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Theorem ntrneix3 44087
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 3984 . . . . . 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 44066 . . . . . . . . . . . 12 (𝜑𝐼 ∈ (𝒫 𝐵m 𝒫 𝐵))
65ad2antrr 726 . . . . . . . . . . 11 (((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) → 𝐼 ∈ (𝒫 𝐵m 𝒫 𝐵))
7 elmapi 8888 . . . . . . . . . . 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 44063 . . . . . . . . . . . 12 (𝜑𝐵 ∈ V)
109ad2antrr 726 . . . . . . . . . . 11 (((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) → 𝐵 ∈ V)
11 simplr 769 . . . . . . . . . . . . 13 (((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) → 𝑠 ∈ 𝒫 𝐵)
1211elpwid 4614 . . . . . . . . . . . 12 (((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) → 𝑠𝐵)
13 simpr 484 . . . . . . . . . . . . 13 (((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) → 𝑡 ∈ 𝒫 𝐵)
1413elpwid 4614 . . . . . . . . . . . 12 (((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) → 𝑡𝐵)
1512, 14unssd 4202 . . . . . . . . . . 11 (((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) → (𝑠𝑡) ⊆ 𝐵)
1610, 15sselpwd 5334 . . . . . . . . . 10 (((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) → (𝑠𝑡) ∈ 𝒫 𝐵)
178, 16ffvelcdmd 7105 . . . . . . . . 9 (((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) → (𝐼‘(𝑠𝑡)) ∈ 𝒫 𝐵)
1817elpwid 4614 . . . . . . . 8 (((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) → (𝐼‘(𝑠𝑡)) ⊆ 𝐵)
19 ralss 4070 . . . . . . . 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 730 . . . . . . . . . 10 ((((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) ∧ 𝑥𝐵) → 𝐼𝐹𝑁)
22 simpr 484 . . . . . . . . . 10 ((((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) ∧ 𝑥𝐵) → 𝑥𝐵)
239ad3antrrr 730 . . . . . . . . . . 11 ((((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) ∧ 𝑥𝐵) → 𝐵 ∈ V)
24 simpllr 776 . . . . . . . . . . . . 13 ((((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) ∧ 𝑥𝐵) → 𝑠 ∈ 𝒫 𝐵)
2524elpwid 4614 . . . . . . . . . . . 12 ((((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) ∧ 𝑥𝐵) → 𝑠𝐵)
26 simplr 769 . . . . . . . . . . . . 13 ((((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) ∧ 𝑥𝐵) → 𝑡 ∈ 𝒫 𝐵)
2726elpwid 4614 . . . . . . . . . . . 12 ((((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) ∧ 𝑥𝐵) → 𝑡𝐵)
2825, 27unssd 4202 . . . . . . . . . . 11 ((((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) ∧ 𝑥𝐵) → (𝑠𝑡) ⊆ 𝐵)
2923, 28sselpwd 5334 . . . . . . . . . 10 ((((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) ∧ 𝑥𝐵) → (𝑠𝑡) ∈ 𝒫 𝐵)
