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Theorem initorcl 17922
Description: Reverse closure for an initial object: If a class has an initial object, the class is a category. (Contributed by AV, 4-Apr-2020.)
Assertion
Ref Expression
initorcl (𝐼 ∈ (InitOβ€˜πΆ) β†’ 𝐢 ∈ Cat)

Proof of Theorem initorcl
Dummy variables π‘Ž 𝑏 𝑐 β„Ž are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-inito 17916 . 2 InitO = (𝑐 ∈ Cat ↦ {π‘Ž ∈ (Baseβ€˜π‘) ∣ βˆ€π‘ ∈ (Baseβ€˜π‘)βˆƒ!β„Ž β„Ž ∈ (π‘Ž(Hom β€˜π‘)𝑏)})
21mptrcl 6993 1 (𝐼 ∈ (InitOβ€˜πΆ) β†’ 𝐢 ∈ Cat)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∈ wcel 2106  βˆƒ!weu 2561  βˆ€wral 3060  {crab 3431  β€˜cfv 6532  (class class class)co 7393  Basecbs 17126  Hom chom 17190  Catccat 17590  InitOcinito 17913
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4523  df-sn 4623  df-pr 4625  df-op 4629  df-uni 4902  df-br 5142  df-opab 5204  df-mpt 5225  df-xp 5675  df-rel 5676  df-cnv 5677  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6484  df-fv 6540  df-inito 17916
This theorem is referenced by: (None)
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