Theorem termorcl 17706

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 17700	. 2 ⊢ TermO = (𝑐 ∈ Cat ↦ {𝑎 ∈ (Base‘𝑐) ∣ ∀𝑏 ∈ (Base‘𝑐)∃!ℎ ℎ ∈ (𝑏(Hom ‘𝑐)𝑎)})
2	1	mptrcl 6884	1 ⊢ (𝑇 ∈ (TermO‘𝐶) → 𝐶 ∈ Cat)

Syntax hints:   → wi 4   ∈ wcel 2106  ∃!weu 2568  ∀wral 3064  {crab 3068  ‘cfv 6433  (class class class)co 7275  Basecbs 16912  Hom chom 16973  Catccat 17373  TermOctermo 17697

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  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 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-mpt 5158  df-xp 5595  df-rel 5596  df-cnv 5597  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fv 6441  df-termo 17700

This theorem is referenced by: (None)