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Theorem ntrneix3 40325
Description: The closure of the union of any pair is a subset of the union of closures if and only if the union of any pair belonging to the convergents of a point implies at least one of the pair belongs to the the convergents of that point. (Contributed by RP, 19-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
ntrneix3 (𝜑 → (∀𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵(𝐼‘(𝑠𝑡)) ⊆ ((𝐼𝑠) ∪ (𝐼𝑡)) ↔ ∀𝑥𝐵𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵((𝑠𝑡) ∈ (𝑁𝑥) → (𝑠 ∈ (𝑁𝑥) ∨ 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 ntrneix3
StepHypRef Expression
1 dfss3 3953 . . . . . 6 ((𝐼‘(𝑠𝑡)) ⊆ ((𝐼𝑠) ∪ (𝐼𝑡)) ↔ ∀𝑥 ∈ (𝐼‘(𝑠𝑡))𝑥 ∈ ((𝐼𝑠) ∪ (𝐼𝑡)))
2 ntrnei.o . . . . . . . . . . . . 13 𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗m 𝑖) ↦ (𝑙𝑗 ↦ {𝑚𝑖𝑙 ∈ (𝑘𝑚)})))
3 ntrnei.f . . . . . . . . . . . . 13 𝐹 = (𝒫 𝐵𝑂𝐵)
4 ntrnei.r . . . . . . . . . . . . 13 (𝜑𝐼𝐹𝑁)
52, 3, 4ntrneiiex 40304 . . . . . . . . . . . 12 (𝜑𝐼 ∈ (𝒫 𝐵m 𝒫 𝐵))
65ad2antrr 722 . . . . . . . . . . 11 (((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) → 𝐼 ∈ (𝒫 𝐵m 𝒫 𝐵))
7 elmapi 8417 . . . . . . . . . . 11 (𝐼 ∈ (𝒫 𝐵m 𝒫 𝐵) → 𝐼:𝒫 𝐵⟶𝒫 𝐵)
86, 7syl 17 . . . . . . . . . 10 (((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) → 𝐼:𝒫 𝐵⟶𝒫 𝐵)
92, 3, 4ntrneibex 40301 . . . . . . . . . . . 12 (𝜑𝐵 ∈ V)
109ad2antrr 722 . . . . . . . . . . 11 (((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) → 𝐵 ∈ V)
11 simplr 765 . . . . . . . . . . . . 13 (((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) → 𝑠 ∈ 𝒫 𝐵)
1211elpwid 4549 . . . . . . . . . . . 12 (((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) → 𝑠𝐵)
13 simpr 485 . . . . . . . . . . . . 13 (((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) → 𝑡 ∈ 𝒫 𝐵)
1413elpwid 4549 . . . . . . . . . . . 12 (((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) → 𝑡𝐵)
1512, 14unssd 4159 . . . . . . . . . . 11 (((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) → (𝑠𝑡) ⊆ 𝐵)
1610, 15sselpwd 5221 . . . . . . . . . 10 (((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) → (𝑠𝑡) ∈ 𝒫 𝐵)
178, 16ffvelrnd 6844 . . . . . . . . 9 (((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) → (𝐼‘(𝑠𝑡)) ∈ 𝒫 𝐵)
