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Theorem neicvgf1o 40804
 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 40802 . . . . 5 (𝜑𝐵 ∈ V)
65pwexd 5248 . . . 4 (𝜑 → 𝒫 𝐵 ∈ V)
7 neicvg.f . . . 4 𝐹 = (𝒫 𝐵𝑂𝐵)
81, 6, 5, 7fsovf1od 40704 . . 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 40709 . . . 4 (𝜑𝐷:(𝒫 𝐵m 𝒫 𝐵)–1-1-onto→(𝒫 𝐵m 𝒫 𝐵))
11 neicvg.g . . . . 5 𝐺 = (𝐵𝑂𝒫 𝐵)
121, 5, 6, 11fsovf1od 40704 . . . 4 (𝜑𝐺:(𝒫 𝒫 𝐵m 𝐵)–1-1-onto→(𝒫 𝐵m 𝒫 𝐵))
13 f1oco 6616 . . . 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 587 . . 3 (𝜑 → (𝐷𝐺):(𝒫 𝒫 𝐵m 𝐵)–1-1-onto→(𝒫 𝐵m 𝒫 𝐵))
15 f1oco 6616 . . 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 587 . 2 (𝜑 → (𝐹 ∘ (𝐷𝐺)):(𝒫 𝒫 𝐵m 𝐵)–1-1-onto→(𝒫 𝒫 𝐵m 𝐵))
17 f1oeq1 6583 . . 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 237 1 (𝜑𝐻:(𝒫 𝒫 𝐵m 𝐵)–1-1-onto→(𝒫 𝒫 𝐵m 𝐵))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   = wceq 1538   ∈ wcel 2112  {crab 3113  Vcvv 3444   ∖ cdif 3881  𝒫 cpw 4500   class class class wbr 5033   ↦ cmpt 5113   ∘ ccom 5527  –1-1-onto→wf1o 6327  ‘cfv 6328  (class class class)co 7139   ∈ cmpo 7141   ↑m cmap 8393 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-reu 3116  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-ov 7142  df-oprab 7143  df-mpo 7144  df-1st 7675  df-2nd 7676  df-map 8395 This theorem is referenced by: (None)
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