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Theorem ntrneik4 39234
Description: Idempotence of the interior function is equivalent to stating a set, 𝑠, is a neighborhood of a point, 𝑥 is equivalent to there existing a special neighborhood, 𝑢, of 𝑥 such that a point is an element of the special neighborhood if and only if 𝑠 is also a neighborhood of the point. (Contributed by RP, 11-Jul-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
ntrneik4 (𝜑 → (∀𝑠 ∈ 𝒫 𝐵(𝐼‘(𝐼𝑠)) = (𝐼𝑠) ↔ ∀𝑥𝐵𝑠 ∈ 𝒫 𝐵(𝑠 ∈ (𝑁𝑥) ↔ ∃𝑢 ∈ (𝑁𝑥)∀𝑦𝐵 (𝑦𝑢𝑠 ∈ (𝑁𝑦)))))
Distinct variable groups:   𝐵,𝑖,𝑗,𝑘,𝑙,𝑚,𝑠,𝑥,𝑦   𝑘,𝐼,𝑙,𝑚,𝑥,𝑦   𝜑,𝑖,𝑗,𝑘,𝑙,𝑠,𝑥   𝑢,𝐵,𝑠,𝑥,𝑦   𝑢,𝑁,𝑦   𝜑,𝑢,𝑦
Allowed substitution hints:   𝜑(𝑚)   𝐹(𝑥,𝑦,𝑢,𝑖,𝑗,𝑘,𝑚,𝑠,𝑙)   𝐼(𝑢,𝑖,𝑗,𝑠)   𝑁(𝑥,𝑖,𝑗,𝑘,𝑚,𝑠,𝑙)   𝑂(𝑥,𝑦,𝑢,𝑖,𝑗,𝑘,𝑚,𝑠,𝑙)

Proof of Theorem ntrneik4
StepHypRef Expression
1 ntrnei.o . . 3 𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗𝑚 𝑖) ↦ (𝑙𝑗 ↦ {𝑚𝑖𝑙 ∈ (𝑘𝑚)})))
2 ntrnei.f . . 3 𝐹 = (𝒫 𝐵𝑂𝐵)
3 ntrnei.r . . 3 (𝜑𝐼𝐹𝑁)
41, 2, 3ntrneik4w 39233 . 2 (𝜑 → (∀𝑠 ∈ 𝒫 𝐵(𝐼‘(𝐼𝑠)) = (𝐼𝑠) ↔ ∀𝑥𝐵𝑠 ∈ 𝒫 𝐵(𝑠 ∈ (𝑁𝑥) ↔ (𝐼𝑠) ∈ (𝑁𝑥))))
53ad2antrr 717 . . . . . . 7 (((𝜑𝑥𝐵) ∧ 𝑠 ∈ 𝒫 𝐵) → 𝐼𝐹𝑁)
6 simplr 785 . . . . . . 7 (((𝜑𝑥𝐵) ∧ 𝑠 ∈ 𝒫 𝐵) → 𝑥𝐵)
71, 2, 3ntrneiiex 39209 . . . . . . . . . 10 (𝜑𝐼 ∈ (𝒫 𝐵𝑚 𝒫 𝐵))
8 elmapi 8149 . . . . . . . . . 10 (𝐼 ∈ (𝒫 𝐵𝑚 𝒫 𝐵) → 𝐼:𝒫 𝐵⟶𝒫 𝐵)
97, 8syl 17 . . . . . . . . 9 (𝜑𝐼:𝒫 𝐵⟶𝒫 𝐵)
109ffvelrnda 6613 . . . . . . . 8 ((𝜑𝑠 ∈ 𝒫 𝐵) → (𝐼𝑠) ∈ 𝒫 𝐵)
1110adantlr 706 . . . . . . 7 (((𝜑𝑥𝐵) ∧ 𝑠 ∈ 𝒫 𝐵) → (𝐼𝑠) ∈ 𝒫 𝐵)
121, 2, 5, 6, 11ntrneiel 39214 . . . . . 6 (((𝜑𝑥𝐵) ∧ 𝑠 ∈ 𝒫 𝐵) → (𝑥 ∈ (𝐼‘(𝐼𝑠)) ↔ (𝐼𝑠) ∈ (𝑁𝑥)))
13 simpr 479 . . . . . . 7 (((𝜑𝑥𝐵) ∧ 𝑠 ∈ 𝒫 𝐵) → 𝑠 ∈ 𝒫 𝐵)
