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Theorem neicvgf1o 39833
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 39831 . . . . 5 (𝜑𝐵 ∈ V)
65pwexd 5133 . . . 4 (𝜑 → 𝒫 𝐵 ∈ V)
7 neicvg.f . . . 4 𝐹 = (𝒫 𝐵𝑂𝐵)
81, 6, 5, 7fsovf1od 39731 . . 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 39736 . . . 4 (𝜑𝐷:(𝒫 𝐵𝑚 𝒫 𝐵)–1-1-onto→(𝒫 𝐵𝑚 𝒫 𝐵))
11 neicvg.g . . . . 5 𝐺 = (𝐵𝑂𝒫 𝐵)
121, 5, 6, 11fsovf1od 39731 . . . 4 (𝜑𝐺:(𝒫 𝒫 𝐵𝑚 𝐵)–1-1-onto→(𝒫 𝐵𝑚 𝒫 𝐵))
13 f1oco 6466 . . . 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 576 . . 3 (𝜑 → (𝐷𝐺):(𝒫 𝒫 𝐵𝑚 𝐵)–1-1-onto→(𝒫 𝐵𝑚 𝒫 𝐵))
15 f1oco 6466 . . 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 576 . 2 (𝜑 → (𝐹 ∘ (𝐷𝐺)):(𝒫 𝒫 𝐵𝑚 𝐵)–1-1-onto→(𝒫 𝒫 𝐵𝑚 𝐵))
17 f1oeq1 6433 . . 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 226 1 (𝜑𝐻:(𝒫 𝒫 𝐵𝑚 𝐵)–1-1-onto→(𝒫 𝒫 𝐵𝑚 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198   = wceq 1507  wcel 2050  {crab 3092  Vcvv 3415  cdif 3826  𝒫 cpw 4422   class class class wbr 4929  cmpt 5008  ccom 5411  1-1-ontowf1o 6187  cfv 6188  (class class class)co 6976  cmpo 6978  𝑚 cmap 8206
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2750  ax-rep 5049  ax-sep 5060  ax-nul 5067  ax-pow 5119  ax-pr 5186  ax-un 7279
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2584  df-clab 2759  df-cleq 2771  df-clel 2846  df-nfc 2918  df-ne 2968  df-ral 3093  df-rex 3094  df-reu 3095  df-rab 3097  df-v 3417  df-sbc 3682  df-csb 3787  df-dif 3832  df-un 3834  df-in 3836  df-ss 3843  df-nul 4179  df-if 4351  df-pw 4424  df-sn 4442  df-pr 4444  df-op 4448  df-uni 4713  df-iun 4794  df-br 4930  df-opab 4992  df-mpt 5009  df-id 5312  df-xp 5413  df-rel 5414  df-cnv 5415  df-co 5416  df-dm 5417  df-rn 5418  df-res 5419  df-ima 5420  df-iota 6152  df-fun 6190  df-fn 6191  df-f 6192  df-f1 6193  df-fo 6194  df-f1o 6195  df-fv 6196  df-ov 6979  df-oprab 6980  df-mpo 6981  df-1st 7501  df-2nd 7502  df-map 8208
This theorem is referenced by: (None)
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