1817elpwid 4549 . . . . . . . 8 (((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) → (𝐼‘(𝑠𝑡)) ⊆ 𝐵)
19 ralss 4034 . . . . . . . 8 ((𝐼‘(𝑠𝑡)) ⊆ 𝐵 → (∀𝑥 ∈ (𝐼‘(𝑠𝑡))𝑥 ∈ ((𝐼𝑠) ∪ (𝐼𝑡)) ↔ ∀𝑥𝐵 (𝑥 ∈ (𝐼‘(𝑠𝑡)) → 𝑥 ∈ ((𝐼𝑠) ∪ (𝐼𝑡)))))
2018, 19syl 17 . . . . . . 7 (((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) → (∀𝑥 ∈ (𝐼‘(𝑠𝑡))𝑥 ∈ ((𝐼𝑠) ∪ (𝐼𝑡)) ↔ ∀𝑥𝐵 (𝑥 ∈ (𝐼‘(𝑠𝑡)) → 𝑥 ∈ ((𝐼𝑠) ∪ (𝐼𝑡)))))
214ad3antrrr 726 . . . . . . . . . 10 ((((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) ∧ 𝑥𝐵) → 𝐼𝐹𝑁)
22 simpr 485 . . . . . . . . . 10 ((((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) ∧ 𝑥𝐵) → 𝑥𝐵)
239ad3antrrr 726 . . . . . . . . . . 11 ((((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) ∧ 𝑥𝐵) → 𝐵 ∈ V)
24 simpllr 772 . . . . . . . . . . . . 13 ((((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) ∧ 𝑥𝐵) → 𝑠 ∈ 𝒫 𝐵)
2524elpwid 4549 . . . . . . . . . . . 12 ((((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) ∧ 𝑥𝐵) → 𝑠𝐵)
26 simplr 765 . . . . . . . . . . . . 13 ((((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) ∧ 𝑥𝐵) → 𝑡 ∈ 𝒫 𝐵)
2726elpwid 4549 . . . . . . . . . . . 12 ((((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) ∧ 𝑥𝐵) → 𝑡𝐵)
2825, 27unssd 4159 . . . . . . . . . . 11 ((((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) ∧ 𝑥𝐵) → (𝑠𝑡) ⊆ 𝐵)
2923, 28sselpwd 5221 . . . . . . . . . 10 ((((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) ∧ 𝑥𝐵) → (𝑠𝑡) ∈ 𝒫 𝐵)
302, 3, 21, 22, 29ntrneiel 40309 . . . . . . . . 9 ((((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) ∧ 𝑥𝐵) → (𝑥 ∈ (𝐼‘(𝑠𝑡)) ↔ (𝑠𝑡) ∈ (𝑁𝑥)))
31 elun 4122 . . . . . . . . . 10 (𝑥 ∈ ((𝐼𝑠) ∪ (𝐼𝑡)) ↔ (𝑥 ∈ (𝐼𝑠) ∨ 𝑥 ∈ (𝐼𝑡)))
322, 3, 21, 22, 24ntrneiel 40309 . . . . . . . . . . 11 ((((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) ∧ 𝑥𝐵) → (𝑥 ∈ (𝐼𝑠) ↔ 𝑠 ∈ (𝑁𝑥)))
332, 3, 21, 22, 26ntrneiel 40309 . . . . . . . . . . 11 ((((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) ∧ 𝑥𝐵) → (𝑥 ∈ (𝐼𝑡) ↔ 𝑡 ∈ (𝑁𝑥)))
3432, 33orbi12d 912 . . . . . . . . . 10 ((((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) ∧ 𝑥𝐵) → ((𝑥 ∈ (𝐼𝑠) ∨ 𝑥 ∈ (𝐼𝑡)) ↔ (𝑠 ∈ (𝑁𝑥) ∨ 𝑡 ∈ (𝑁𝑥))))
