Theorem thincc 49415

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 2730	. . 3 ⊢ (Base‘𝐶) = (Base‘𝐶)
2		eqid 2730	. . 3 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
3	1, 2	isthinc 49412	. 2 ⊢ (𝐶 ∈ ThinCat ↔ (𝐶 ∈ Cat ∧ ∀𝑥 ∈ (Base‘𝐶)∀𝑦 ∈ (Base‘𝐶)∃*𝑓 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)))
4	3	simplbi 497	1 ⊢ (𝐶 ∈ ThinCat → 𝐶 ∈ Cat)

Syntax hints:   → wi 4   ∈ wcel 2109  ∃*wmo 2532  ∀wral 3045  ‘cfv 6514  (class class class)co 7390  Basecbs 17186  Hom chom 17238  Catccat 17632  ThinCatcthinc 49410

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2702  ax-nul 5264

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-mo 2534  df-clab 2709  df-cleq 2722  df-clel 2804  df-ne 2927  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-sbc 3757  df-dif 3920  df-un 3922  df-ss 3934  df-nul 4300  df-if 4492  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-br 5111  df-iota 6467  df-fv 6522  df-ov 7393  df-thinc 49411

This theorem is referenced by:  thinccd  49416  thincssc  49417  oppcthin  49431