Users' Mathboxes Mathbox for Zhi Wang < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  isthinc2 Structured version   Visualization version   GIF version

Theorem isthinc2 47913
Description: A thin category is a category in which all hom-sets have cardinality less than or equal to the cardinality of 1o. (Contributed by Zhi Wang, 17-Sep-2024.)
Hypotheses
Ref Expression
isthinc.b 𝐵 = (Base‘𝐶)
isthinc.h 𝐻 = (Hom ‘𝐶)
Assertion
Ref Expression
isthinc2 (𝐶 ∈ ThinCat ↔ (𝐶 ∈ Cat ∧ ∀𝑥𝐵𝑦𝐵 (𝑥𝐻𝑦) ≼ 1o))
Distinct variable groups:   𝑥,𝐵,𝑦   𝑥,𝐶,𝑦   𝑥,𝐻,𝑦

Proof of Theorem isthinc2
Dummy variable 𝑓 is distinct from all other variables.
StepHypRef Expression
1 isthinc.b . . 3 𝐵 = (Base‘𝐶)
2 isthinc.h . . 3 𝐻 = (Hom ‘𝐶)
31, 2isthinc 47912 . 2 (𝐶 ∈ ThinCat ↔ (𝐶 ∈ Cat ∧ ∀𝑥𝐵𝑦𝐵 ∃*𝑓 𝑓 ∈ (𝑥𝐻𝑦)))
4 modom2 9247 . . . 4 (∃*𝑓 𝑓 ∈ (𝑥𝐻𝑦) ↔ (𝑥𝐻𝑦) ≼ 1o)
542ralbii 3122 . . 3 (∀𝑥𝐵𝑦𝐵 ∃*𝑓 𝑓 ∈ (𝑥𝐻𝑦) ↔ ∀𝑥𝐵𝑦𝐵 (𝑥𝐻𝑦) ≼ 1o)
65anbi2i 622 . 2 ((𝐶 ∈ Cat ∧ ∀𝑥𝐵𝑦𝐵 ∃*𝑓 𝑓 ∈ (𝑥𝐻𝑦)) ↔ (𝐶 ∈ Cat ∧ ∀𝑥𝐵𝑦𝐵 (𝑥𝐻𝑦) ≼ 1o))
73, 6bitri 275 1 (𝐶 ∈ ThinCat ↔ (𝐶 ∈ Cat ∧ ∀𝑥𝐵𝑦𝐵 (𝑥𝐻𝑦) ≼ 1o))
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 395   = wceq 1533  wcel 2098  ∃*wmo 2526  wral 3055   class class class wbr 5141  cfv 6537  (class class class)co 7405  1oc1o 8460  cdom 8939  Basecbs 17153  Hom chom 17217  Catccat 17617  ThinCatcthinc 47910
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-ne 2935  df-ral 3056  df-rex 3065  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-suc 6364  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7408  df-1o 8467  df-en 8942  df-dom 8943  df-sdom 8944  df-thinc 47911
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator