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Theorem ntrneiel2 44099
Description: Membership in iterated interior of a set is equivalent to there existing a particular neighborhood of that member such that points are members of that neighborhood if and only if the set is a neighborhood of each of those points. (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)
ntrneiel2.x (𝜑𝑋𝐵)
ntrneiel2.s (𝜑𝑆 ∈ 𝒫 𝐵)
Assertion
Ref Expression
ntrneiel2 (𝜑 → (𝑋 ∈ (𝐼‘(𝐼𝑆)) ↔ ∃𝑢 ∈ (𝑁𝑋)∀𝑦𝐵 (𝑦𝑢𝑆 ∈ (𝑁𝑦))))
Distinct variable groups:   𝐵,𝑖,𝑗,𝑘,𝑙,𝑚,𝑦   𝑢,𝐵,𝑦   𝑘,𝐼,𝑙,𝑚,𝑦   𝑢,𝑁,𝑦   𝑆,𝑚,𝑦   𝑢,𝑆   𝑋,𝑙,𝑚,𝑦   𝑢,𝑋   𝜑,𝑖,𝑗,𝑘,𝑙,𝑦   𝜑,𝑢
Allowed substitution hints:   𝜑(𝑚)   𝑆(𝑖,𝑗,𝑘,𝑙)   𝐹(𝑦,𝑢,𝑖,𝑗,𝑘,𝑚,𝑙)   𝐼(𝑢,𝑖,𝑗)   𝑁(𝑖,𝑗,𝑘,𝑚,𝑙)   𝑂(𝑦,𝑢,𝑖,𝑗,𝑘,𝑚,𝑙)   𝑋(𝑖,𝑗,𝑘)

Proof of Theorem ntrneiel2
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)
4 ntrneiel2.x . . 3 (𝜑𝑋𝐵)
51, 2, 3ntrneiiex 44089 . . . . 5 (𝜑𝐼 ∈ (𝒫 𝐵m 𝒫 𝐵))
6 elmapi 8889 . . . . 5 (𝐼 ∈ (𝒫 𝐵m 𝒫 𝐵) → 𝐼:𝒫 𝐵⟶𝒫 𝐵)
75, 6syl 17 . . . 4 (𝜑𝐼:𝒫 𝐵⟶𝒫 𝐵)
8 ntrneiel2.s . . . 4 (𝜑𝑆 ∈ 𝒫 𝐵)
97, 8ffvelcdmd 7105 . . 3 (𝜑 → (𝐼𝑆) ∈ 𝒫 𝐵)
101, 2, 3, 4, 9ntrneiel 44094 . 2 (𝜑 → (𝑋 ∈ (𝐼‘(𝐼𝑆)) ↔ (𝐼𝑆) ∈ (𝑁𝑋)))
111, 2, 3, 8ntrneifv4 44098 . . . 4 (𝜑 → (𝐼𝑆) = {𝑦𝐵𝑆 ∈ (𝑁𝑦)})
12 df-rab 3437 . . . 4 {𝑦𝐵𝑆 ∈ (𝑁𝑦)} = {𝑦 ∣ (𝑦𝐵𝑆 ∈ (𝑁𝑦))}
1311, 12eqtrdi 2793 . . 3 (𝜑 → (𝐼𝑆) = {𝑦 ∣ (𝑦𝐵𝑆 ∈ (𝑁𝑦))})
1413eleq1d 2826 . 2 (𝜑 → ((𝐼𝑆) ∈ (𝑁𝑋) ↔ {𝑦 ∣ (𝑦𝐵𝑆 ∈ (𝑁𝑦))} ∈ (𝑁𝑋)))
15 clabel 2888 . . . 4 ({𝑦 ∣ (𝑦𝐵𝑆 ∈ (𝑁𝑦))} ∈ (𝑁𝑋) ↔ ∃𝑢(𝑢 ∈ (𝑁𝑋) ∧ ∀𝑦(𝑦𝑢 ↔ (𝑦𝐵𝑆 ∈ (𝑁𝑦)))))
