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Theorem ntrneiiso 43493
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 3964 . . . . . . . 8 ((𝐼𝑠) ⊆ (𝐼𝑡) ↔ ∀𝑥(𝑥 ∈ (𝐼𝑠) → 𝑥 ∈ (𝐼𝑡)))
21imbi2i 336 . . . . . . 7 ((𝑠𝑡 → (𝐼𝑠) ⊆ (𝐼𝑡)) ↔ (𝑠𝑡 → ∀𝑥(𝑥 ∈ (𝐼𝑠) → 𝑥 ∈ (𝐼𝑡))))
3 19.21v 1935 . . . . . . 7 (∀𝑥(𝑠𝑡 → (𝑥 ∈ (𝐼𝑠) → 𝑥 ∈ (𝐼𝑡))) ↔ (𝑠𝑡 → ∀𝑥(𝑥 ∈ (𝐼𝑠) → 𝑥 ∈ (𝐼𝑡))))
42, 3bitr4i 278 . . . . . 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 43478 . . . . . . . . . . . . . . . . . . 19 (𝜑𝐼 ∈ (𝒫 𝐵m 𝒫 𝐵))
11 elmapi 8861 . . . . . . . . . . . . . . . . . . 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, 13ffvelcdmd 7089 . . . . . . . . . . . . . . . . 17 (((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) → (𝐼𝑠) ∈ 𝒫 𝐵)
1514elpwid 4607 . . . . . . . . . . . . . . . 16 (((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) → (𝐼𝑠) ⊆ 𝐵)
1615sselda 3978 . . . . . . . . . . . . . . 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 412 . . . . . . . . . . . . 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 1916 . . . . . . . 8 (((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) → (∀𝑥(𝑠𝑡 → (𝑥 ∈ (𝐼𝑠) → 𝑥 ∈ (𝐼𝑡))) ↔ ∀𝑥(𝑥𝐵 → (𝑠𝑡 → (𝑥 ∈ (𝐼𝑠) → 𝑥 ∈ (𝐼𝑡))))))
25 df-ral 3057 . . . . . . . 8 (∀𝑥𝐵 (𝑠𝑡 → (𝑥 ∈ (𝐼𝑠) → 𝑥 ∈ (𝐼𝑡))) ↔ ∀𝑥(𝑥𝐵 → (𝑠𝑡 → (𝑥 ∈ (𝐼𝑠) → 𝑥 ∈ (𝐼𝑡)))))
2624, 25bitr4di 289 . . . . . . 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 484 . . . . . . . . . . . 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 43483 . . . . . . . . . . 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 43483 . . . . . . . . . . 11 ((((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) ∧ 𝑥𝐵) → (𝑥 ∈ (𝐼𝑡) ↔ 𝑡 ∈ (𝑁𝑥)))
3330, 32imbi12d 344 . . . . . . . . . 10 ((((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) ∧ 𝑥𝐵) → ((𝑥 ∈ (𝐼𝑠) → 𝑥 ∈ (𝐼𝑡)) ↔ (𝑠 ∈ (𝑁𝑥) → 𝑡 ∈ (𝑁𝑥))))
3433imbi2d 340 . . . . . . . . 9 ((((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) ∧ 𝑥𝐵) → ((𝑠𝑡 → (𝑥 ∈ (𝐼𝑠) → 𝑥 ∈ (𝐼𝑡))) ↔ (𝑠𝑡 → (𝑠 ∈ (𝑁𝑥) → 𝑡 ∈ (𝑁𝑥)))))
35 impexp 450 . . . . . . . . . 10 (((𝑠𝑡𝑠 ∈ (𝑁𝑥)) → 𝑡 ∈ (𝑁𝑥)) ↔ (𝑠𝑡 → (𝑠 ∈ (𝑁𝑥) → 𝑡 ∈ (𝑁𝑥))))
36 ancomst 464 . . . . . . . . . 10 (((𝑠𝑡𝑠 ∈ (𝑁𝑥)) → 𝑡 ∈ (𝑁𝑥)) ↔ ((𝑠 ∈ (𝑁𝑥) ∧ 𝑠𝑡) → 𝑡 ∈ (𝑁𝑥)))
