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Theorem neicvgfv 39252
Description: The value of the neighborhoods (convergents) in terms of the the convergents (neighborhoods) function. (Contributed by RP, 27-Jun-2021.)
Hypotheses
Ref Expression
neicvg.o 𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗𝑚 𝑖) ↦ (𝑙𝑗 ↦ {𝑚𝑖𝑙 ∈ (𝑘𝑚)})))
neicvg.p 𝑃 = (𝑛 ∈ V ↦ (𝑝 ∈ (𝒫 𝑛𝑚 𝒫 𝑛) ↦ (𝑜 ∈ 𝒫 𝑛 ↦ (𝑛 ∖ (𝑝‘(𝑛𝑜))))))
neicvg.d 𝐷 = (𝑃𝐵)
neicvg.f 𝐹 = (𝒫 𝐵𝑂𝐵)
neicvg.g 𝐺 = (𝐵𝑂𝒫 𝐵)
neicvg.h 𝐻 = (𝐹 ∘ (𝐷𝐺))
neicvg.r (𝜑𝑁𝐻𝑀)
neicvgfv.x (𝜑𝑋𝐵)
Assertion
Ref Expression
neicvgfv (𝜑 → (𝑁𝑋) = {𝑠 ∈ 𝒫 𝐵 ∣ ¬ (𝐵𝑠) ∈ (𝑀𝑋)})
Distinct variable groups:   𝐵,𝑖,𝑗,𝑘,𝑙,𝑚,𝑠   𝐵,𝑛,𝑜,𝑝,𝑠   𝐷,𝑖,𝑗,𝑘,𝑙,𝑚   𝐷,𝑛,𝑜,𝑝   𝑖,𝐹,𝑗,𝑘,𝑙   𝑛,𝐹,𝑜,𝑝   𝑖,𝐺,𝑗,𝑘,𝑙,𝑚   𝑛,𝐺,𝑜,𝑝   𝑖,𝑀,𝑗,𝑘,𝑙   𝑛,𝑀,𝑜,𝑝   𝑖,𝑁,𝑗,𝑘,𝑙,𝑚,𝑠   𝑛,𝑁,𝑜,𝑝   𝑋,𝑙,𝑚,𝑠   𝜑,𝑖,𝑗,𝑘,𝑙,𝑠   𝜑,𝑛,𝑜,𝑝
Allowed substitution hints:   𝜑(𝑚)   𝐷(𝑠)   𝑃(𝑖,𝑗,𝑘,𝑚,𝑛,𝑜,𝑠,𝑝,𝑙)   𝐹(𝑚,𝑠)   𝐺(𝑠)   𝐻(𝑖,𝑗,𝑘,𝑚,𝑛,𝑜,𝑠,𝑝,𝑙)   𝑀(𝑚,𝑠)   𝑂(𝑖,𝑗,𝑘,𝑚,𝑛,𝑜,𝑠,𝑝,𝑙)   𝑋(𝑖,𝑗,𝑘,𝑛,𝑜,𝑝)

Proof of Theorem neicvgfv
StepHypRef Expression
1 dfin5 3806 . 2 (𝒫 𝐵 ∩ (𝑁𝑋)) = {𝑠 ∈ 𝒫 𝐵𝑠 ∈ (𝑁𝑋)}
2 neicvg.o . . . . . . 7 𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗𝑚 𝑖) ↦ (𝑙𝑗 ↦ {𝑚𝑖𝑙 ∈ (𝑘𝑚)})))
3 neicvg.p . . . . . . 7 𝑃 = (𝑛 ∈ V ↦ (𝑝 ∈ (𝒫 𝑛𝑚 𝒫 𝑛) ↦ (𝑜 ∈ 𝒫 𝑛 ↦ (𝑛 ∖ (𝑝‘(𝑛𝑜))))))
4 neicvg.d . . . . . . 7 𝐷 = (𝑃𝐵)
5 neicvg.f . . . . . . 7 𝐹 = (𝒫 𝐵𝑂𝐵)
6 neicvg.g . . . . . . 7 𝐺 = (𝐵𝑂𝒫 𝐵)
7 neicvg.h . . . . . . 7 𝐻 = (𝐹 ∘ (𝐷𝐺))
8 neicvg.r . . . . . . 7 (𝜑𝑁𝐻𝑀)
92, 3, 4, 5, 6, 7, 8neicvgnex 39249 . . . . . 6 (𝜑𝑁 ∈ (𝒫 𝒫 𝐵𝑚 𝐵))
10 elmapi 8144 . . . . . 6 (𝑁 ∈ (𝒫 𝒫 𝐵𝑚 𝐵) → 𝑁:𝐵⟶𝒫 𝒫 𝐵)
119, 10syl 17 . . . . 5 (𝜑𝑁:𝐵⟶𝒫 𝒫 𝐵)
12 neicvgfv.x . . . . 5 (𝜑𝑋𝐵)
