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Theorem termorcl 16411
Description: Reverse closure for a terminal object: If a class has a terminal object, the class is a category. (Contributed by AV, 4-Apr-2020.)
Assertion
Ref Expression
termorcl (𝑇 ∈ (TermO‘𝐶) → 𝐶 ∈ Cat)

Proof of Theorem termorcl
Dummy variables 𝑎 𝑏 𝑐 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-termo 16408 . 2 TermO = (𝑐 ∈ Cat ↦ {𝑎 ∈ (Base‘𝑐) ∣ ∀𝑏 ∈ (Base‘𝑐)∃! ∈ (𝑏(Hom ‘𝑐)𝑎)})
21mptrcl 6180 1 (𝑇 ∈ (TermO‘𝐶) → 𝐶 ∈ Cat)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 1976  ∃!weu 2454  wral 2892  {crab 2896  cfv 5787  (class class class)co 6524  Basecbs 15638  Hom chom 15722  Catccat 16091  TermOctermo 16405
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2032  ax-13 2229  ax-ext 2586  ax-sep 4700  ax-nul 4709  ax-pow 4761  ax-pr 4825
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2458  df-mo 2459  df-clab 2593  df-cleq 2599  df-clel 2602  df-nfc 2736  df-ral 2897  df-rex 2898  df-rab 2901  df-v 3171  df-dif 3539  df-un 3541  df-in 3543  df-ss 3550  df-nul 3871  df-if 4033  df-sn 4122  df-pr 4124  df-op 4128  df-uni 4364  df-br 4575  df-opab 4635  df-mpt 4636  df-xp 5031  df-rel 5032  df-cnv 5033  df-dm 5035  df-rn 5036  df-res 5037  df-ima 5038  df-iota 5751  df-fv 5795  df-termo 16408
This theorem is referenced by: (None)
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