302, 3, 21, 22, 29ntrneiel 44071 . . . . . . . . 9 ((((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) ∧ 𝑥𝐵) → (𝑥 ∈ (𝐼‘(𝑠𝑡)) ↔ (𝑠𝑡) ∈ (𝑁𝑥)))
31 elun 4163 . . . . . . . . . 10 (𝑥 ∈ ((𝐼𝑠) ∪ (𝐼𝑡)) ↔ (𝑥 ∈ (𝐼𝑠) ∨ 𝑥 ∈ (𝐼𝑡)))
322, 3, 21, 22, 24ntrneiel 44071 . . . . . . . . . . 11 ((((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) ∧ 𝑥𝐵) → (𝑥 ∈ (𝐼𝑠) ↔ 𝑠 ∈ (𝑁𝑥)))
332, 3, 21, 22, 26ntrneiel 44071 . . . . . . . . . . 11 ((((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) ∧ 𝑥𝐵) → (𝑥 ∈ (𝐼𝑡) ↔ 𝑡 ∈ (𝑁𝑥)))
3432, 33orbi12d 918 . . . . . . . . . 10 ((((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) ∧ 𝑥𝐵) → ((𝑥 ∈ (𝐼𝑠) ∨ 𝑥 ∈ (𝐼𝑡)) ↔ (𝑠 ∈ (𝑁𝑥) ∨ 𝑡 ∈ (𝑁𝑥))))
3531, 34bitrid 283 . . . . . . . . 9 ((((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) ∧ 𝑥𝐵) → (𝑥 ∈ ((𝐼𝑠) ∪ (𝐼𝑡)) ↔ (𝑠 ∈ (𝑁𝑥) ∨ 𝑡 ∈ (𝑁𝑥))))
3630, 35imbi12d 344 . . . . . . . 8 ((((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) ∧ 𝑥𝐵) → ((𝑥 ∈ (𝐼‘(𝑠𝑡)) → 𝑥 ∈ ((𝐼𝑠) ∪ (𝐼𝑡))) ↔ ((𝑠𝑡) ∈ (𝑁𝑥) → (𝑠 ∈ (𝑁𝑥) ∨ 𝑡 ∈ (𝑁𝑥)))))
3736ralbidva 3174 . . . . . . 7 (((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) → (∀𝑥𝐵 (𝑥 ∈ (𝐼‘(𝑠𝑡)) → 𝑥 ∈ ((𝐼𝑠) ∪ (𝐼𝑡))) ↔ ∀𝑥𝐵 ((𝑠𝑡) ∈ (𝑁𝑥) → (𝑠 ∈ (𝑁𝑥) ∨ 𝑡 ∈ (𝑁𝑥)))))
3820, 37bitrd 279 . . . . . 6 (((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) → (∀𝑥 ∈ (𝐼‘(𝑠𝑡))𝑥 ∈ ((𝐼𝑠) ∪ (𝐼𝑡)) ↔ ∀𝑥𝐵 ((𝑠𝑡) ∈ (𝑁𝑥) → (𝑠 ∈ (𝑁𝑥) ∨ 𝑡 ∈ (𝑁𝑥)))))
391, 38bitrid 283 . . . . 5 (((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) → ((𝐼‘(𝑠𝑡)) ⊆ ((𝐼𝑠) ∪ (𝐼𝑡)) ↔ ∀𝑥𝐵 ((𝑠𝑡) ∈ (𝑁𝑥) → (𝑠 ∈ (𝑁𝑥) ∨ 𝑡 ∈ (𝑁𝑥)))))
4039ralbidva 3174 . . . 4 ((𝜑𝑠 ∈ 𝒫 𝐵) → (∀𝑡 ∈ 𝒫 𝐵(𝐼‘(𝑠𝑡)) ⊆ ((𝐼𝑠) ∪ (𝐼𝑡)) ↔ ∀𝑡 ∈ 𝒫 𝐵𝑥𝐵 ((𝑠𝑡) ∈ (𝑁𝑥) → (𝑠 ∈ (𝑁𝑥) ∨ 𝑡 ∈ (𝑁𝑥)))))
41 ralcom 3287 . . . 4 (∀𝑡 ∈ 𝒫 𝐵𝑥𝐵 ((𝑠𝑡) ∈ (𝑁𝑥) → (𝑠 ∈ (𝑁𝑥) ∨ 𝑡 ∈ (𝑁𝑥))) ↔ ∀𝑥𝐵𝑡 ∈ 𝒫 𝐵((𝑠𝑡) ∈ (𝑁𝑥) → (𝑠 ∈ (𝑁𝑥) ∨ 𝑡 ∈ (𝑁𝑥))))
4240, 41bitrdi 287 . . 3 ((𝜑𝑠 ∈ 𝒫 𝐵) → (∀𝑡 ∈ 𝒫 𝐵(𝐼‘(𝑠𝑡)) ⊆ ((𝐼𝑠) ∪ (𝐼𝑡)) ↔ ∀𝑥𝐵𝑡 ∈ 𝒫 𝐵((𝑠𝑡) ∈ (𝑁𝑥) → (𝑠 ∈ (𝑁𝑥) ∨ 𝑡 ∈ (𝑁𝑥)))))
4342ralbidva 3174 . 2 (𝜑 → (∀𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵(𝐼‘(𝑠𝑡)) ⊆ ((𝐼𝑠) ∪ (𝐼𝑡)) ↔ ∀𝑠 ∈ 𝒫 𝐵𝑥𝐵𝑡 ∈ 𝒫 𝐵((𝑠𝑡) ∈ (𝑁𝑥) → (𝑠 ∈ (𝑁𝑥) ∨ 𝑡 ∈ (𝑁𝑥)))))
44 ralcom 3287 . 2 (∀𝑠 ∈ 𝒫 𝐵𝑥𝐵𝑡 ∈ 𝒫 𝐵((𝑠𝑡) ∈ (𝑁𝑥) → (𝑠 ∈ (𝑁𝑥) ∨ 𝑡 ∈ (𝑁𝑥))) ↔ ∀𝑥𝐵𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵((𝑠𝑡) ∈ (𝑁𝑥) → (𝑠 ∈ (𝑁𝑥) ∨ 𝑡 ∈ (𝑁𝑥))))
4543, 44bitrdi 287 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 206  wa 395  wo 847   = wceq 1537  wcel 2106  wral 3059  {crab 3433  Vcvv 3478  cun 3961  wss 3963  𝒫 cpw 4605   class class class wbr 5148  cmpt 5231  wf 6559  cfv 6563  (class class class)co 7431  cmpo 7433  m cmap 8865
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8013  df-2nd 8014  df-map 8867
This theorem is referenced by: (None)
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