1311, 12ffvelrnd 6609 . . . 4 (𝜑 → (𝑁𝑋) ∈ 𝒫 𝒫 𝐵)
1413elpwid 4390 . . 3 (𝜑 → (𝑁𝑋) ⊆ 𝒫 𝐵)
15 sseqin2 4044 . . 3 ((𝑁𝑋) ⊆ 𝒫 𝐵 ↔ (𝒫 𝐵 ∩ (𝑁𝑋)) = (𝑁𝑋))
1614, 15sylib 210 . 2 (𝜑 → (𝒫 𝐵 ∩ (𝑁𝑋)) = (𝑁𝑋))
178adantr 474 . . . 4 ((𝜑𝑠 ∈ 𝒫 𝐵) → 𝑁𝐻𝑀)
1812adantr 474 . . . 4 ((𝜑𝑠 ∈ 𝒫 𝐵) → 𝑋𝐵)
19 simpr 479 . . . 4 ((𝜑𝑠 ∈ 𝒫 𝐵) → 𝑠 ∈ 𝒫 𝐵)
202, 3, 4, 5, 6, 7, 17, 18, 19neicvgel1 39250 . . 3 ((𝜑𝑠 ∈ 𝒫 𝐵) → (𝑠 ∈ (𝑁𝑋) ↔ ¬ (𝐵𝑠) ∈ (𝑀𝑋)))
2120rabbidva 3401 . 2 (𝜑 → {𝑠 ∈ 𝒫 𝐵𝑠 ∈ (𝑁𝑋)} = {𝑠 ∈ 𝒫 𝐵 ∣ ¬ (𝐵𝑠) ∈ (𝑀𝑋)})
221, 16, 213eqtr3a 2885 1 (𝜑 → (𝑁𝑋) = {𝑠 ∈ 𝒫 𝐵 ∣ ¬ (𝐵𝑠) ∈ (𝑀𝑋)})
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 386   = wceq 1656  wcel 2164  {crab 3121  Vcvv 3414  cdif 3795  cin 3797  wss 3798  𝒫 cpw 4378   class class class wbr 4873  cmpt 4952  ccom 5346  wf 6119  cfv 6123  (class class class)co 6905  cmpt2 6907  𝑚 cmap 8122
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-8 2166  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-rep 4994  ax-sep 5005  ax-nul 5013  ax-pow 5065  ax-pr 5127  ax-un 7209
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-reu 3124  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4145  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4659  df-iun 4742  df-br 4874  df-opab 4936  df-mpt 4953  df-id 5250  df-xp 5348  df-rel 5349  df-cnv 5350  df-co 5351  df-dm 5352  df-rn 5353  df-res 5354  df-ima 5355  df-iota 6086  df-fun 6125  df-fn 6126  df-f 6127  df-f1 6128  df-fo 6129  df-f1o 6130  df-fv 6131  df-ov 6908  df-oprab 6909  df-mpt2 6910  df-1st 7428  df-2nd 7429  df-map 8124
This theorem is referenced by: (None)
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