16 df-rex 3071 . . . 4 (∃𝑢 ∈ (𝑁𝑋)∀𝑦(𝑦𝑢 ↔ (𝑦𝐵𝑆 ∈ (𝑁𝑦))) ↔ ∃𝑢(𝑢 ∈ (𝑁𝑋) ∧ ∀𝑦(𝑦𝑢 ↔ (𝑦𝐵𝑆 ∈ (𝑁𝑦)))))
1715, 16bitr4i 278 . . 3 ({𝑦 ∣ (𝑦𝐵𝑆 ∈ (𝑁𝑦))} ∈ (𝑁𝑋) ↔ ∃𝑢 ∈ (𝑁𝑋)∀𝑦(𝑦𝑢 ↔ (𝑦𝐵𝑆 ∈ (𝑁𝑦))))
18 ibar 528 . . . . . . . 8 (𝑦𝐵 → (𝑆 ∈ (𝑁𝑦) ↔ (𝑦𝐵𝑆 ∈ (𝑁𝑦))))
1918bibi2d 342 . . . . . . 7 (𝑦𝐵 → ((𝑦𝑢𝑆 ∈ (𝑁𝑦)) ↔ (𝑦𝑢 ↔ (𝑦𝐵𝑆 ∈ (𝑁𝑦)))))
2019ralbiia 3091 . . . . . 6 (∀𝑦𝐵 (𝑦𝑢𝑆 ∈ (𝑁𝑦)) ↔ ∀𝑦𝐵 (𝑦𝑢 ↔ (𝑦𝐵𝑆 ∈ (𝑁𝑦))))
21 ssv 4008 . . . . . . . 8 𝐵 ⊆ V
2221a1i 11 . . . . . . 7 ((𝜑𝑢 ∈ (𝑁𝑋)) → 𝐵 ⊆ V)
23 vex 3484 . . . . . . . . . 10 𝑦 ∈ V
24 eldif 3961 . . . . . . . . . 10 (𝑦 ∈ (V ∖ 𝐵) ↔ (𝑦 ∈ V ∧ ¬ 𝑦𝐵))
2523, 24mpbiran 709 . . . . . . . . 9 (𝑦 ∈ (V ∖ 𝐵) ↔ ¬ 𝑦𝐵)
261, 2, 3ntrneinex 44090 . . . . . . . . . . . . . . . . . 18 (𝜑𝑁 ∈ (𝒫 𝒫 𝐵m 𝐵))
27 elmapi 8889 . . . . . . . . . . . . . . . . . 18 (𝑁 ∈ (𝒫 𝒫 𝐵m 𝐵) → 𝑁:𝐵⟶𝒫 𝒫 𝐵)
2826, 27syl 17 . . . . . . . . . . . . . . . . 17 (𝜑𝑁:𝐵⟶𝒫 𝒫 𝐵)
2928, 4ffvelcdmd 7105 . . . . . . . . . . . . . . . 16 (𝜑 → (𝑁𝑋) ∈ 𝒫 𝒫 𝐵)
3029elpwid 4609 . . . . . . . . . . . . . . 15 (𝜑 → (𝑁𝑋) ⊆ 𝒫 𝐵)
3130sselda 3983 . . . . . . . . . . . . . 14 ((𝜑𝑢 ∈ (𝑁𝑋)) → 𝑢 ∈ 𝒫 𝐵)
3231elpwid 4609 . . . . . . . . . . . . 13 ((𝜑𝑢 ∈ (𝑁𝑋)) → 𝑢𝐵)
3332sseld 3982 . . . . . . . . . . . 12 ((𝜑𝑢 ∈ (𝑁𝑋)) → (𝑦𝑢𝑦𝐵))
3433con3dimp 408 . . . . . . . . . . 11 (((𝜑𝑢 ∈ (𝑁𝑋)) ∧ ¬ 𝑦𝐵) → ¬ 𝑦𝑢)
35 pm3.14 998 . . . . . . . . . . . . 13 ((¬ 𝑦𝐵 ∨ ¬ 𝑆 ∈ (𝑁𝑦)) → ¬ (𝑦𝐵𝑆 ∈ (𝑁𝑦)))
3635orcs 876 . . . . . . . . . . . 12 𝑦𝐵 → ¬ (𝑦𝐵𝑆 ∈ (𝑁𝑦)))
3736adantl 481 . . . . . . . . . . 11 (((𝜑𝑢 ∈ (𝑁𝑋)) ∧ ¬ 𝑦𝐵) → ¬ (𝑦𝐵𝑆 ∈ (𝑁𝑦)))
