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Theorem clsneif1o 38222
 Description: If a (pseudo-)closure function and a (pseudo-)neighborhood function are related by the 𝐻 operator, then the operator is a one-to-one, onto mapping. (Contributed by RP, 5-Jun-2021.)
Hypotheses
Ref Expression
clsnei.o 𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗𝑚 𝑖) ↦ (𝑙𝑗 ↦ {𝑚𝑖𝑙 ∈ (𝑘𝑚)})))
clsnei.p 𝑃 = (𝑛 ∈ V ↦ (𝑝 ∈ (𝒫 𝑛𝑚 𝒫 𝑛) ↦ (𝑜 ∈ 𝒫 𝑛 ↦ (𝑛 ∖ (𝑝‘(𝑛𝑜))))))
clsnei.d 𝐷 = (𝑃𝐵)
clsnei.f 𝐹 = (𝒫 𝐵𝑂𝐵)
clsnei.h 𝐻 = (𝐹𝐷)
clsnei.r (𝜑𝐾𝐻𝑁)
Assertion
Ref Expression
clsneif1o (𝜑𝐻:(𝒫 𝐵𝑚 𝒫 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)   𝐻(𝑖,𝑗,𝑘,𝑚,𝑛,𝑜,𝑝,𝑙)   𝐾(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 clsneif1o
StepHypRef Expression
1 clsnei.d . . . 4 𝐷 = (𝑃𝐵)
2 clsnei.h . . . 4 𝐻 = (𝐹𝐷)
3 clsnei.r . . . 4 (𝜑𝐾𝐻𝑁)
41, 2, 3clsneibex 38220 . . 3 (𝜑𝐵 ∈ V)
5 clsnei.o . . . . 5 𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗𝑚 𝑖) ↦ (𝑙𝑗 ↦ {𝑚𝑖𝑙 ∈ (𝑘𝑚)})))
6 pwexg 4841 . . . . . 6 (𝐵 ∈ V → 𝒫 𝐵 ∈ V)
76adantl 482 . . . . 5 ((𝜑𝐵 ∈ V) → 𝒫 𝐵 ∈ V)
8 simpr 477 . . . . 5 ((𝜑𝐵 ∈ V) → 𝐵 ∈ V)
9 eqid 2620 . . . . 5 (𝒫 𝐵𝑂𝐵) = (𝒫 𝐵𝑂𝐵)
105, 7, 8, 9fsovf1od 38130 . . . 4 ((𝜑𝐵 ∈ V) → (𝒫 𝐵𝑂𝐵):(𝒫 𝐵𝑚 𝒫 𝐵)–1-1-onto→(𝒫 𝒫 𝐵𝑚 𝐵))
11 clsnei.p . . . . 5 𝑃 = (𝑛 ∈ V ↦ (𝑝 ∈ (𝒫 𝑛𝑚 𝒫 𝑛) ↦ (𝑜 ∈ 𝒫 𝑛 ↦ (𝑛 ∖ (𝑝‘(𝑛𝑜))))))
12 eqid 2620 . . . . 5 (𝑃𝐵) = (𝑃𝐵)
1311, 12, 8dssmapf1od 38135 . . . 4 ((𝜑𝐵 ∈ V) → (𝑃𝐵):(𝒫 𝐵𝑚 𝒫 𝐵)–1-1-onto→(𝒫 𝐵𝑚 𝒫 𝐵))
14 f1oco 6146 . . . 4 (((𝒫 𝐵𝑂𝐵):(𝒫 𝐵𝑚 𝒫 𝐵)–1-1-onto→(𝒫 𝒫 𝐵𝑚 𝐵) ∧ (𝑃𝐵):(𝒫 𝐵𝑚 𝒫 𝐵)–1-1-onto→(𝒫 𝐵𝑚 𝒫 𝐵)) → ((𝒫 𝐵𝑂𝐵) ∘ (𝑃𝐵)):(𝒫 𝐵𝑚 𝒫 𝐵)–1-1-onto→(𝒫 𝒫 𝐵𝑚 𝐵))
1510, 13, 14syl2anc 692 . . 3 ((𝜑𝐵 ∈ V) → ((𝒫 𝐵𝑂𝐵) ∘ (𝑃𝐵)):(𝒫 𝐵𝑚 𝒫 𝐵)–1-1-onto→(𝒫 𝒫 𝐵𝑚 𝐵))
164, 15mpdan 701 . 2 (𝜑 → ((𝒫 𝐵𝑂𝐵) ∘ (𝑃𝐵)):(𝒫 𝐵𝑚 𝒫 𝐵)–1-1-onto→(𝒫 𝒫 𝐵𝑚 𝐵))
17 clsnei.f . . . . 5 𝐹 = (𝒫 𝐵𝑂𝐵)
1817, 1coeq12i 5274 . . . 4 (𝐹𝐷) = ((𝒫 𝐵𝑂𝐵) ∘ (𝑃𝐵))
192, 18eqtri 2642 . . 3 𝐻 = ((𝒫 𝐵𝑂𝐵) ∘ (𝑃𝐵))
20 f1oeq1 6114 . . 3 (𝐻 = ((𝒫 𝐵𝑂𝐵) ∘ (𝑃𝐵)) → (𝐻:(𝒫 𝐵𝑚 𝒫 𝐵)–1-1-onto→(𝒫 𝒫 𝐵𝑚 𝐵) ↔ ((𝒫 𝐵𝑂𝐵) ∘ (𝑃𝐵)):(𝒫 𝐵𝑚 𝒫 𝐵)–1-1-onto→(𝒫 𝒫 𝐵𝑚 𝐵)))
2119, 20ax-mp 5 . 2 (𝐻:(𝒫 𝐵𝑚 𝒫 𝐵)–1-1-onto→(𝒫 𝒫 𝐵𝑚 𝐵) ↔ ((𝒫 𝐵𝑂𝐵) ∘ (𝑃𝐵)):(𝒫 𝐵𝑚 𝒫 𝐵)–1-1-onto→(𝒫 𝒫 𝐵𝑚 𝐵))
2216, 21sylibr 224 1 (𝜑𝐻:(𝒫 𝐵𝑚 𝒫 𝐵)–1-1-onto→(𝒫 𝒫 𝐵𝑚 𝐵))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1481   ∈ wcel 1988  {crab 2913  Vcvv 3195   ∖ cdif 3564  𝒫 cpw 4149   class class class wbr 4644   ↦ cmpt 4720   ∘ ccom 5108  –1-1-onto→wf1o 5875  ‘cfv 5876  (class class class)co 6635   ↦ cmpt2 6637   ↑𝑚 cmap 7842 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-rep 4762  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-reu 2916  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-iun 4513  df-br 4645  df-opab 4704  df-mpt 4721  df-id 5014  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-fv 5884  df-ov 6638  df-oprab 6639  df-mpt2 6640  df-1st 7153  df-2nd 7154  df-map 7844 This theorem is referenced by: (None)
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