Theorem initorcl 18047

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 ∈ (InitO‘𝐶) → 𝐶 ∈ Cat)

Proof of Theorem initorcl

Dummy variables 𝑎 𝑏 𝑐 ℎ are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-inito 18041	. 2 ⊢ InitO = (𝑐 ∈ Cat ↦ {𝑎 ∈ (Base‘𝑐) ∣ ∀𝑏 ∈ (Base‘𝑐)∃!ℎ ℎ ∈ (𝑎(Hom ‘𝑐)𝑏)})
2	1	mptrcl 7000	1 ⊢ (𝐼 ∈ (InitO‘𝐶) → 𝐶 ∈ Cat)

Syntax hints:   → wi 4   ∈ wcel 2149  ∃!weu 2602  ∀wral 3085  {crab 3423  ‘cfv 6537  (class class class)co 7411  Basecbs 17269  Hom chom 17321  Catccat 17720  InitOcinito 18038

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5261  ax-nul 5271  ax-pr 5405

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-br 5114  df-opab 5178  df-mpt 5197  df-xp 5668  df-rel 5669  df-cnv 5670  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-iota 6493  df-fv 6545  df-inito 18041

This theorem is referenced by:  initoo2  49895  oppcinito  49898  oppctermo  49899  isinito2  50162  isinito4a  50211  initocmd  50332