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Theorem ntrneiel2 39082
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 39072 . . . . 5 (𝜑𝐼 ∈ (𝒫 𝐵𝑚 𝒫 𝐵))
6 elmapi 8086 . . . . 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 6554 . . 3 (𝜑 → (𝐼𝑆) ∈ 𝒫 𝐵)
101, 2, 3, 4, 9ntrneiel 39077 . 2 (𝜑 → (𝑋 ∈ (𝐼‘(𝐼𝑆)) ↔ (𝐼𝑆) ∈ (𝑁𝑋)))
111, 2, 3, 8ntrneifv4 39081 . . . 4 (𝜑 → (𝐼𝑆) = {𝑦𝐵𝑆 ∈ (𝑁𝑦)})
12 df-rab 3064 . . . 4 {𝑦𝐵𝑆 ∈ (𝑁𝑦)} = {𝑦 ∣ (𝑦𝐵𝑆 ∈ (𝑁𝑦))}
1311, 12syl6eq 2815 . . 3 (𝜑 → (𝐼𝑆) = {𝑦 ∣ (𝑦𝐵𝑆 ∈ (𝑁𝑦))})
1413eleq1d 2829 . 2 (𝜑 → ((𝐼𝑆) ∈ (𝑁𝑋) ↔ {𝑦 ∣ (𝑦𝐵𝑆 ∈ (𝑁𝑦))} ∈ (𝑁𝑋)))
15 clabel 2892 . . . 4 ({𝑦 ∣ (𝑦𝐵𝑆 ∈ (𝑁𝑦))} ∈ (𝑁𝑋) ↔ ∃𝑢(𝑢 ∈ (𝑁𝑋) ∧ ∀𝑦(𝑦𝑢 ↔ (𝑦𝐵𝑆 ∈ (𝑁𝑦)))))
16 df-rex 3061 . . . 4 (∃𝑢 ∈ (𝑁𝑋)∀𝑦(𝑦𝑢 ↔ (𝑦𝐵𝑆 ∈ (𝑁𝑦))) ↔ ∃𝑢(𝑢 ∈ (𝑁𝑋) ∧ ∀𝑦(𝑦𝑢 ↔ (𝑦𝐵𝑆 ∈ (𝑁𝑦)))))
1715, 16bitr4i 269 . . 3 ({𝑦 ∣ (𝑦𝐵𝑆 ∈ (𝑁𝑦))} ∈ (𝑁𝑋) ↔ ∃𝑢 ∈ (𝑁𝑋)∀𝑦(𝑦𝑢 ↔ (𝑦𝐵𝑆 ∈ (𝑁𝑦))))
18 ibar 524 . . . . . . . 8 (𝑦𝐵 → (𝑆 ∈ (𝑁𝑦) ↔ (𝑦𝐵𝑆 ∈ (𝑁𝑦))))
1918bibi2d 333 . . . . . . 7 (𝑦𝐵 → ((𝑦𝑢𝑆 ∈ (𝑁𝑦)) ↔ (𝑦𝑢 ↔ (𝑦𝐵𝑆 ∈ (𝑁𝑦)))))
2019ralbiia 3126 . . . . . 6 (∀𝑦𝐵 (𝑦𝑢𝑆 ∈ (𝑁𝑦)) ↔ ∀𝑦𝐵 (𝑦𝑢 ↔ (𝑦𝐵𝑆 ∈ (𝑁𝑦))))
21 ssv 3787 . . . . . . . 8 𝐵 ⊆ V
2221a1i 11 . . . . . . 7 ((𝜑𝑢 ∈ (𝑁𝑋)) → 𝐵 ⊆ V)
23 vex 3353 . . . . . . . . . 10 𝑦 ∈ V
24 eldif 3744 . . . . . . . . . 10 (𝑦 ∈ (V ∖ 𝐵) ↔ (𝑦 ∈ V ∧ ¬ 𝑦𝐵))
2523, 24mpbiran 700 . . . . . . . . 9 (𝑦 ∈ (V ∖ 𝐵) ↔ ¬ 𝑦𝐵)
261, 2, 3ntrneinex 39073 . . . . . . . . . . . . . . . . . 18 (𝜑𝑁 ∈ (𝒫 𝒫 𝐵𝑚 𝐵))
27 elmapi 8086 . . . . . . . . . . . . . . . . . 18 (𝑁 ∈ (𝒫 𝒫 𝐵𝑚 𝐵) → 𝑁:𝐵⟶𝒫 𝒫 𝐵)
2826, 27syl 17 . . . . . . . . . . . . . . . . 17 (𝜑𝑁:𝐵⟶𝒫 𝒫 𝐵)
2928, 4ffvelrnd 6554 . . . . . . . . . . . . . . . 16 (𝜑 → (𝑁𝑋) ∈ 𝒫 𝒫 𝐵)
3029elpwid 4329 . . . . . . . . . . . . . . 15 (𝜑 → (𝑁𝑋) ⊆ 𝒫 𝐵)
3130sselda 3763 . . . . . . . . . . . . . 14 ((𝜑𝑢 ∈ (𝑁𝑋)) → 𝑢 ∈ 𝒫 𝐵)
3231elpwid 4329 . . . . . . . . . . . . 13 ((𝜑𝑢 ∈ (𝑁𝑋)) → 𝑢𝐵)
3332sseld 3762 . . . . . . . . . . . 12 ((𝜑𝑢 ∈ (𝑁𝑋)) → (𝑦𝑢𝑦𝐵))
3433con3dimp 397 . . . . . . . . . . 11 (((𝜑𝑢 ∈ (𝑁𝑋)) ∧ ¬ 𝑦𝐵) → ¬ 𝑦𝑢)
35 pm3.14 1018 . . . . . . . . . . . . 13 ((¬ 𝑦𝐵 ∨ ¬ 𝑆 ∈ (𝑁𝑦)) → ¬ (𝑦𝐵𝑆 ∈ (𝑁𝑦)))
3635orcs 901 . . . . . . . . . . . 12 𝑦𝐵 → ¬ (𝑦𝐵𝑆 ∈ (𝑁𝑦)))
