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Theorem isthinc3 49071
Description: A thin category is a category in which, given a pair of objects 𝑥 and 𝑦 and any two morphisms 𝑓, 𝑔 from 𝑥 to 𝑦, the morphisms are equal. (Contributed by Zhi Wang, 17-Sep-2024.)
Hypotheses
Ref Expression
isthinc.b 𝐵 = (Base‘𝐶)
isthinc.h 𝐻 = (Hom ‘𝐶)
Assertion
Ref Expression
isthinc3 (𝐶 ∈ ThinCat ↔ (𝐶 ∈ Cat ∧ ∀𝑥𝐵𝑦𝐵𝑓 ∈ (𝑥𝐻𝑦)∀𝑔 ∈ (𝑥𝐻𝑦)𝑓 = 𝑔))
Distinct variable groups:   𝐵,𝑓,𝑔,𝑥,𝑦   𝐶,𝑓,𝑔,𝑥,𝑦   𝑓,𝐻,𝑔,𝑥,𝑦

Proof of Theorem isthinc3
StepHypRef Expression
1 isthinc.b . . 3 𝐵 = (Base‘𝐶)
2 isthinc.h . . 3 𝐻 = (Hom ‘𝐶)
31, 2isthinc 49069 . 2 (𝐶 ∈ ThinCat ↔ (𝐶 ∈ Cat ∧ ∀𝑥𝐵𝑦𝐵 ∃*𝑓 𝑓 ∈ (𝑥𝐻𝑦)))
4 moel 3402 . . . 4 (∃*𝑓 𝑓 ∈ (𝑥𝐻𝑦) ↔ ∀𝑓 ∈ (𝑥𝐻𝑦)∀𝑔 ∈ (𝑥𝐻𝑦)𝑓 = 𝑔)
542ralbii 3128 . . 3 (∀𝑥𝐵𝑦𝐵 ∃*𝑓 𝑓 ∈ (𝑥𝐻𝑦) ↔ ∀𝑥𝐵𝑦𝐵𝑓 ∈ (𝑥𝐻𝑦)∀𝑔 ∈ (𝑥𝐻𝑦)𝑓 = 𝑔)
65anbi2i 623 . 2 ((𝐶 ∈ Cat ∧ ∀𝑥𝐵𝑦𝐵 ∃*𝑓 𝑓 ∈ (𝑥𝐻𝑦)) ↔ (𝐶 ∈ Cat ∧ ∀𝑥𝐵𝑦𝐵𝑓 ∈ (𝑥𝐻𝑦)∀𝑔 ∈ (𝑥𝐻𝑦)𝑓 = 𝑔))
73, 6bitri 275 1 (𝐶 ∈ ThinCat ↔ (𝐶 ∈ Cat ∧ ∀𝑥𝐵𝑦𝐵𝑓 ∈ (𝑥𝐻𝑦)∀𝑔 ∈ (𝑥𝐻𝑦)𝑓 = 𝑔))
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395   = wceq 1540  wcel 2108  ∃*wmo 2538  wral 3061  cfv 6561  (class class class)co 7431  Basecbs 17247  Hom chom 17308  Catccat 17707  ThinCatcthinc 49067
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 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-nul 5306
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-mo 2540  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-iota 6514  df-fv 6569  df-ov 7434  df-thinc 49068
This theorem is referenced by: (None)
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