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Theorem ntrneiel2 38910
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 38900 . . . . 5 (𝜑𝐼 ∈ (𝒫 𝐵𝑚 𝒫 𝐵))
6 elmapi 8031 . . . . 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, 8ffvelrnd 6503 . . 3 (𝜑 → (𝐼𝑆) ∈ 𝒫 𝐵)
101, 2, 3, 4, 9ntrneiel 38905 . 2 (𝜑 → (𝑋 ∈ (𝐼‘(𝐼𝑆)) ↔ (𝐼𝑆) ∈ (𝑁𝑋)))
111, 2, 3, 8ntrneifv4 38909 . . . 4 (𝜑 → (𝐼𝑆) = {𝑦𝐵𝑆 ∈ (𝑁𝑦)})
12 df-rab 3070 . . . 4 {𝑦𝐵𝑆 ∈ (𝑁𝑦)} = {𝑦 ∣ (𝑦𝐵𝑆 ∈ (𝑁𝑦))}
1311, 12syl6eq 2821 . . 3 (𝜑 → (𝐼𝑆) = {𝑦 ∣ (𝑦𝐵𝑆 ∈ (𝑁𝑦))})
1413eleq1d 2835 . 2 (𝜑 → ((𝐼𝑆) ∈ (𝑁𝑋) ↔ {𝑦 ∣ (𝑦𝐵𝑆 ∈ (𝑁𝑦))} ∈ (𝑁𝑋)))
15 clabel 2898 . . . 4 ({𝑦 ∣ (𝑦𝐵𝑆 ∈ (𝑁𝑦))} ∈ (𝑁𝑋) ↔ ∃𝑢(𝑢 ∈ (𝑁𝑋) ∧ ∀𝑦(𝑦𝑢 ↔ (𝑦𝐵𝑆 ∈ (𝑁𝑦)))))
16 df-rex 3067 . . . 4 (∃𝑢 ∈ (𝑁𝑋)∀𝑦(𝑦𝑢 ↔ (𝑦𝐵𝑆 ∈ (𝑁𝑦))) ↔ ∃𝑢(𝑢 ∈ (𝑁𝑋) ∧ ∀𝑦(𝑦𝑢 ↔ (𝑦𝐵𝑆 ∈ (𝑁𝑦)))))
1715, 16bitr4i 267 . . 3 ({𝑦 ∣ (𝑦𝐵𝑆 ∈ (𝑁𝑦))} ∈ (𝑁𝑋) ↔ ∃𝑢 ∈ (𝑁𝑋)∀𝑦(𝑦𝑢 ↔ (𝑦𝐵𝑆 ∈ (𝑁𝑦))))
18 ibar 512 . . . . . . . 8 (𝑦𝐵 → (𝑆 ∈ (𝑁𝑦) ↔ (𝑦𝐵𝑆 ∈ (𝑁𝑦))))
1918bibi2d 331 . . . . . . 7 (𝑦𝐵 → ((𝑦𝑢𝑆 ∈ (𝑁𝑦)) ↔ (𝑦𝑢 ↔ (𝑦𝐵𝑆 ∈ (𝑁𝑦)))))
2019ralbiia 3128 . . . . . 6 (∀𝑦𝐵 (𝑦𝑢𝑆 ∈ (𝑁𝑦)) ↔ ∀𝑦𝐵 (𝑦𝑢 ↔ (𝑦𝐵𝑆 ∈ (𝑁𝑦))))
21 ssv 3774 . . . . . . . 8 𝐵 ⊆ V
2221a1i 11 . . . . . . 7 ((𝜑𝑢 ∈ (𝑁𝑋)) → 𝐵 ⊆ V)
23 vex 3354 . . . . . . . . . 10 𝑦 ∈ V
24 eldif 3733 . . . . . . . . . 10 (𝑦 ∈ (V ∖ 𝐵) ↔ (𝑦 ∈ V ∧ ¬ 𝑦𝐵))
2523, 24mpbiran 680 . . . . . . . . 9 (𝑦 ∈ (V ∖ 𝐵) ↔ ¬ 𝑦𝐵)
261, 2, 3ntrneinex 38901 . . . . . . . . . . . . . . . . . 18 (𝜑𝑁 ∈ (𝒫 𝒫 𝐵𝑚 𝐵))
27 elmapi 8031 . . . . . . . . . . . . . . . . . 18 (𝑁 ∈ (𝒫 𝒫 𝐵𝑚 𝐵) → 𝑁:𝐵⟶𝒫 𝒫 𝐵)
2826, 27syl 17 . . . . . . . . . . . . . . . . 17 (𝜑𝑁:𝐵⟶𝒫 𝒫 𝐵)
2928, 4ffvelrnd 6503 . . . . . . . . . . . . . . . 16 (𝜑 → (𝑁𝑋) ∈ 𝒫 𝒫 𝐵)
3029elpwid 4309 . . . . . . . . . . . . . . 15 (𝜑 → (𝑁𝑋) ⊆ 𝒫 𝐵)
3130sselda 3752 . . . . . . . . . . . . . 14 ((𝜑𝑢 ∈ (𝑁𝑋)) → 𝑢 ∈ 𝒫 𝐵)
3231elpwid 4309 . . . . . . . . . . . . 13 ((𝜑𝑢 ∈ (𝑁𝑋)) → 𝑢𝐵)
