Theorem thincc 45921

Description: A thin category is a category. (Contributed by Zhi Wang, 17-Sep-2024.)

Assertion

Ref	Expression
thincc	⊢ (𝐶 ∈ ThinCat → 𝐶 ∈ Cat)

Proof of Theorem thincc

Dummy variables 𝑓 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2736	. . 3 ⊢ (Base‘𝐶) = (Base‘𝐶)
2		eqid 2736	. . 3 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
3	1, 2	isthinc 45918	. 2 ⊢ (𝐶 ∈ ThinCat ↔ (𝐶 ∈ Cat ∧ ∀𝑥 ∈ (Base‘𝐶)∀𝑦 ∈ (Base‘𝐶)∃*𝑓 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)))
4	3	simplbi 501	1 ⊢ (𝐶 ∈ ThinCat → 𝐶 ∈ Cat)

Syntax hints:   → wi 4   ∈ wcel 2112  ∃*wmo 2537  ∀wral 3051  ‘cfv 6358  (class class class)co 7191  Basecbs 16666  Hom chom 16760  Catccat 17121  ThinCatcthinc 45916

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2018  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2160  ax-12 2177  ax-ext 2708  ax-nul 5184

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2073  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2728  df-clel 2809  df-ral 3056  df-rex 3057  df-rab 3060  df-v 3400  df-sbc 3684  df-dif 3856  df-un 3858  df-in 3860  df-ss 3870  df-nul 4224  df-if 4426  df-sn 4528  df-pr 4530  df-op 4534  df-uni 4806  df-br 5040  df-iota 6316  df-fv 6366  df-ov 7194  df-thinc 45917

This theorem is referenced by:  thinccd  45922  thincssc  45923  oppcthin  45936