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Theorem ntrneiel2 43658
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 43648 . . . . 5 (𝜑𝐼 ∈ (𝒫 𝐵m 𝒫 𝐵))
6 elmapi 8868 . . . . 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 7094 . . 3 (𝜑 → (𝐼𝑆) ∈ 𝒫 𝐵)
101, 2, 3, 4, 9ntrneiel 43653 . 2 (𝜑 → (𝑋 ∈ (𝐼‘(𝐼𝑆)) ↔ (𝐼𝑆) ∈ (𝑁𝑋)))
111, 2, 3, 8ntrneifv4 43657 . . . 4 (𝜑 → (𝐼𝑆) = {𝑦𝐵𝑆 ∈ (𝑁𝑦)})
12 df-rab 3419 . . . 4 {𝑦𝐵𝑆 ∈ (𝑁𝑦)} = {𝑦 ∣ (𝑦𝐵𝑆 ∈ (𝑁𝑦))}
1311, 12eqtrdi 2781 . . 3 (𝜑 → (𝐼𝑆) = {𝑦 ∣ (𝑦𝐵𝑆 ∈ (𝑁𝑦))})
1413eleq1d 2810 . 2 (𝜑 → ((𝐼𝑆) ∈ (𝑁𝑋) ↔ {𝑦 ∣ (𝑦𝐵𝑆 ∈ (𝑁𝑦))} ∈ (𝑁𝑋)))
15 clabel 2873 . . . 4 ({𝑦 ∣ (𝑦𝐵𝑆 ∈ (𝑁𝑦))} ∈ (𝑁𝑋) ↔ ∃𝑢(𝑢 ∈ (𝑁𝑋) ∧ ∀𝑦(𝑦𝑢 ↔ (𝑦𝐵𝑆 ∈ (𝑁𝑦)))))
16 df-rex 3060 . . . 4 (∃𝑢 ∈ (𝑁𝑋)∀𝑦(𝑦𝑢 ↔ (𝑦𝐵𝑆 ∈ (𝑁𝑦))) ↔ ∃𝑢(𝑢 ∈ (𝑁𝑋) ∧ ∀𝑦(𝑦𝑢 ↔ (𝑦𝐵𝑆 ∈ (𝑁𝑦)))))
1715, 16bitr4i 277 . . 3 ({𝑦 ∣ (𝑦𝐵𝑆 ∈ (𝑁𝑦))} ∈ (𝑁𝑋) ↔ ∃𝑢 ∈ (𝑁𝑋)∀𝑦(𝑦𝑢 ↔ (𝑦𝐵𝑆 ∈ (𝑁𝑦))))
18 ibar 527 . . . . . . . 8 (𝑦𝐵 → (𝑆 ∈ (𝑁𝑦) ↔ (𝑦𝐵𝑆 ∈ (𝑁𝑦))))
1918bibi2d 341 . . . . . . 7 (𝑦𝐵 → ((𝑦𝑢𝑆 ∈ (𝑁𝑦)) ↔ (𝑦𝑢 ↔ (𝑦𝐵𝑆 ∈ (𝑁𝑦)))))
2019ralbiia 3080 . . . . . 6 (∀𝑦𝐵 (𝑦𝑢𝑆 ∈ (𝑁𝑦)) ↔ ∀𝑦𝐵 (𝑦𝑢 ↔ (𝑦𝐵𝑆 ∈ (𝑁𝑦))))
21 ssv 4001 . . . . . . . 8 𝐵 ⊆ V
2221a1i 11 . . . . . . 7 ((𝜑𝑢 ∈ (𝑁𝑋)) → 𝐵 ⊆ V)
23 vex 3465 . . . . . . . . . 10 𝑦 ∈ V
24 eldif 3954 . . . . . . . . . 10 (𝑦 ∈ (V ∖ 𝐵) ↔ (𝑦 ∈ V ∧ ¬ 𝑦𝐵))
2523, 24mpbiran 707 . . . . . . . . 9 (𝑦 ∈ (V ∖ 𝐵) ↔ ¬ 𝑦𝐵)
261, 2, 3ntrneinex 43649 . . . . . . . . . . . . . . . . . 18 (𝜑𝑁 ∈ (𝒫 𝒫 𝐵m 𝐵))
27 elmapi 8868 . . . . . . . . . . . . . . . . . 18 (𝑁 ∈ (𝒫 𝒫 𝐵m 𝐵) → 𝑁:𝐵⟶𝒫 𝒫 𝐵)
2826, 27syl 17 . . . . . . . . . . . . . . . . 17 (𝜑𝑁:𝐵⟶𝒫 𝒫 𝐵)
2928, 4ffvelcdmd 7094 . . . . . . . . . . . . . . . 16 (𝜑 → (𝑁𝑋) ∈ 𝒫 𝒫 𝐵)
3029elpwid 4613 . . . . . . . . . . . . . . 15 (𝜑 → (𝑁𝑋) ⊆ 𝒫 𝐵)
3130sselda 3976 . . . . . . . . . . . . . 14 ((𝜑𝑢 ∈ (𝑁𝑋)) → 𝑢 ∈ 𝒫 𝐵)
3231elpwid 4613 . . . . . . . . . . . . 13 ((𝜑𝑢 ∈ (𝑁𝑋)) → 𝑢𝐵)
3332sseld 3975 . . . . . . . . . . . 12 ((𝜑𝑢 ∈ (𝑁𝑋)) → (𝑦𝑢𝑦𝐵))
3433con3dimp 407 . . . . . . . . . . 11 (((𝜑𝑢 ∈ (𝑁𝑋)) ∧ ¬ 𝑦𝐵) → ¬ 𝑦𝑢)
