Theorem thinccat 50699

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

Assertion

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

Proof of Theorem thinccat

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

Step	Hyp	Ref	Expression
1		eqid 2763	. . 3 ⊢ (Base‘𝐶) = (Base‘𝐶)
2		eqid 2763	. . 3 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
3	1, 2	isthinc 50696	. 2 ⊢ (𝐶 ∈ ThinCat ↔ (𝐶 ∈ Cat ∧ ∀𝑥 ∈ (Base‘𝐶)∀𝑦 ∈ (Base‘𝐶)∃*𝑓 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)))
4	3	simplbi 502	1 ⊢ (𝐶 ∈ ThinCat → 𝐶 ∈ Cat)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2146  ∃*wmo 2565  ∀wral 3080  ‘cfv 6547  (class class class)co 7430  Basecbs 17402  Hom chom 17454  Catccat 17853  ThinCatcthinc 50694

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-ext 2735  ax-nul 5263

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2101  df-mo 2567  df-clab 2742  df-cleq 2755  df-clel 2839  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-sbc 3743  df-ss 3901  df-dif 4050  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-br 5107  df-iota 6501  df-fv 6555  df-ov 7433  df-thinc 50695

This theorem is used by:  thinccatd  50700  thincssc  50701  oppcthin  50715  fuconuniqex2  50929