141, 2, 5, 6, 13ntrneiel2 39219 . . . . . 6 (((𝜑𝑥𝐵) ∧ 𝑠 ∈ 𝒫 𝐵) → (𝑥 ∈ (𝐼‘(𝐼𝑠)) ↔ ∃𝑢 ∈ (𝑁𝑥)∀𝑦𝐵 (𝑦𝑢𝑠 ∈ (𝑁𝑦))))
1512, 14bitr3d 273 . . . . 5 (((𝜑𝑥𝐵) ∧ 𝑠 ∈ 𝒫 𝐵) → ((𝐼𝑠) ∈ (𝑁𝑥) ↔ ∃𝑢 ∈ (𝑁𝑥)∀𝑦𝐵 (𝑦𝑢𝑠 ∈ (𝑁𝑦))))
1615bibi2d 334 . . . 4 (((𝜑𝑥𝐵) ∧ 𝑠 ∈ 𝒫 𝐵) → ((𝑠 ∈ (𝑁𝑥) ↔ (𝐼𝑠) ∈ (𝑁𝑥)) ↔ (𝑠 ∈ (𝑁𝑥) ↔ ∃𝑢 ∈ (𝑁𝑥)∀𝑦𝐵 (𝑦𝑢𝑠 ∈ (𝑁𝑦)))))
1716ralbidva 3194 . . 3 ((𝜑𝑥𝐵) → (∀𝑠 ∈ 𝒫 𝐵(𝑠 ∈ (𝑁𝑥) ↔ (𝐼𝑠) ∈ (𝑁𝑥)) ↔ ∀𝑠 ∈ 𝒫 𝐵(𝑠 ∈ (𝑁𝑥) ↔ ∃𝑢 ∈ (𝑁𝑥)∀𝑦𝐵 (𝑦𝑢𝑠 ∈ (𝑁𝑦)))))
1817ralbidva 3194 . 2 (𝜑 → (∀𝑥𝐵𝑠 ∈ 𝒫 𝐵(𝑠 ∈ (𝑁𝑥) ↔ (𝐼𝑠) ∈ (𝑁𝑥)) ↔ ∀𝑥𝐵𝑠 ∈ 𝒫 𝐵(𝑠 ∈ (𝑁𝑥) ↔ ∃𝑢 ∈ (𝑁𝑥)∀𝑦𝐵 (𝑦𝑢𝑠 ∈ (𝑁𝑦)))))
194, 18bitrd 271 1 (𝜑 → (∀𝑠 ∈ 𝒫 𝐵(𝐼‘(𝐼𝑠)) = (𝐼𝑠) ↔ ∀𝑥𝐵𝑠 ∈ 𝒫 𝐵(𝑠 ∈ (𝑁𝑥) ↔ ∃𝑢 ∈ (𝑁𝑥)∀𝑦𝐵 (𝑦𝑢𝑠 ∈ (𝑁𝑦)))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 386   = wceq 1656  wcel 2164  wral 3117  wrex 3118  {crab 3121  Vcvv 3414  𝒫 cpw 4380   class class class wbr 4875  cmpt 4954  wf 6123  cfv 6127  (class class class)co 6910  cmpt2 6912  𝑚 cmap 8127
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-8 2166  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-rep 4996  ax-sep 5007  ax-nul 5015  ax-pow 5067  ax-pr 5129  ax-un 7214
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-reu 3124  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4147  df-if 4309  df-pw 4382  df-sn 4400  df-pr 4402  df-op 4406  df-uni 4661  df-iun 4744  df-br 4876  df-opab 4938  df-mpt 4955  df-id 5252  df-xp 5352  df-rel 5353  df-cnv 5354  df-co 5355  df-dm 5356  df-rn 5357  df-res 5358  df-ima 5359  df-iota 6090  df-fun 6129  df-fn 6130  df-f 6131  df-f1 6132  df-fo 6133  df-f1o 6134  df-fv 6135  df-ov 6913  df-oprab 6914  df-mpt2 6915  df-1st 7433  df-2nd 7434  df-map 8129
This theorem is referenced by: (None)
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