35 pm3.14 993 . . . . . . . . . . . . 13 ((¬ 𝑦𝐵 ∨ ¬ 𝑆 ∈ (𝑁𝑦)) → ¬ (𝑦𝐵𝑆 ∈ (𝑁𝑦)))
3635orcs 873 . . . . . . . . . . . 12 𝑦𝐵 → ¬ (𝑦𝐵𝑆 ∈ (𝑁𝑦)))
3736adantl 480 . . . . . . . . . . 11 (((𝜑𝑢 ∈ (𝑁𝑋)) ∧ ¬ 𝑦𝐵) → ¬ (𝑦𝐵𝑆 ∈ (𝑁𝑦)))
3834, 372falsed 375 . . . . . . . . . 10 (((𝜑𝑢 ∈ (𝑁𝑋)) ∧ ¬ 𝑦𝐵) → (𝑦𝑢 ↔ (𝑦𝐵𝑆 ∈ (𝑁𝑦))))
3938ex 411 . . . . . . . . 9 ((𝜑𝑢 ∈ (𝑁𝑋)) → (¬ 𝑦𝐵 → (𝑦𝑢 ↔ (𝑦𝐵𝑆 ∈ (𝑁𝑦)))))
4025, 39biimtrid 241 . . . . . . . 8 ((𝜑𝑢 ∈ (𝑁𝑋)) → (𝑦 ∈ (V ∖ 𝐵) → (𝑦𝑢 ↔ (𝑦𝐵𝑆 ∈ (𝑁𝑦)))))
4140ralrimiv 3134 . . . . . . 7 ((𝜑𝑢 ∈ (𝑁𝑋)) → ∀𝑦 ∈ (V ∖ 𝐵)(𝑦𝑢 ↔ (𝑦𝐵𝑆 ∈ (𝑁𝑦))))
4222, 41raldifeq 4495 . . . . . 6 ((𝜑𝑢 ∈ (𝑁𝑋)) → (∀𝑦𝐵 (𝑦𝑢 ↔ (𝑦𝐵𝑆 ∈ (𝑁𝑦))) ↔ ∀𝑦 ∈ V (𝑦𝑢 ↔ (𝑦𝐵𝑆 ∈ (𝑁𝑦)))))
4320, 42bitrid 282 . . . . 5 ((𝜑𝑢 ∈ (𝑁𝑋)) → (∀𝑦𝐵 (𝑦𝑢𝑆 ∈ (𝑁𝑦)) ↔ ∀𝑦 ∈ V (𝑦𝑢 ↔ (𝑦𝐵𝑆 ∈ (𝑁𝑦)))))
44 ralv 3487 . . . . 5 (∀𝑦 ∈ V (𝑦𝑢 ↔ (𝑦𝐵𝑆 ∈ (𝑁𝑦))) ↔ ∀𝑦(𝑦𝑢 ↔ (𝑦𝐵𝑆 ∈ (𝑁𝑦))))
4543, 44bitr2di 287 . . . 4 ((𝜑𝑢 ∈ (𝑁𝑋)) → (∀𝑦(𝑦𝑢 ↔ (𝑦𝐵𝑆 ∈ (𝑁𝑦))) ↔ ∀𝑦𝐵 (𝑦𝑢𝑆 ∈ (𝑁𝑦))))
4645rexbidva 3166 . . 3 (𝜑 → (∃𝑢 ∈ (𝑁𝑋)∀𝑦(𝑦𝑢 ↔ (𝑦𝐵𝑆 ∈ (𝑁𝑦))) ↔ ∃𝑢 ∈ (𝑁𝑋)∀𝑦𝐵 (𝑦𝑢𝑆 ∈ (𝑁𝑦))))
4717, 46bitrid 282 . 2 (𝜑 → ({𝑦 ∣ (𝑦𝐵𝑆 ∈ (𝑁𝑦))} ∈ (𝑁𝑋) ↔ ∃𝑢 ∈ (𝑁𝑋)∀𝑦𝐵 (𝑦𝑢𝑆 ∈ (𝑁𝑦))))
4810, 14, 473bitrd 304 1 (𝜑 → (𝑋 ∈ (𝐼‘(𝐼𝑆)) ↔ ∃𝑢 ∈ (𝑁𝑋)∀𝑦𝐵 (𝑦𝑢𝑆 ∈ (𝑁𝑦))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 394  wal 1531   = wceq 1533  wex 1773  wcel 2098  {cab 2702  wral 3050  wrex 3059  {crab 3418  Vcvv 3461  cdif 3941  wss 3944  𝒫 cpw 4604   class class class wbr 5149  cmpt 5232  wf 6545  cfv 6549  (class class class)co 7419  cmpo 7421  m cmap 8845
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5365  ax-pr 5429  ax-un 7741
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-reu 3364  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-iun 4999  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-f1 6554  df-fo 6555  df-f1o 6556  df-fv 6557  df-ov 7422  df-oprab 7423  df-mpo 7424  df-1st 7994  df-2nd 7995  df-map 8847
This theorem is referenced by:  ntrneik4  43673
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