Theorem termorcl 17958

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 17952	. 2 ⊢ TermO = (𝑐 ∈ Cat ↦ {𝑎 ∈ (Base‘𝑐) ∣ ∀𝑏 ∈ (Base‘𝑐)∃!ℎ ℎ ∈ (𝑏(Hom ‘𝑐)𝑎)})
2	1	mptrcl 6957	1 ⊢ (𝑇 ∈ (TermO‘𝐶) → 𝐶 ∈ Cat)

Syntax hints:   → wi 4   ∈ wcel 2114  ∃!weu 2568  ∀wral 3051  {crab 3389  ‘cfv 6498  (class class class)co 7367  Basecbs 17179  Hom chom 17231  Catccat 17630  TermOctermo 17949

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-sep 5231  ax-nul 5241  ax-pr 5375

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-rab 3390  df-v 3431  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-br 5086  df-opab 5148  df-mpt 5167  df-xp 5637  df-rel 5638  df-cnv 5639  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6454  df-fv 6506  df-termo 17952

This theorem is referenced by:  termoo2  49708  oppcinito  49710  oppctermo  49711  termcterm2  49989  termolmd  50145