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Theorem ntrneiel2 37893
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 37883 . . . . 5 (𝜑𝐼 ∈ (𝒫 𝐵𝑚 𝒫 𝐵))
6 elmapi 7830 . . . . 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 6321 . . 3 (𝜑 → (𝐼𝑆) ∈ 𝒫 𝐵)
101, 2, 3, 4, 9ntrneiel 37888 . 2 (𝜑 → (𝑋 ∈ (𝐼‘(𝐼𝑆)) ↔ (𝐼𝑆) ∈ (𝑁𝑋)))
111, 2, 3, 8ntrneifv4 37892 . . . 4 (𝜑 → (𝐼𝑆) = {𝑦𝐵𝑆 ∈ (𝑁𝑦)})
12 df-rab 2916 . . . 4 {𝑦𝐵𝑆 ∈ (𝑁𝑦)} = {𝑦 ∣ (𝑦𝐵𝑆 ∈ (𝑁𝑦))}
1311, 12syl6eq 2671 . . 3 (𝜑 → (𝐼𝑆) = {𝑦 ∣ (𝑦𝐵𝑆 ∈ (𝑁𝑦))})
1413eleq1d 2683 . 2 (𝜑 → ((𝐼𝑆) ∈ (𝑁𝑋) ↔ {𝑦 ∣ (𝑦𝐵𝑆 ∈ (𝑁𝑦))} ∈ (𝑁𝑋)))
15 clabel 2746 . . . 4 ({𝑦 ∣ (𝑦𝐵𝑆 ∈ (𝑁𝑦))} ∈ (𝑁𝑋) ↔ ∃𝑢(𝑢 ∈ (𝑁𝑋) ∧ ∀𝑦(𝑦𝑢 ↔ (𝑦𝐵𝑆 ∈ (𝑁𝑦)))))
16 df-rex 2913 . . . 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 525 . . . . . . . 8 (𝑦𝐵 → (𝑆 ∈ (𝑁𝑦) ↔ (𝑦𝐵𝑆 ∈ (𝑁𝑦))))
1918bibi2d 332 . . . . . . 7 (𝑦𝐵 → ((𝑦𝑢𝑆 ∈ (𝑁𝑦)) ↔ (𝑦𝑢 ↔ (𝑦𝐵𝑆 ∈ (𝑁𝑦)))))
2019ralbiia 2974 . . . . . 6 (∀𝑦𝐵 (𝑦𝑢𝑆 ∈ (𝑁𝑦)) ↔ ∀𝑦𝐵 (𝑦𝑢 ↔ (𝑦𝐵𝑆 ∈ (𝑁𝑦))))
21 ssv 3609 . . . . . . . 8 𝐵 ⊆ V
2221a1i 11 . . . . . . 7 ((𝜑𝑢 ∈ (𝑁𝑋)) → 𝐵 ⊆ V)
23 vex 3192 . . . . . . . . . 10 𝑦 ∈ V
24 eldif 3569 . . . . . . . . . 10 (𝑦 ∈ (V ∖ 𝐵) ↔ (𝑦 ∈ V ∧ ¬ 𝑦𝐵))
2523, 24mpbiran 952 . . . . . . . . 9 (𝑦 ∈ (V ∖ 𝐵) ↔ ¬ 𝑦𝐵)
261, 2, 3ntrneinex 37884 . . . . . . . . . . . . . . . . . 18 (𝜑𝑁 ∈ (𝒫 𝒫 𝐵𝑚 𝐵))
27 elmapi 7830 . . . . . . . . . . . . . . . . . 18 (𝑁 ∈ (𝒫 𝒫 𝐵𝑚 𝐵) → 𝑁:𝐵⟶𝒫 𝒫 𝐵)
2826, 27syl 17 . . . . . . . . . . . . . . . . 17 (𝜑𝑁:𝐵⟶𝒫 𝒫 𝐵)
2928, 4ffvelrnd 6321 . . . . . . . . . . . . . . . 16 (𝜑 → (𝑁𝑋) ∈ 𝒫 𝒫 𝐵)
3029elpwid 4146 . . . . . . . . . . . . . . 15 (𝜑 → (𝑁𝑋) ⊆ 𝒫 𝐵)
