Theorem thincc 50051

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 2765	. . 3 ⊢ (Base‘𝐶) = (Base‘𝐶)
2		eqid 2765	. . 3 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
3	1, 2	isthinc 50048	. 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 2145  ∃*wmo 2567  ∀wral 3079  ‘cfv 6525  (class class class)co 7400  Basecbs 17259  Hom chom 17311  Catccat 17710  ThinCatcthinc 50046

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-ext 2737  ax-nul 5261

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-sb 2094  df-mo 2569  df-clab 2744  df-cleq 2757  df-clel 2840  df-ne 2961  df-ral 3080  df-rex 3090  df-rab 3418  df-v 3459  df-sbc 3748  df-dif 3910  df-un 3912  df-ss 3924  df-nul 4289  df-if 4484  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4869  df-br 5106  df-iota 6481  df-fv 6533  df-ov 7403  df-thinc 50047

This theorem is referenced by:  thinccd  50052  thincssc  50053  oppcthin  50067