3736adantl 473 . . . . . . . . . . 11 (((𝜑𝑢 ∈ (𝑁𝑋)) ∧ ¬ 𝑦𝐵) → ¬ (𝑦𝐵𝑆 ∈ (𝑁𝑦)))
3834, 372falsed 367 . . . . . . . . . 10 (((𝜑𝑢 ∈ (𝑁𝑋)) ∧ ¬ 𝑦𝐵) → (𝑦𝑢 ↔ (𝑦𝐵𝑆 ∈ (𝑁𝑦))))
3938ex 401 . . . . . . . . 9 ((𝜑𝑢 ∈ (𝑁𝑋)) → (¬ 𝑦𝐵 → (𝑦𝑢 ↔ (𝑦𝐵𝑆 ∈ (𝑁𝑦)))))
4025, 39syl5bi 233 . . . . . . . 8 ((𝜑𝑢 ∈ (𝑁𝑋)) → (𝑦 ∈ (V ∖ 𝐵) → (𝑦𝑢 ↔ (𝑦𝐵𝑆 ∈ (𝑁𝑦)))))
4140ralrimiv 3112 . . . . . . 7 ((𝜑𝑢 ∈ (𝑁𝑋)) → ∀𝑦 ∈ (V ∖ 𝐵)(𝑦𝑢 ↔ (𝑦𝐵𝑆 ∈ (𝑁𝑦))))
4222, 41raldifeq 4220 . . . . . 6 ((𝜑𝑢 ∈ (𝑁𝑋)) → (∀𝑦𝐵 (𝑦𝑢 ↔ (𝑦𝐵𝑆 ∈ (𝑁𝑦))) ↔ ∀𝑦 ∈ V (𝑦𝑢 ↔ (𝑦𝐵𝑆 ∈ (𝑁𝑦)))))
4320, 42syl5bb 274 . . . . 5 ((𝜑𝑢 ∈ (𝑁𝑋)) → (∀𝑦𝐵 (𝑦𝑢𝑆 ∈ (𝑁𝑦)) ↔ ∀𝑦 ∈ V (𝑦𝑢 ↔ (𝑦𝐵𝑆 ∈ (𝑁𝑦)))))
44 ralv 3372 . . . . 5 (∀𝑦 ∈ V (𝑦𝑢 ↔ (𝑦𝐵𝑆 ∈ (𝑁𝑦))) ↔ ∀𝑦(𝑦𝑢 ↔ (𝑦𝐵𝑆 ∈ (𝑁𝑦))))
4543, 44syl6rbb 279 . . . 4 ((𝜑𝑢 ∈ (𝑁𝑋)) → (∀𝑦(𝑦𝑢 ↔ (𝑦𝐵𝑆 ∈ (𝑁𝑦))) ↔ ∀𝑦𝐵 (𝑦𝑢𝑆 ∈ (𝑁𝑦))))
4645rexbidva 3196 . . 3 (𝜑 → (∃𝑢 ∈ (𝑁𝑋)∀𝑦(𝑦𝑢 ↔ (𝑦𝐵𝑆 ∈ (𝑁𝑦))) ↔ ∃𝑢 ∈ (𝑁𝑋)∀𝑦𝐵 (𝑦𝑢𝑆 ∈ (𝑁𝑦))))
4717, 46syl5bb 274 . 2 (𝜑 → ({𝑦 ∣ (𝑦𝐵𝑆 ∈ (𝑁𝑦))} ∈ (𝑁𝑋) ↔ ∃𝑢 ∈ (𝑁𝑋)∀𝑦𝐵 (𝑦𝑢𝑆 ∈ (𝑁𝑦))))
4810, 14, 473bitrd 296 1 (𝜑 → (𝑋 ∈ (𝐼‘(𝐼𝑆)) ↔ ∃𝑢 ∈ (𝑁𝑋)∀𝑦𝐵 (𝑦𝑢𝑆 ∈ (𝑁𝑦))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 197  wa 384  wal 1650   = wceq 1652  wex 1874  wcel 2155  {cab 2751  wral 3055  wrex 3056  {crab 3059  Vcvv 3350  cdif 3731  wss 3734  𝒫 cpw 4317   class class class wbr 4811  cmpt 4890  wf 6066  cfv 6070  (class class class)co 6846  cmpt2 6848  𝑚 cmap 8064
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4932  ax-sep 4943  ax-nul 4951  ax-pow 5003  ax-pr 5064  ax-un 7151
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-reu 3062  df-rab 3064  df-v 3352  df-sbc 3599  df-csb 3694  df-dif 3737  df-un 3739  df-in 3741  df-ss 3748  df-nul 4082  df-if 4246  df-pw 4319  df-sn 4337  df-pr 4339  df-op 4343  df-uni 4597  df-iun 4680  df-br 4812  df-opab 4874  df-mpt 4891  df-id 5187  df-xp 5285  df-rel 5286  df-cnv 5287  df-co 5288  df-dm 5289  df-rn 5290  df-res 5291  df-ima 5292  df-iota 6033  df-fun 6072  df-fn 6073  df-f 6074  df-f1 6075  df-fo 6076  df-f1o 6077  df-fv 6078  df-ov 6849  df-oprab 6850  df-mpt2 6851  df-1st 7370  df-2nd 7371  df-map 8066
This theorem is referenced by:  ntrneik4  39097
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