Mathbox for Richard Penner < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  ntrneik3 Structured version   Visualization version   GIF version

Theorem ntrneik3 41172
 Description: The intersection of interiors of any pair is a subset of the interior of the intersection if and only if the intersection of any two neighborhoods of a point is also a neighborhood. (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
ntrneik3 (𝜑 → (∀𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵((𝐼𝑠) ∩ (𝐼𝑡)) ⊆ (𝐼‘(𝑠𝑡)) ↔ ∀𝑥𝐵𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵((𝑠 ∈ (𝑁𝑥) ∧ 𝑡 ∈ (𝑁𝑥)) → (𝑠𝑡) ∈ (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 ntrneik3
StepHypRef Expression
1 dfss3 3880 . . . . . 6 (((𝐼𝑠) ∩ (𝐼𝑡)) ⊆ (𝐼‘(𝑠𝑡)) ↔ ∀𝑥 ∈ ((𝐼𝑠) ∩ (𝐼𝑡))𝑥 ∈ (𝐼‘(𝑠𝑡)))
2 ntrnei.o . . . . . . . . . . . . . 14 𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗m 𝑖) ↦ (𝑙𝑗 ↦ {𝑚𝑖𝑙 ∈ (𝑘𝑚)})))
3 ntrnei.f . . . . . . . . . . . . . 14 𝐹 = (𝒫 𝐵𝑂𝐵)
4 ntrnei.r . . . . . . . . . . . . . 14 (𝜑𝐼𝐹𝑁)
52, 3, 4ntrneiiex 41152 . . . . . . . . . . . . 13 (𝜑𝐼 ∈ (𝒫 𝐵m 𝒫 𝐵))
6 elmapi 8438 . . . . . . . . . . . . 13 (𝐼 ∈ (𝒫 𝐵m 𝒫 𝐵) → 𝐼:𝒫 𝐵⟶𝒫 𝐵)
75, 6syl 17 . . . . . . . . . . . 12 (𝜑𝐼:𝒫 𝐵⟶𝒫 𝐵)
87ffvelrnda 6842 . . . . . . . . . . 11 ((𝜑𝑠 ∈ 𝒫 𝐵) → (𝐼𝑠) ∈ 𝒫 𝐵)
98elpwid 4505 . . . . . . . . . 10 ((𝜑𝑠 ∈ 𝒫 𝐵) → (𝐼𝑠) ⊆ 𝐵)
10 ssinss1 4142 . . . . . . . . . 10 ((𝐼𝑠) ⊆ 𝐵 → ((𝐼𝑠) ∩ (𝐼𝑡)) ⊆ 𝐵)
119, 10syl 17 . . . . . . . . 9 ((𝜑𝑠 ∈ 𝒫 𝐵) → ((𝐼𝑠) ∩ (𝐼𝑡)) ⊆ 𝐵)
1211adantr 484 . . . . . . . 8 (((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) → ((𝐼𝑠) ∩ (𝐼𝑡)) ⊆ 𝐵)
13 ralss 3962 . . . . . . . 8 (((𝐼𝑠) ∩ (𝐼𝑡)) ⊆ 𝐵 → (∀𝑥 ∈ ((𝐼𝑠) ∩ (𝐼𝑡))𝑥 ∈ (𝐼‘(𝑠𝑡)) ↔ ∀𝑥𝐵 (𝑥 ∈ ((𝐼𝑠) ∩ (𝐼𝑡)) → 𝑥 ∈ (𝐼‘(𝑠𝑡)))))
1412, 13syl 17 . . . . . . 7 (((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) → (∀𝑥 ∈ ((𝐼𝑠) ∩ (𝐼𝑡))𝑥 ∈ (𝐼‘(𝑠𝑡)) ↔ ∀𝑥𝐵 (𝑥 ∈ ((𝐼𝑠) ∩ (𝐼𝑡)) → 𝑥 ∈ (𝐼‘(𝑠𝑡)))))
15 elin 3874 . . . . . . . . . 10 (𝑥 ∈ ((𝐼𝑠) ∩ (𝐼𝑡)) ↔ (𝑥 ∈ (𝐼𝑠) ∧ 𝑥 ∈ (𝐼𝑡)))
164ad3antrrr 729 . . . . . . . . . . . 12 ((((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) ∧ 𝑥𝐵) → 𝐼𝐹𝑁)
17 simpr 488 . . . . . . . . . . . 12 ((((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) ∧ 𝑥𝐵) → 𝑥𝐵)