3834, 372falsed 376 . . . . . . . . . 10 (((𝜑𝑢 ∈ (𝑁𝑋)) ∧ ¬ 𝑦𝐵) → (𝑦𝑢 ↔ (𝑦𝐵𝑆 ∈ (𝑁𝑦))))
3938ex 412 . . . . . . . . 9 ((𝜑𝑢 ∈ (𝑁𝑋)) → (¬ 𝑦𝐵 → (𝑦𝑢 ↔ (𝑦𝐵𝑆 ∈ (𝑁𝑦)))))
4025, 39biimtrid 242 . . . . . . . 8 ((𝜑𝑢 ∈ (𝑁𝑋)) → (𝑦 ∈ (V ∖ 𝐵) → (𝑦𝑢 ↔ (𝑦𝐵𝑆 ∈ (𝑁𝑦)))))
4140ralrimiv 3145 . . . . . . 7 ((𝜑𝑢 ∈ (𝑁𝑋)) → ∀𝑦 ∈ (V ∖ 𝐵)(𝑦𝑢 ↔ (𝑦𝐵𝑆 ∈ (𝑁𝑦))))
4222, 41raldifeq 4494 . . . . . 6 ((𝜑𝑢 ∈ (𝑁𝑋)) → (∀𝑦𝐵 (𝑦𝑢 ↔ (𝑦𝐵𝑆 ∈ (𝑁𝑦))) ↔ ∀𝑦 ∈ V (𝑦𝑢 ↔ (𝑦𝐵𝑆 ∈ (𝑁𝑦)))))
4320, 42bitrid 283 . . . . 5 ((𝜑𝑢 ∈ (𝑁𝑋)) → (∀𝑦𝐵 (𝑦𝑢𝑆 ∈ (𝑁𝑦)) ↔ ∀𝑦 ∈ V (𝑦𝑢 ↔ (𝑦𝐵𝑆 ∈ (𝑁𝑦)))))
44 ralv 3508 . . . . 5 (∀𝑦 ∈ V (𝑦𝑢 ↔ (𝑦𝐵𝑆 ∈ (𝑁𝑦))) ↔ ∀𝑦(𝑦𝑢 ↔ (𝑦𝐵𝑆 ∈ (𝑁𝑦))))
4543, 44bitr2di 288 . . . 4 ((𝜑𝑢 ∈ (𝑁𝑋)) → (∀𝑦(𝑦𝑢 ↔ (𝑦𝐵𝑆 ∈ (𝑁𝑦))) ↔ ∀𝑦𝐵 (𝑦𝑢𝑆 ∈ (𝑁𝑦))))
4645rexbidva 3177 . . 3 (𝜑 → (∃𝑢 ∈ (𝑁𝑋)∀𝑦(𝑦𝑢 ↔ (𝑦𝐵𝑆 ∈ (𝑁𝑦))) ↔ ∃𝑢 ∈ (𝑁𝑋)∀𝑦𝐵 (𝑦𝑢𝑆 ∈ (𝑁𝑦))))
4717, 46bitrid 283 . 2 (𝜑 → ({𝑦 ∣ (𝑦𝐵𝑆 ∈ (𝑁𝑦))} ∈ (𝑁𝑋) ↔ ∃𝑢 ∈ (𝑁𝑋)∀𝑦𝐵 (𝑦𝑢𝑆 ∈ (𝑁𝑦))))
4810, 14, 473bitrd 305 1 (𝜑 → (𝑋 ∈ (𝐼‘(𝐼𝑆)) ↔ ∃𝑢 ∈ (𝑁𝑋)∀𝑦𝐵 (𝑦𝑢𝑆 ∈ (𝑁𝑦))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395  wal 1538   = wceq 1540  wex 1779  wcel 2108  {cab 2714  wral 3061  wrex 3070  {crab 3436  Vcvv 3480  cdif 3948  wss 3951  𝒫 cpw 4600   class class class wbr 5143  cmpt 5225  wf 6557  cfv 6561  (class class class)co 7431  cmpo 7433  m cmap 8866
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8014  df-2nd 8015  df-map 8868
This theorem is referenced by:  ntrneik4  44114
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