Theorem isthinc2 49924

Description: A thin category is a category in which all hom-sets have cardinality less than or equal to the cardinality of 1_o. (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 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (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.

Step	Hyp	Ref	Expression
1		isthinc.b	. . 3 ⊢ 𝐵 = (Base‘𝐶)
2		isthinc.h	. . 3 ⊢ 𝐻 = (Hom ‘𝐶)
3	1, 2	isthinc 49923	. 2 ⊢ (𝐶 ∈ ThinCat ↔ (𝐶 ∈ Cat ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∃*𝑓 𝑓 ∈ (𝑥𝐻𝑦)))
4		modom2 9156	. . . 4 ⊢ (∃*𝑓 𝑓 ∈ (𝑥𝐻𝑦) ↔ (𝑥𝐻𝑦) ≼ 1o)
5	4	2ralbii 3116	. . 3 ⊢ (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∃*𝑓 𝑓 ∈ (𝑥𝐻𝑦) ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥𝐻𝑦) ≼ 1o)
6	5	anbi2i 630	. 2 ⊢ ((𝐶 ∈ Cat ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∃*𝑓 𝑓 ∈ (𝑥𝐻𝑦)) ↔ (𝐶 ∈ Cat ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥𝐻𝑦) ≼ 1o))
7	3, 6	bitri 277	1 ⊢ (𝐶 ∈ ThinCat ↔ (𝐶 ∈ Cat ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥𝐻𝑦) ≼ 1o))

Syntax hints:   ↔ wb 208   ∧ wa 397   = wceq 1548   ∈ wcel 2121  ∃*wmo 2543  ∀wral 3055   class class class wbr 5075  ‘cfv 6489  (class class class)co 7360  1_oc1o 8392   ≼ cdom 8885  Basecbs 17174  Hom chom 17226  Catccat 17625  ThinCatcthinc 49921

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-10 2154  ax-11 2170  ax-12 2191  ax-ext 2713  ax-sep 5221  ax-nul 5231  ax-pr 5365

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-nf 1792  df-sb 2075  df-mo 2545  df-eu 2575  df-clab 2720  df-cleq 2733  df-clel 2816  df-ne 2937  df-ral 3056  df-rex 3066  df-reu 3347  df-rab 3394  df-v 3435  df-sbc 3726  df-dif 3888  df-un 3890  df-in 3892  df-ss 3902  df-nul 4265  df-if 4458  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4842  df-br 5076  df-opab 5138  df-id 5516  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-suc 6320  df-iota 6445  df-fun 6491  df-fn 6492  df-f 6493  df-f1 6494  df-fo 6495  df-f1o 6496  df-fv 6497  df-ov 7363  df-1o 8399  df-en 8888  df-dom 8889  df-sdom 8890  df-thinc 49922

This theorem is referenced by: (None)