18 simpllr 775 . . . . . . . . . . . 12 ((((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) ∧ 𝑥𝐵) → 𝑠 ∈ 𝒫 𝐵)
192, 3, 16, 17, 18ntrneiel 41157 . . . . . . . . . . 11 ((((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) ∧ 𝑥𝐵) → (𝑥 ∈ (𝐼𝑠) ↔ 𝑠 ∈ (𝑁𝑥)))
20 simplr 768 . . . . . . . . . . . 12 ((((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) ∧ 𝑥𝐵) → 𝑡 ∈ 𝒫 𝐵)
212, 3, 16, 17, 20ntrneiel 41157 . . . . . . . . . . 11 ((((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) ∧ 𝑥𝐵) → (𝑥 ∈ (𝐼𝑡) ↔ 𝑡 ∈ (𝑁𝑥)))
2219, 21anbi12d 633 . . . . . . . . . 10 ((((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) ∧ 𝑥𝐵) → ((𝑥 ∈ (𝐼𝑠) ∧ 𝑥 ∈ (𝐼𝑡)) ↔ (𝑠 ∈ (𝑁𝑥) ∧ 𝑡 ∈ (𝑁𝑥))))
2315, 22syl5bb 286 . . . . . . . . 9 ((((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) ∧ 𝑥𝐵) → (𝑥 ∈ ((𝐼𝑠) ∩ (𝐼𝑡)) ↔ (𝑠 ∈ (𝑁𝑥) ∧ 𝑡 ∈ (𝑁𝑥))))
242, 3, 4ntrneibex 41149 . . . . . . . . . . . 12 (𝜑𝐵 ∈ V)
2524ad3antrrr 729 . . . . . . . . . . 11 ((((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) ∧ 𝑥𝐵) → 𝐵 ∈ V)
2618elpwid 4505 . . . . . . . . . . . 12 ((((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) ∧ 𝑥𝐵) → 𝑠𝐵)
27 ssinss1 4142 . . . . . . . . . . . 12 (𝑠𝐵 → (𝑠𝑡) ⊆ 𝐵)
2826, 27syl 17 . . . . . . . . . . 11 ((((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) ∧ 𝑥𝐵) → (𝑠𝑡) ⊆ 𝐵)
2925, 28sselpwd 5196 . . . . . . . . . 10 ((((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) ∧ 𝑥𝐵) → (𝑠𝑡) ∈ 𝒫 𝐵)
302, 3, 16, 17, 29ntrneiel 41157 . . . . . . . . 9 ((((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) ∧ 𝑥𝐵) → (𝑥 ∈ (𝐼‘(𝑠𝑡)) ↔ (𝑠𝑡) ∈ (𝑁𝑥)))
3123, 30imbi12d 348 . . . . . . . 8 ((((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) ∧ 𝑥𝐵) → ((𝑥 ∈ ((𝐼𝑠) ∩ (𝐼𝑡)) → 𝑥 ∈ (𝐼‘(𝑠𝑡))) ↔ ((𝑠 ∈ (𝑁𝑥) ∧ 𝑡 ∈ (𝑁𝑥)) → (𝑠𝑡) ∈ (𝑁𝑥))))
3231ralbidva 3125 . . . . . . 7 (((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) → (∀𝑥𝐵 (𝑥 ∈ ((𝐼𝑠) ∩ (𝐼𝑡)) → 𝑥 ∈ (𝐼‘(𝑠𝑡))) ↔ ∀𝑥𝐵 ((𝑠 ∈ (𝑁𝑥) ∧ 𝑡 ∈ (𝑁𝑥)) → (𝑠𝑡) ∈ (𝑁𝑥))))
3314, 32bitrd 282 . . . . . 6 (((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) → (∀𝑥 ∈ ((𝐼𝑠) ∩ (𝐼𝑡))𝑥 ∈ (𝐼‘(𝑠𝑡)) ↔ ∀𝑥𝐵 ((𝑠 ∈ (𝑁𝑥) ∧ 𝑡 ∈ (𝑁𝑥)) → (𝑠𝑡) ∈ (𝑁𝑥))))
