Theorem thinccat 50351

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 2760	. . 3 ⊢ (Base‘𝐶) = (Base‘𝐶)
2		eqid 2760	. . 3 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
3	1, 2	isthinc 50348	. 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 2145  ∃*wmo 2562  ∀wral 3076  ‘cfv 6533  (class class class)co 7414  Basecbs 17304  Hom chom 17356  Catccat 17755  ThinCatcthinc 50346

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2732  ax-nul 5263

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-mo 2564  df-clab 2739  df-cleq 2752  df-clel 2835  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-sbc 3740  df-dif 3902  df-un 3904  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-iota 6489  df-fv 6541  df-ov 7417  df-thinc 50347

This theorem is used by:  thinccatd  50352  thincssc  50353  oppcthin  50367