . . . . . . . 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 190 . . . . . . . . . 10 (((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) → ((𝑥𝐵 → (𝑠𝑡 → (𝑥 ∈ (𝐼𝑠) → 𝑥 ∈ (𝐼𝑡)))) → (𝑠𝑡 → (𝑥 ∈ (𝐼𝑠) → 𝑥 ∈ (𝐼𝑡)))))
235, 22impbid2 229 . . . . . . . . 9 (((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) → ((𝑠𝑡 → (𝑥 ∈ (𝐼𝑠) → 𝑥 ∈ (𝐼𝑡))) ↔ (𝑥𝐵 → (𝑠𝑡 → (𝑥 ∈ (𝐼𝑠) → 𝑥 ∈ (𝐼𝑡))))))
2423albidv 1922 . . . . . . . 8 (((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) → (∀𝑥(𝑠𝑡 → (𝑥 ∈ (𝐼𝑠) → 𝑥 ∈ (𝐼𝑡))) ↔ ∀𝑥(𝑥𝐵 → (𝑠𝑡 → (𝑥 ∈ (𝐼𝑠) → 𝑥 ∈ (𝐼𝑡))))))
25 df-ral 3138 . . . . . . . 8 (∀𝑥𝐵 (𝑠𝑡 → (𝑥 ∈ (𝐼𝑠) → 𝑥 ∈ (𝐼𝑡))) ↔ ∀𝑥(𝑥𝐵 → (𝑠𝑡 → (𝑥 ∈ (𝐼𝑠) → 𝑥 ∈ (𝐼𝑡)))))
2624, 25syl6bbr 292 . . . . . . 7 (((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) → (∀𝑥(𝑠𝑡 → (𝑥 ∈ (𝐼𝑠) → 𝑥 ∈ (𝐼𝑡))) ↔ ∀𝑥𝐵 (𝑠𝑡 → (𝑥 ∈ (𝐼𝑠) → 𝑥 ∈ (𝐼𝑡)))))
279ad3antrrr 729 . . . . . . . . . . . 12 ((((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) ∧ 𝑥𝐵) → 𝐼𝐹𝑁)
28 simpr 488 . . . . . . . . . . . 12 ((((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) ∧ 𝑥𝐵) → 𝑥𝐵)
29 simpllr 775 . . . . . . . . . . . 12 ((((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) ∧ 𝑥𝐵) → 𝑠 ∈ 𝒫 𝐵)
307, 8, 27, 28, 29ntrneiel 40731 . . . . . . . . . . 11 ((((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) ∧ 𝑥𝐵) → (𝑥 ∈ (𝐼𝑠) ↔ 𝑠 ∈ (𝑁𝑥)))
31 simplr 768 . . . . . . . . . . . 12 ((((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) ∧ 𝑥𝐵) → 𝑡 ∈ 𝒫 𝐵)
327, 8, 27, 28, 31ntrneiel 40731 . . . . . . . . . . 11 ((((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) ∧ 𝑥𝐵) → (𝑥 ∈ (𝐼𝑡) ↔ 𝑡 ∈ (𝑁𝑥)))
3330, 32imbi12d 348 . . . . . . . . . 10 ((((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) ∧ 𝑥𝐵) → ((𝑥 ∈ (𝐼𝑠) → 𝑥 ∈ (𝐼𝑡)) ↔ (𝑠 ∈ (𝑁𝑥) → 𝑡 ∈ (𝑁𝑥))))
3433imbi2d 344 . . . . . . . . 9 ((((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) ∧ 𝑥𝐵) → ((𝑠𝑡 → (𝑥 ∈ (𝐼𝑠) → 𝑥 ∈ (𝐼𝑡))) ↔ (𝑠𝑡 → (𝑠 ∈ (𝑁𝑥) → 𝑡 ∈ (𝑁𝑥)))))
35 impexp 454 . . . . . . . . . 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 280 . . . . . . . . 9 ((𝑠𝑡 → (𝑠 ∈ (𝑁𝑥) → 𝑡 ∈ (𝑁𝑥))) ↔ ((𝑠 ∈ (𝑁𝑥) ∧ 𝑠𝑡) → 𝑡 ∈ (𝑁𝑥)))
3834, 37syl6bb 290 . . . . . . . 8 ((((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) ∧ 𝑥𝐵) → ((𝑠𝑡 → (𝑥 ∈ (𝐼𝑠) → 𝑥 ∈ (𝐼𝑡))) ↔ ((𝑠 ∈ (𝑁𝑥) ∧ 𝑠𝑡) → 𝑡 ∈ (𝑁𝑥))))