3332sseld 3751 . . . . . . . . . . . 12 ((𝜑𝑢 ∈ (𝑁𝑋)) → (𝑦𝑢𝑦𝐵))
3433con3dimp 395 . . . . . . . . . . 11 (((𝜑𝑢 ∈ (𝑁𝑋)) ∧ ¬ 𝑦𝐵) → ¬ 𝑦𝑢)
35 pm3.14 954 . . . . . . . . . . . . 13 ((¬ 𝑦𝐵 ∨ ¬ 𝑆 ∈ (𝑁𝑦)) → ¬ (𝑦𝐵𝑆 ∈ (𝑁𝑦)))
3635orcs 854 . . . . . . . . . . . 12 𝑦𝐵 → ¬ (𝑦𝐵𝑆 ∈ (𝑁𝑦)))
3736adantl 467 . . . . . . . . . . 11 (((𝜑𝑢 ∈ (𝑁𝑋)) ∧ ¬ 𝑦𝐵) → ¬ (𝑦𝐵𝑆 ∈ (𝑁𝑦)))
3834, 372falsed 365 . . . . . . . . . 10 (((𝜑𝑢 ∈ (𝑁𝑋)) ∧ ¬ 𝑦𝐵) → (𝑦𝑢 ↔ (𝑦𝐵𝑆 ∈ (𝑁𝑦))))
3938ex 397 . . . . . . . . 9 ((𝜑𝑢 ∈ (𝑁𝑋)) → (¬ 𝑦𝐵 → (𝑦𝑢 ↔ (𝑦𝐵𝑆 ∈ (𝑁𝑦)))))
4025, 39syl5bi 232 . . . . . . . 8 ((𝜑𝑢 ∈ (𝑁𝑋)) → (𝑦 ∈ (V ∖ 𝐵) → (𝑦𝑢 ↔ (𝑦𝐵𝑆 ∈ (𝑁𝑦)))))
4140ralrimiv 3114 . . . . . . 7 ((𝜑𝑢 ∈ (𝑁𝑋)) → ∀𝑦 ∈ (V ∖ 𝐵)(𝑦𝑢 ↔ (𝑦𝐵𝑆 ∈ (𝑁𝑦))))
4222, 41raldifeq 4200 . . . . . 6 ((𝜑𝑢 ∈ (𝑁𝑋)) → (∀𝑦𝐵 (𝑦𝑢 ↔ (𝑦𝐵𝑆 ∈ (𝑁𝑦))) ↔ ∀𝑦 ∈ V (𝑦𝑢 ↔ (𝑦𝐵𝑆 ∈ (𝑁𝑦)))))
4320, 42syl5bb 272 . . . . 5 ((𝜑𝑢 ∈ (𝑁𝑋)) → (∀𝑦𝐵 (𝑦𝑢𝑆 ∈ (𝑁𝑦)) ↔ ∀𝑦 ∈ V (𝑦𝑢 ↔ (𝑦𝐵𝑆 ∈ (𝑁𝑦)))))
44 ralv 3370 . . . . 5 (∀𝑦 ∈ V (𝑦𝑢 ↔ (𝑦𝐵𝑆 ∈ (𝑁𝑦))) ↔ ∀𝑦(𝑦𝑢 ↔ (𝑦𝐵𝑆 ∈ (𝑁𝑦))))
4543, 44syl6rbb 277 . . . 4 ((𝜑𝑢 ∈ (𝑁𝑋)) → (∀𝑦(𝑦𝑢 ↔ (𝑦𝐵𝑆 ∈ (𝑁𝑦))) ↔ ∀𝑦𝐵 (𝑦𝑢𝑆 ∈ (𝑁𝑦))))
4645rexbidva 3197 . . 3 (𝜑 → (∃𝑢 ∈ (𝑁𝑋)∀𝑦(𝑦𝑢 ↔ (𝑦𝐵𝑆 ∈ (𝑁𝑦))) ↔ ∃𝑢 ∈ (𝑁𝑋)∀𝑦𝐵 (𝑦𝑢𝑆 ∈ (𝑁𝑦))))
4717, 46syl5bb 272 . 2 (𝜑 → ({𝑦 ∣ (𝑦𝐵𝑆 ∈ (𝑁𝑦))} ∈ (𝑁𝑋) ↔ ∃𝑢 ∈ (𝑁𝑋)∀𝑦𝐵 (𝑦𝑢𝑆 ∈ (𝑁𝑦))))
4810, 14, 473bitrd 294 1 (𝜑 → (𝑋 ∈ (𝐼‘(𝐼𝑆)) ↔ ∃𝑢 ∈ (𝑁𝑋)∀𝑦𝐵 (𝑦𝑢𝑆 ∈ (𝑁𝑦))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 382  wal 1629   = wceq 1631  wex 1852  wcel 2145  {cab 2757  wral 3061  wrex 3062  {crab 3065  Vcvv 3351  cdif 3720  wss 3723  𝒫 cpw 4297   class class class wbr 4786  cmpt 4863  wf 6027  cfv 6031  (class class class)co 6793  cmpt2 6795  𝑚 cmap 8009
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-1st 7315  df-2nd 7316  df-map 8011
This theorem is referenced by:  ntrneik4  38925
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