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Theorem isthinc2 48378
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 48377 . 2 (𝐶 ∈ ThinCat ↔ (𝐶 ∈ Cat ∧ ∀𝑥𝐵𝑦𝐵 ∃*𝑓 𝑓 ∈ (𝑥𝐻𝑦)))
4 modom2 9271 . . . 4 (∃*𝑓 𝑓 ∈ (𝑥𝐻𝑦) ↔ (𝑥𝐻𝑦) ≼ 1o)
542ralbii 3118 . . 3 (∀𝑥𝐵𝑦𝐵 ∃*𝑓 𝑓 ∈ (𝑥𝐻𝑦) ↔ ∀𝑥𝐵𝑦𝐵 (𝑥𝐻𝑦) ≼ 1o)
65anbi2i 621 . 2 ((𝐶 ∈ Cat ∧ ∀𝑥𝐵𝑦𝐵 ∃*𝑓 𝑓 ∈ (𝑥𝐻𝑦)) ↔ (𝐶 ∈ Cat ∧ ∀𝑥𝐵𝑦𝐵 (𝑥𝐻𝑦) ≼ 1o))
73, 6bitri 274 1 (𝐶 ∈ ThinCat ↔ (𝐶 ∈ Cat ∧ ∀𝑥𝐵𝑦𝐵 (𝑥𝐻𝑦) ≼ 1o))
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 394   = wceq 1534  wcel 2099  ∃*wmo 2527  wral 3051   class class class wbr 5145  cfv 6545  (class class class)co 7415  1oc1o 8480  cdom 8963  Basecbs 17207  Hom chom 17271  Catccat 17671  ThinCatcthinc 48375
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-sep 5296  ax-nul 5303  ax-pr 5425
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-ne 2931  df-ral 3052  df-rex 3061  df-reu 3366  df-rab 3421  df-v 3466  df-sbc 3778  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4325  df-if 4526  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4908  df-br 5146  df-opab 5208  df-id 5572  df-xp 5680  df-rel 5681  df-cnv 5682  df-co 5683  df-dm 5684  df-rn 5685  df-res 5686  df-ima 5687  df-suc 6373  df-iota 6497  df-fun 6547  df-fn 6548  df-f 6549  df-f1 6550  df-fo 6551  df-f1o 6552  df-fv 6553  df-ov 7418  df-1o 8487  df-en 8966  df-dom 8967  df-sdom 8968  df-thinc 48376
This theorem is referenced by: (None)
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