3938ralbidva 3191 . . . . . . 7 (((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) → (∀𝑥𝐵 (𝑠𝑡 → (𝑥 ∈ (𝐼𝑠) → 𝑥 ∈ (𝐼𝑡))) ↔ ∀𝑥𝐵 ((𝑠 ∈ (𝑁𝑥) ∧ 𝑠𝑡) → 𝑡 ∈ (𝑁𝑥))))
4026, 39bitrd 282 . . . . . 6 (((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) → (∀𝑥(𝑠𝑡 → (𝑥 ∈ (𝐼𝑠) → 𝑥 ∈ (𝐼𝑡))) ↔ ∀𝑥𝐵 ((𝑠 ∈ (𝑁𝑥) ∧ 𝑠𝑡) → 𝑡 ∈ (𝑁𝑥))))
414, 40syl5bb 286 . . . . 5 (((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) → ((𝑠𝑡 → (𝐼𝑠) ⊆ (𝐼𝑡)) ↔ ∀𝑥𝐵 ((𝑠 ∈ (𝑁𝑥) ∧ 𝑠𝑡) → 𝑡 ∈ (𝑁𝑥))))
4241ralbidva 3191 . . . 4 ((𝜑𝑠 ∈ 𝒫 𝐵) → (∀𝑡 ∈ 𝒫 𝐵(𝑠𝑡 → (𝐼𝑠) ⊆ (𝐼𝑡)) ↔ ∀𝑡 ∈ 𝒫 𝐵𝑥𝐵 ((𝑠 ∈ (𝑁𝑥) ∧ 𝑠𝑡) → 𝑡 ∈ (𝑁𝑥))))
43 ralcom 3345 . . . 4 (∀𝑡 ∈ 𝒫 𝐵𝑥𝐵 ((𝑠 ∈ (𝑁𝑥) ∧ 𝑠𝑡) → 𝑡 ∈ (𝑁𝑥)) ↔ ∀𝑥𝐵𝑡 ∈ 𝒫 𝐵((𝑠 ∈ (𝑁𝑥) ∧ 𝑠𝑡) → 𝑡 ∈ (𝑁𝑥)))
4442, 43syl6bb 290 . . 3 ((𝜑𝑠 ∈ 𝒫 𝐵) → (∀𝑡 ∈ 𝒫 𝐵(𝑠𝑡 → (𝐼𝑠) ⊆ (𝐼𝑡)) ↔ ∀𝑥𝐵𝑡 ∈ 𝒫 𝐵((𝑠 ∈ (𝑁𝑥) ∧ 𝑠𝑡) → 𝑡 ∈ (𝑁𝑥))))
4544ralbidva 3191 . 2 (𝜑 → (∀𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵(𝑠𝑡 → (𝐼𝑠) ⊆ (𝐼𝑡)) ↔ ∀𝑠 ∈ 𝒫 𝐵𝑥𝐵𝑡 ∈ 𝒫 𝐵((𝑠 ∈ (𝑁𝑥) ∧ 𝑠𝑡) → 𝑡 ∈ (𝑁𝑥))))
46 ralcom 3345 . 2 (∀𝑠 ∈ 𝒫 𝐵𝑥𝐵𝑡 ∈ 𝒫 𝐵((𝑠 ∈ (𝑁𝑥) ∧ 𝑠𝑡) → 𝑡 ∈ (𝑁𝑥)) ↔ ∀𝑥𝐵𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵((𝑠 ∈ (𝑁𝑥) ∧ 𝑠𝑡) → 𝑡 ∈ (𝑁𝑥)))
4745, 46syl6bb 290 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 209  wa 399  wal 1536   = wceq 1538  wcel 2115  wral 3133  {crab 3137  Vcvv 3480  wss 3919  𝒫 cpw 4522   class class class wbr 5053  cmpt 5133  wf 6341  cfv 6345  (class class class)co 7151  cmpo 7153  m cmap 8404
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5177  ax-sep 5190  ax-nul 5197  ax-pow 5254  ax-pr 5318  ax-un 7457
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 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-reu 3140  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-iun 4907  df-br 5054  df-opab 5116  df-mpt 5134  df-id 5448  df-xp 5549  df-rel 5550  df-cnv 5551  df-co 5552  df-dm 5553  df-rn 5554  df-res 5555  df-ima 5556  df-iota 6304  df-fun 6347  df-fn 6348  df-f 6349  df-f1 6350  df-fo 6351  df-f1o 6352  df-fv 6353  df-ov 7154  df-oprab 7155  df-mpo 7156  df-1st 7686  df-2nd 7687  df-map 8406
This theorem is referenced by: (None)
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