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Theorem neicvgf1o 44127
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 44125 . . . . 5 (𝜑𝐵 ∈ V)
65pwexd 5379 . . . 4 (𝜑 → 𝒫 𝐵 ∈ V)
7 neicvg.f . . . 4 𝐹 = (𝒫 𝐵𝑂𝐵)
81, 6, 5, 7fsovf1od 44029 . . 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 44034 . . . 4 (𝜑𝐷:(𝒫 𝐵m 𝒫 𝐵)–1-1-onto→(𝒫 𝐵m 𝒫 𝐵))
11 neicvg.g . . . . 5 𝐺 = (𝐵𝑂𝒫 𝐵)
121, 5, 6, 11fsovf1od 44029 . . . 4 (𝜑𝐺:(𝒫 𝒫 𝐵m 𝐵)–1-1-onto→(𝒫 𝐵m 𝒫 𝐵))
13 f1oco 6871 . . . 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 584 . . 3 (𝜑 → (𝐷𝐺):(𝒫 𝒫 𝐵m 𝐵)–1-1-onto→(𝒫 𝐵m 𝒫 𝐵))
15 f1oco 6871 . . 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 584 . 2 (𝜑 → (𝐹 ∘ (𝐷𝐺)):(𝒫 𝒫 𝐵m 𝐵)–1-1-onto→(𝒫 𝒫 𝐵m 𝐵))
17 f1oeq1 6836 . . 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 234 1 (𝜑𝐻:(𝒫 𝒫 𝐵m 𝐵)–1-1-onto→(𝒫 𝒫 𝐵m 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206   = wceq 1540  wcel 2108  {crab 3436  Vcvv 3480  cdif 3948  𝒫 cpw 4600   class class class wbr 5143  cmpt 5225  ccom 5689  1-1-ontowf1o 6560  cfv 6561  (class class class)co 7431  cmpo 7433  m cmap 8866
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8014  df-2nd 8015  df-map 8868
This theorem is referenced by: (None)
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