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Theorem initoo 17885
Description: An initial object is an object. (Contributed by AV, 14-Apr-2020.)
Assertion
Ref Expression
initoo (𝐶 ∈ Cat → (𝑂 ∈ (InitO‘𝐶) → 𝑂 ∈ (Base‘𝐶)))

Proof of Theorem initoo
Dummy variables 𝑏 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2736 . . . 4 (Base‘𝐶) = (Base‘𝐶)
2 eqid 2736 . . . 4 (Hom ‘𝐶) = (Hom ‘𝐶)
3 id 22 . . . 4 (𝐶 ∈ Cat → 𝐶 ∈ Cat)
41, 2, 3isinitoi 17877 . . 3 ((𝐶 ∈ Cat ∧ 𝑂 ∈ (InitO‘𝐶)) → (𝑂 ∈ (Base‘𝐶) ∧ ∀𝑏 ∈ (Base‘𝐶)∃! ∈ (𝑂(Hom ‘𝐶)𝑏)))
54simpld 495 . 2 ((𝐶 ∈ Cat ∧ 𝑂 ∈ (InitO‘𝐶)) → 𝑂 ∈ (Base‘𝐶))
65ex 413 1 (𝐶 ∈ Cat → (𝑂 ∈ (InitO‘𝐶) → 𝑂 ∈ (Base‘𝐶)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  wcel 2106  ∃!weu 2566  wral 3062  cfv 6493  (class class class)co 7353  Basecbs 17075  Hom chom 17136  Catccat 17536  InitOcinito 17859
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 2707  ax-sep 5254  ax-nul 5261  ax-pr 5382
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 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3406  df-v 3445  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4281  df-if 4485  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4864  df-br 5104  df-opab 5166  df-mpt 5187  df-id 5529  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-iota 6445  df-fun 6495  df-fv 6501  df-ov 7356  df-inito 17862
This theorem is referenced by:  iszeroi  17887  2initoinv  17888  initoeu1w  17890
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