3130sselda 3587 . . . . . . . . . . . . . 14 ((𝜑𝑢 ∈ (𝑁𝑋)) → 𝑢 ∈ 𝒫 𝐵)
3231elpwid 4146 . . . . . . . . . . . . 13 ((𝜑𝑢 ∈ (𝑁𝑋)) → 𝑢𝐵)
3332sseld 3586 . . . . . . . . . . . 12 ((𝜑𝑢 ∈ (𝑁𝑋)) → (𝑦𝑢𝑦𝐵))
3433con3dimp 457 . . . . . . . . . . 11 (((𝜑𝑢 ∈ (𝑁𝑋)) ∧ ¬ 𝑦𝐵) → ¬ 𝑦𝑢)
35 pm3.14 523 . . . . . . . . . . . . 13 ((¬ 𝑦𝐵 ∨ ¬ 𝑆 ∈ (𝑁𝑦)) → ¬ (𝑦𝐵𝑆 ∈ (𝑁𝑦)))
3635orcs 409 . . . . . . . . . . . 12 𝑦𝐵 → ¬ (𝑦𝐵𝑆 ∈ (𝑁𝑦)))
3736adantl 482 . . . . . . . . . . 11 (((𝜑𝑢 ∈ (𝑁𝑋)) ∧ ¬ 𝑦𝐵) → ¬ (𝑦𝐵𝑆 ∈ (𝑁𝑦)))
3834, 372falsed 366 . . . . . . . . . 10 (((𝜑𝑢 ∈ (𝑁𝑋)) ∧ ¬ 𝑦𝐵) → (𝑦𝑢 ↔ (𝑦𝐵𝑆 ∈ (𝑁𝑦))))
3938ex 450 . . . . . . . . 9 ((𝜑𝑢 ∈ (𝑁𝑋)) → (¬ 𝑦𝐵 → (𝑦𝑢 ↔ (𝑦𝐵𝑆 ∈ (𝑁𝑦)))))
4025, 39syl5bi 232 . . . . . . . 8 ((𝜑𝑢 ∈ (𝑁𝑋)) → (𝑦 ∈ (V ∖ 𝐵) → (𝑦𝑢 ↔ (𝑦𝐵𝑆 ∈ (𝑁𝑦)))))
4140ralrimiv 2960 . . . . . . 7 ((𝜑𝑢 ∈ (𝑁𝑋)) → ∀𝑦 ∈ (V ∖ 𝐵)(𝑦𝑢 ↔ (𝑦𝐵𝑆 ∈ (𝑁𝑦))))
4222, 41raldifeq 4036 . . . . . 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 3208 . . . . 5 (∀𝑦 ∈ V (𝑦𝑢 ↔ (𝑦𝐵𝑆 ∈ (𝑁𝑦))) ↔ ∀𝑦(𝑦𝑢 ↔ (𝑦𝐵𝑆 ∈ (𝑁𝑦))))
4543, 44syl6rbb 277 . . . 4 ((𝜑𝑢 ∈ (𝑁𝑋)) → (∀𝑦(𝑦𝑢 ↔ (𝑦𝐵𝑆 ∈ (𝑁𝑦))) ↔ ∀𝑦𝐵 (𝑦𝑢𝑆 ∈ (𝑁𝑦))))
4645rexbidva 3043 . . 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 384  wal 1478   = wceq 1480  wex 1701  wcel 1987  {cab 2607  wral 2907  wrex 2908  {crab 2911  Vcvv 3189  cdif 3556  wss 3559  𝒫 cpw 4135   class class class wbr 4618  cmpt 4678  wf 5848  cfv 5852  (class class class)co 6610  cmpt2 6612  𝑚 cmap 7809
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-ov 6613  df-oprab 6614  df-mpt2 6615  df-1st 7120  df-2nd 7121  df-map 7811
This theorem is referenced by:  ntrneik4  37908
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