3531, 34syl5bb 284 . . . . . . . . 9 ((((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) ∧ 𝑥𝐵) → (𝑥 ∈ ((𝐼𝑠) ∪ (𝐼𝑡)) ↔ (𝑠 ∈ (𝑁𝑥) ∨ 𝑡 ∈ (𝑁𝑥))))
3630, 35imbi12d 346 . . . . . . . 8 ((((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) ∧ 𝑥𝐵) → ((𝑥 ∈ (𝐼‘(𝑠𝑡)) → 𝑥 ∈ ((𝐼𝑠) ∪ (𝐼𝑡))) ↔ ((𝑠𝑡) ∈ (𝑁𝑥) → (𝑠 ∈ (𝑁𝑥) ∨ 𝑡 ∈ (𝑁𝑥)))))
3736ralbidva 3193 . . . . . . 7 (((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) → (∀𝑥𝐵 (𝑥 ∈ (𝐼‘(𝑠𝑡)) → 𝑥 ∈ ((𝐼𝑠) ∪ (𝐼𝑡))) ↔ ∀𝑥𝐵 ((𝑠𝑡) ∈ (𝑁𝑥) → (𝑠 ∈ (𝑁𝑥) ∨ 𝑡 ∈ (𝑁𝑥)))))
3820, 37bitrd 280 . . . . . 6 (((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) → (∀𝑥 ∈ (𝐼‘(𝑠𝑡))𝑥 ∈ ((𝐼𝑠) ∪ (𝐼𝑡)) ↔ ∀𝑥𝐵 ((𝑠𝑡) ∈ (𝑁𝑥) → (𝑠 ∈ (𝑁𝑥) ∨ 𝑡 ∈ (𝑁𝑥)))))
391, 38syl5bb 284 . . . . 5 (((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) → ((𝐼‘(𝑠𝑡)) ⊆ ((𝐼𝑠) ∪ (𝐼𝑡)) ↔ ∀𝑥𝐵 ((𝑠𝑡) ∈ (𝑁𝑥) → (𝑠 ∈ (𝑁𝑥) ∨ 𝑡 ∈ (𝑁𝑥)))))
4039ralbidva 3193 . . . 4 ((𝜑𝑠 ∈ 𝒫 𝐵) → (∀𝑡 ∈ 𝒫 𝐵(𝐼‘(𝑠𝑡)) ⊆ ((𝐼𝑠) ∪ (𝐼𝑡)) ↔ ∀𝑡 ∈ 𝒫 𝐵𝑥𝐵 ((𝑠𝑡) ∈ (𝑁𝑥) → (𝑠 ∈ (𝑁𝑥) ∨ 𝑡 ∈ (𝑁𝑥)))))
41 ralcom 3351 . . . 4 (∀𝑡 ∈ 𝒫 𝐵𝑥𝐵 ((𝑠𝑡) ∈ (𝑁𝑥) → (𝑠 ∈ (𝑁𝑥) ∨ 𝑡 ∈ (𝑁𝑥))) ↔ ∀𝑥𝐵𝑡 ∈ 𝒫 𝐵((𝑠𝑡) ∈ (𝑁𝑥) → (𝑠 ∈ (𝑁𝑥) ∨ 𝑡 ∈ (𝑁𝑥))))
4240, 41syl6bb 288 . . 3 ((𝜑𝑠 ∈ 𝒫 𝐵) → (∀𝑡 ∈ 𝒫 𝐵(𝐼‘(𝑠𝑡)) ⊆ ((𝐼𝑠) ∪ (𝐼𝑡)) ↔ ∀𝑥𝐵𝑡 ∈ 𝒫 𝐵((𝑠𝑡) ∈ (𝑁𝑥) → (𝑠 ∈ (𝑁𝑥) ∨ 𝑡 ∈ (𝑁𝑥)))))
4342ralbidva 3193 . 2 (𝜑 → (∀𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵(𝐼‘(𝑠𝑡)) ⊆ ((𝐼𝑠) ∪ (𝐼𝑡)) ↔ ∀𝑠 ∈ 𝒫 𝐵𝑥𝐵𝑡 ∈ 𝒫 𝐵((𝑠𝑡) ∈ (𝑁𝑥) → (𝑠 ∈ (𝑁𝑥) ∨ 𝑡 ∈ (𝑁𝑥)))))
44 ralcom 3351 . 2 (∀𝑠 ∈ 𝒫 𝐵𝑥𝐵𝑡 ∈ 𝒫 𝐵((𝑠𝑡) ∈ (𝑁𝑥) → (𝑠 ∈ (𝑁𝑥) ∨ 𝑡 ∈ (𝑁𝑥))) ↔ ∀𝑥𝐵𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵((𝑠𝑡) ∈ (𝑁𝑥) → (𝑠 ∈ (𝑁𝑥) ∨ 𝑡 ∈ (𝑁𝑥))))
4543, 44syl6bb 288 1 (𝜑 → (∀𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵(𝐼‘(𝑠𝑡)) ⊆ ((𝐼𝑠) ∪ (𝐼𝑡)) ↔ ∀𝑥𝐵𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵((𝑠𝑡) ∈ (𝑁𝑥) → (𝑠 ∈ (𝑁𝑥) ∨ 𝑡 ∈ (𝑁𝑥)))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396  wo 841   = wceq 1528  wcel 2105  wral 3135  {crab 3139  Vcvv 3492  cun 3931  wss 3933  𝒫 cpw 4535   class class class wbr 5057  cmpt 5137  wf 6344  cfv 6348  (class class class)co 7145  cmpo 7147  m cmap 8395
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-ov 7148  df-oprab 7149  df-mpo 7150  df-1st 7678  df-2nd 7679  df-map 8397
This theorem is referenced by: (None)
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