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Theorem neicvgmex 40345
Description: If the neighborhoods and convergents functions are related, the convergents function exists. (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)
Assertion
Ref Expression
neicvgmex (𝜑𝑀 ∈ (𝒫 𝒫 𝐵m 𝐵))
Distinct variable groups:   𝐵,𝑖,𝑗,𝑘,𝑙,𝑚   𝐵,𝑛,𝑜,𝑝   𝜑,𝑖,𝑗,𝑘,𝑙   𝜑,𝑛,𝑜,𝑝
Allowed substitution hints:   𝜑(𝑚)   𝐷(𝑖,𝑗,𝑘,𝑚,𝑛,𝑜,𝑝,𝑙)   𝑃(𝑖,𝑗,𝑘,𝑚,𝑛,𝑜,𝑝,𝑙)   𝐹(𝑖,𝑗,𝑘,𝑚,𝑛,𝑜,𝑝,𝑙)   𝐺(𝑖,𝑗,𝑘,𝑚,𝑛,𝑜,𝑝,𝑙)   𝐻(𝑖,𝑗,𝑘,𝑚,𝑛,𝑜,𝑝,𝑙)   𝑀(𝑖,𝑗,𝑘,𝑚,𝑛,𝑜,𝑝,𝑙)   𝑁(𝑖,𝑗,𝑘,𝑚,𝑛,𝑜,𝑝,𝑙)   𝑂(𝑖,𝑗,𝑘,𝑚,𝑛,𝑜,𝑝,𝑙)

Proof of Theorem neicvgmex
StepHypRef Expression
1 neicvg.o . 2 𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗m 𝑖) ↦ (𝑙𝑗 ↦ {𝑚𝑖𝑙 ∈ (𝑘𝑚)})))
2 neicvg.f . 2 𝐹 = (𝒫 𝐵𝑂𝐵)
3 neicvg.d . . . . 5 𝐷 = (𝑃𝐵)
4 neicvg.h . . . . 5 𝐻 = (𝐹 ∘ (𝐷𝐺))
5 neicvg.r . . . . 5 (𝜑𝑁𝐻𝑀)
63, 4, 5neicvgbex 40340 . . . 4 (𝜑𝐵 ∈ V)
7 pwexg 5270 . . . . . . . 8 (𝐵 ∈ V → 𝒫 𝐵 ∈ V)
87adantl 482 . . . . . . 7 ((𝜑𝐵 ∈ V) → 𝒫 𝐵 ∈ V)
9 simpr 485 . . . . . . 7 ((𝜑𝐵 ∈ V) → 𝐵 ∈ V)
101, 8, 9, 2fsovf1od 40240 . . . . . 6 ((𝜑𝐵 ∈ V) → 𝐹:(𝒫 𝐵m 𝒫 𝐵)–1-1-onto→(𝒫 𝒫 𝐵m 𝐵))
11 f1ofn 6609 . . . . . 6 (𝐹:(𝒫 𝐵m 𝒫 𝐵)–1-1-onto→(𝒫 𝒫 𝐵m 𝐵) → 𝐹 Fn (𝒫 𝐵m 𝒫 𝐵))
1210, 11syl 17 . . . . 5 ((𝜑𝐵 ∈ V) → 𝐹 Fn (𝒫 𝐵m 𝒫 𝐵))
13 neicvg.p . . . . . . 7 𝑃 = (𝑛 ∈ V ↦ (𝑝 ∈ (𝒫 𝑛m 𝒫 𝑛) ↦ (𝑜 ∈ 𝒫 𝑛 ↦ (𝑛 ∖ (𝑝‘(𝑛𝑜))))))
1413, 3, 9dssmapf1od 40245 . . . . . 6 ((𝜑𝐵 ∈ V) → 𝐷:(𝒫 𝐵m 𝒫 𝐵)–1-1-onto→(𝒫 𝐵m 𝒫 𝐵))
15 f1of 6608 . . . . . 6 (𝐷:(𝒫 𝐵m 𝒫 𝐵)–1-1-onto→(𝒫 𝐵m 𝒫 𝐵) → 𝐷:(𝒫 𝐵m 𝒫 𝐵)⟶(𝒫 𝐵m 𝒫 𝐵))
1614, 15syl 17 . . . . 5 ((𝜑𝐵 ∈ V) → 𝐷:(𝒫 𝐵m 𝒫 𝐵)⟶(𝒫 𝐵m 𝒫 𝐵))
17 neicvg.g . . . . . 6 𝐺 = (𝐵𝑂𝒫 𝐵)
181, 9, 8, 17fsovfd 40236 . . . . 5 ((𝜑𝐵 ∈ V) → 𝐺:(𝒫 𝒫 𝐵m 𝐵)⟶(𝒫 𝐵m 𝒫 𝐵))
194breqi 5063 . . . . . . 7 (𝑁𝐻𝑀𝑁(𝐹 ∘ (𝐷𝐺))𝑀)
205, 19sylib 219 . . . . . 6 (𝜑𝑁(𝐹 ∘ (𝐷𝐺))𝑀)
2120adantr 481 . . . . 5 ((𝜑𝐵 ∈ V) → 𝑁(𝐹 ∘ (𝐷𝐺))𝑀)
2212, 16, 18, 21brcofffn 40259 . . . 4 ((𝜑𝐵 ∈ V) → (𝑁𝐺(𝐺𝑁) ∧ (𝐺𝑁)𝐷(𝐷‘(𝐺𝑁)) ∧ (𝐷‘(𝐺𝑁))𝐹𝑀))
236, 22mpdan 683 . . 3 (𝜑 → (𝑁𝐺(𝐺𝑁) ∧ (𝐺𝑁)𝐷(𝐷‘(𝐺𝑁)) ∧ (𝐷‘(𝐺𝑁))𝐹𝑀))
2423simp3d 1136 . 2 (𝜑 → (𝐷‘(𝐺𝑁))𝐹𝑀)
251, 2, 24ntrneinex 40305 1 (𝜑𝑀 ∈ (𝒫 𝒫 𝐵m 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1079   = wceq 1528  wcel 2105  {crab 3139  Vcvv 3492  cdif 3930  𝒫 cpw 4535   class class class wbr 5057  cmpt 5137  ccom 5552   Fn wfn 6343  wf 6344  1-1-ontowf1o 6347  cfv 6348  (class class class)co 7145  cmpo 7147  m cmap 8395
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-ov 7148  df-oprab 7149  df-mpo 7150  df-1st 7678  df-2nd 7679  df-map 8397
This theorem is referenced by:  neicvgnex  40346
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