Theorem termorcl 17622

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 ℎ are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-termo 17616	. 2 ⊢ TermO = (𝑐 ∈ Cat ↦ {𝑎 ∈ (Base‘𝑐) ∣ ∀𝑏 ∈ (Base‘𝑐)∃!ℎ ℎ ∈ (𝑏(Hom ‘𝑐)𝑎)})
2	1	mptrcl 6866	1 ⊢ (𝑇 ∈ (TermO‘𝐶) → 𝐶 ∈ Cat)

Syntax hints:   → wi 4   ∈ wcel 2108  ∃!weu 2568  ∀wral 3063  {crab 3067  ‘cfv 6418  (class class class)co 7255  Basecbs 16840  Hom chom 16899  Catccat 17290  TermOctermo 17613

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-opab 5133  df-mpt 5154  df-xp 5586  df-rel 5587  df-cnv 5588  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-iota 6376  df-fv 6426  df-termo 17616

This theorem is referenced by: (None)