3735, 36bitr3i 277 . . . . . . . . 9 ((𝑠𝑡 → (𝑠 ∈ (𝑁𝑥) → 𝑡 ∈ (𝑁𝑥))) ↔ ((𝑠 ∈ (𝑁𝑥) ∧ 𝑠𝑡) → 𝑡 ∈ (𝑁𝑥)))
3834, 37bitrdi 287 . . . . . . . 8 ((((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) ∧ 𝑥𝐵) → ((𝑠𝑡 → (𝑥 ∈ (𝐼𝑠) → 𝑥 ∈ (𝐼𝑡))) ↔ ((𝑠 ∈ (𝑁𝑥) ∧ 𝑠𝑡) → 𝑡 ∈ (𝑁𝑥))))
3938ralbidva 3170 . . . . . . 7 (((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) → (∀𝑥𝐵 (𝑠𝑡 → (𝑥 ∈ (𝐼𝑠) → 𝑥 ∈ (𝐼𝑡))) ↔ ∀𝑥𝐵 ((𝑠 ∈ (𝑁𝑥) ∧ 𝑠𝑡) → 𝑡 ∈ (𝑁𝑥))))
4026, 39bitrd 279 . . . . . 6 (((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) → (∀𝑥(𝑠𝑡 → (𝑥 ∈ (𝐼𝑠) → 𝑥 ∈ (𝐼𝑡))) ↔ ∀𝑥𝐵 ((𝑠 ∈ (𝑁𝑥) ∧ 𝑠𝑡) → 𝑡 ∈ (𝑁𝑥))))
414, 40bitrid 283 . . . . 5 (((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) → ((𝑠𝑡 → (𝐼𝑠) ⊆ (𝐼𝑡)) ↔ ∀𝑥𝐵 ((𝑠 ∈ (𝑁𝑥) ∧ 𝑠𝑡) → 𝑡 ∈ (𝑁𝑥))))
4241ralbidva 3170 . . . 4 ((𝜑𝑠 ∈ 𝒫 𝐵) → (∀𝑡 ∈ 𝒫 𝐵(𝑠𝑡 → (𝐼𝑠) ⊆ (𝐼𝑡)) ↔ ∀𝑡 ∈ 𝒫 𝐵𝑥𝐵 ((𝑠 ∈ (𝑁𝑥) ∧ 𝑠𝑡) → 𝑡 ∈ (𝑁𝑥))))
43 ralcom 3281 . . . 4 (∀𝑡 ∈ 𝒫 𝐵𝑥𝐵 ((𝑠 ∈ (𝑁𝑥) ∧ 𝑠𝑡) → 𝑡 ∈ (𝑁𝑥)) ↔ ∀𝑥𝐵𝑡 ∈ 𝒫 𝐵((𝑠 ∈ (𝑁𝑥) ∧ 𝑠𝑡) → 𝑡 ∈ (𝑁𝑥)))
4442, 43bitrdi 287 . . 3 ((𝜑𝑠 ∈ 𝒫 𝐵) → (∀𝑡 ∈ 𝒫 𝐵(𝑠𝑡 → (𝐼𝑠) ⊆ (𝐼𝑡)) ↔ ∀𝑥𝐵𝑡 ∈ 𝒫 𝐵((𝑠 ∈ (𝑁𝑥) ∧ 𝑠𝑡) → 𝑡 ∈ (𝑁𝑥))))
4544ralbidva 3170 . 2 (𝜑 → (∀𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵(𝑠𝑡 → (𝐼𝑠) ⊆ (𝐼𝑡)) ↔ ∀𝑠 ∈ 𝒫 𝐵𝑥𝐵𝑡 ∈ 𝒫 𝐵((𝑠 ∈ (𝑁𝑥) ∧ 𝑠𝑡) → 𝑡 ∈ (𝑁𝑥))))
46 ralcom 3281 . 2 (∀𝑠 ∈ 𝒫 𝐵𝑥𝐵𝑡 ∈ 𝒫 𝐵((𝑠 ∈ (𝑁𝑥) ∧ 𝑠𝑡) → 𝑡 ∈ (𝑁𝑥)) ↔ ∀𝑥𝐵𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵((𝑠 ∈ (𝑁𝑥) ∧ 𝑠𝑡) → 𝑡 ∈ (𝑁𝑥)))
4745, 46bitrdi 287 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 395  wal 1532   = wceq 1534  wcel 2099  wral 3056  {crab 3427  Vcvv 3469  wss 3944  𝒫 cpw 4598   class class class wbr 5142  cmpt 5225  wf 6538  cfv 6542  (class class class)co 7414  cmpo 7416  m cmap 8838
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-rep 5279  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7734
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-ral 3057  df-rex 3066  df-reu 3372  df-rab 3428  df-v 3471  df-sbc 3775  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-iun 4993  df-br 5143  df-opab 5205  df-mpt 5226  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-ov 7417  df-oprab 7418  df-mpo 7419  df-1st 7987  df-2nd 7988  df-map 8840
This theorem is referenced by: (None)
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