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Theorem neicvgf1o 41677
Description: If neighborhood and convergent functions are related by operator 𝐻, it is a one-to-one onto relation. (Contributed by RP, 11-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
neicvgf1o (𝜑𝐻:(𝒫 𝒫 𝐵m 𝐵)–1-1-onto→(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 neicvgf1o
StepHypRef Expression
1 neicvg.o . . . 4 𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗m 𝑖) ↦ (𝑙𝑗 ↦ {𝑚𝑖𝑙 ∈ (𝑘𝑚)})))
2 neicvg.d . . . . . 6 𝐷 = (𝑃𝐵)
3 neicvg.h . . . . . 6 𝐻 = (𝐹 ∘ (𝐷𝐺))
4 neicvg.r . . . . . 6 (𝜑𝑁𝐻𝑀)
52, 3, 4neicvgbex 41675 . . . . 5 (𝜑𝐵 ∈ V)
65pwexd 5305 . . . 4 (𝜑 → 𝒫 𝐵 ∈ V)
7 neicvg.f . . . 4 𝐹 = (𝒫 𝐵𝑂𝐵)
81, 6, 5, 7fsovf1od 41577 . . 3 (𝜑𝐹:(𝒫 𝐵m 𝒫 𝐵)–1-1-onto→(𝒫 𝒫 𝐵m 𝐵))
9 neicvg.p . . . . 5 𝑃 = (𝑛 ∈ V ↦ (𝑝 ∈ (𝒫 𝑛m 𝒫 𝑛) ↦ (𝑜 ∈ 𝒫 𝑛 ↦ (𝑛 ∖ (𝑝‘(𝑛𝑜))))))
109, 2, 5dssmapf1od 41582 . . . 4 (𝜑𝐷:(𝒫 𝐵m 𝒫 𝐵)–1-1-onto→(𝒫 𝐵m 𝒫 𝐵))
11 neicvg.g . . . . 5 𝐺 = (𝐵𝑂𝒫 𝐵)
121, 5, 6, 11fsovf1od 41577 . . . 4 (𝜑𝐺:(𝒫 𝒫 𝐵m 𝐵)–1-1-onto→(𝒫 𝐵m 𝒫 𝐵))
13 f1oco 6734 . . . 4 ((𝐷:(𝒫 𝐵m 𝒫 𝐵)–1-1-onto→(𝒫 𝐵m 𝒫 𝐵) ∧ 𝐺:(𝒫 𝒫 𝐵m 𝐵)–1-1-onto→(𝒫 𝐵m 𝒫 𝐵)) → (𝐷𝐺):(𝒫 𝒫 𝐵m 𝐵)–1-1-onto→(𝒫 𝐵m 𝒫 𝐵))
1410, 12, 13syl2anc 583 . . 3 (𝜑 → (𝐷𝐺):(𝒫 𝒫 𝐵m 𝐵)–1-1-onto→(𝒫 𝐵m 𝒫 𝐵))
15 f1oco 6734 . . 3 ((𝐹:(𝒫 𝐵m 𝒫 𝐵)–1-1-onto→(𝒫 𝒫 𝐵m 𝐵) ∧ (𝐷𝐺):(𝒫 𝒫 𝐵m 𝐵)–1-1-onto→(𝒫 𝐵m 𝒫 𝐵)) → (𝐹 ∘ (𝐷𝐺)):(𝒫 𝒫 𝐵m 𝐵)–1-1-onto→(𝒫 𝒫 𝐵m 𝐵))
168, 14, 15syl2anc 583 . 2 (𝜑 → (𝐹 ∘ (𝐷𝐺)):(𝒫 𝒫 𝐵m 𝐵)–1-1-onto→(𝒫 𝒫 𝐵m 𝐵))
17 f1oeq1 6700 . . 3 (𝐻 = (𝐹 ∘ (𝐷𝐺)) → (𝐻:(𝒫 𝒫 𝐵m 𝐵)–1-1-onto→(𝒫 𝒫 𝐵m 𝐵) ↔ (𝐹 ∘ (𝐷𝐺)):(𝒫 𝒫 𝐵m 𝐵)–1-1-onto→(𝒫 𝒫 𝐵m 𝐵)))
183, 17ax-mp 5 . 2 (𝐻:(𝒫 𝒫 𝐵m 𝐵)–1-1-onto→(𝒫 𝒫 𝐵m 𝐵) ↔ (𝐹 ∘ (𝐷𝐺)):(𝒫 𝒫 𝐵m 𝐵)–1-1-onto→(𝒫 𝒫 𝐵m 𝐵))
1916, 18sylibr 233 1 (𝜑𝐻:(𝒫 𝒫 𝐵m 𝐵)–1-1-onto→(𝒫 𝒫 𝐵m 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205   = wceq 1541  wcel 2109  {crab 3069  Vcvv 3430  cdif 3888  𝒫 cpw 4538   class class class wbr 5078  cmpt 5161  ccom 5592  1-1-ontowf1o 6429  cfv 6430  (class class class)co 7268  cmpo 7270  m cmap 8589
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1801  ax-4 1815  ax-5 1916  ax-6 1974  ax-7 2014  ax-8 2111  ax-9 2119  ax-10 2140  ax-11 2157  ax-12 2174  ax-ext 2710  ax-rep 5213  ax-sep 5226  ax-nul 5233  ax-pow 5291  ax-pr 5355  ax-un 7579
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1544  df-fal 1554  df-ex 1786  df-nf 1790  df-sb 2071  df-mo 2541  df-eu 2570  df-clab 2717  df-cleq 2731  df-clel 2817  df-nfc 2890  df-ne 2945  df-ral 3070  df-rex 3071  df-reu 3072  df-rab 3074  df-v 3432  df-sbc 3720  df-csb 3837  df-dif 3894  df-un 3896  df-in 3898  df-ss 3908  df-nul 4262  df-if 4465  df-pw 4540  df-sn 4567  df-pr 4569  df-op 4573  df-uni 4845  df-iun 4931  df-br 5079  df-opab 5141  df-mpt 5162  df-id 5488  df-xp 5594  df-rel 5595  df-cnv 5596  df-co 5597  df-dm 5598  df-rn 5599  df-res 5600  df-ima 5601  df-iota 6388  df-fun 6432  df-fn 6433  df-f 6434  df-f1 6435  df-fo 6436  df-f1o 6437  df-fv 6438  df-ov 7271  df-oprab 7272  df-mpo 7273  df-1st 7817  df-2nd 7818  df-map 8591
This theorem is referenced by: (None)
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