341, 33syl5bb 286 . . . . 5 (((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) → (((𝐼𝑠) ∩ (𝐼𝑡)) ⊆ (𝐼‘(𝑠𝑡)) ↔ ∀𝑥𝐵 ((𝑠 ∈ (𝑁𝑥) ∧ 𝑡 ∈ (𝑁𝑥)) → (𝑠𝑡) ∈ (𝑁𝑥))))
3534ralbidva 3125 . . . 4 ((𝜑𝑠 ∈ 𝒫 𝐵) → (∀𝑡 ∈ 𝒫 𝐵((𝐼𝑠) ∩ (𝐼𝑡)) ⊆ (𝐼‘(𝑠𝑡)) ↔ ∀𝑡 ∈ 𝒫 𝐵𝑥𝐵 ((𝑠 ∈ (𝑁𝑥) ∧ 𝑡 ∈ (𝑁𝑥)) → (𝑠𝑡) ∈ (𝑁𝑥))))
36 ralcom 3272 . . . 4 (∀𝑡 ∈ 𝒫 𝐵𝑥𝐵 ((𝑠 ∈ (𝑁𝑥) ∧ 𝑡 ∈ (𝑁𝑥)) → (𝑠𝑡) ∈ (𝑁𝑥)) ↔ ∀𝑥𝐵𝑡 ∈ 𝒫 𝐵((𝑠 ∈ (𝑁𝑥) ∧ 𝑡 ∈ (𝑁𝑥)) → (𝑠𝑡) ∈ (𝑁𝑥)))
3735, 36bitrdi 290 . . 3 ((𝜑𝑠 ∈ 𝒫 𝐵) → (∀𝑡 ∈ 𝒫 𝐵((𝐼𝑠) ∩ (𝐼𝑡)) ⊆ (𝐼‘(𝑠𝑡)) ↔ ∀𝑥𝐵𝑡 ∈ 𝒫 𝐵((𝑠 ∈ (𝑁𝑥) ∧ 𝑡 ∈ (𝑁𝑥)) → (𝑠𝑡) ∈ (𝑁𝑥))))
3837ralbidva 3125 . 2 (𝜑 → (∀𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵((𝐼𝑠) ∩ (𝐼𝑡)) ⊆ (𝐼‘(𝑠𝑡)) ↔ ∀𝑠 ∈ 𝒫 𝐵𝑥𝐵𝑡 ∈ 𝒫 𝐵((𝑠 ∈ (𝑁𝑥) ∧ 𝑡 ∈ (𝑁𝑥)) → (𝑠𝑡) ∈ (𝑁𝑥))))
39 ralcom 3272 . 2 (∀𝑠 ∈ 𝒫 𝐵𝑥𝐵𝑡 ∈ 𝒫 𝐵((𝑠 ∈ (𝑁𝑥) ∧ 𝑡 ∈ (𝑁𝑥)) → (𝑠𝑡) ∈ (𝑁𝑥)) ↔ ∀𝑥𝐵𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵((𝑠 ∈ (𝑁𝑥) ∧ 𝑡 ∈ (𝑁𝑥)) → (𝑠𝑡) ∈ (𝑁𝑥)))
4038, 39bitrdi 290 1 (𝜑 → (∀𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵((𝐼𝑠) ∩ (𝐼𝑡)) ⊆ (𝐼‘(𝑠𝑡)) ↔ ∀𝑥𝐵𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵((𝑠 ∈ (𝑁𝑥) ∧ 𝑡 ∈ (𝑁𝑥)) → (𝑠𝑡) ∈ (𝑁𝑥))))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2111  ∀wral 3070  {crab 3074  Vcvv 3409   ∩ cin 3857   ⊆ wss 3858  𝒫 cpw 4494   class class class wbr 5032   ↦ cmpt 5112  ⟶wf 6331  ‘cfv 6335  (class class class)co 7150   ∈ cmpo 7152   ↑m cmap 8416 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-rep 5156  ax-sep 5169  ax-nul 5176  ax-pow 5234  ax-pr 5298  ax-un 7459 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-reu 3077  df-rab 3079  df-v 3411  df-sbc 3697  df-csb 3806  df-dif 3861  df-un 3863  df-in 3865  df-ss 3875  df-nul 4226  df-if 4421  df-pw 4496  df-sn 4523  df-pr 4525  df-op 4529  df-uni 4799  df-iun 4885  df-br 5033  df-opab 5095  df-mpt 5113  df-id 5430  df-xp 5530  df-rel 5531  df-cnv 5532  df-co 5533  df-dm 5534  df-rn 5535  df-res 5536  df-ima 5537  df-iota 6294  df-fun 6337  df-fn 6338  df-f 6339  df-f1 6340  df-fo 6341  df-f1o 6342  df-fv 6343  df-ov 7153  df-oprab 7154  df-mpo 7155  df-1st 7693  df-2nd 7694  df-map 8418 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator