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Mirrors > Home > MPE Home > Th. List > Mathboxes > isthinc2 | Structured version Visualization version GIF version |
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.) |
Ref | Expression |
---|---|
isthinc.b | ⊢ 𝐵 = (Base‘𝐶) |
isthinc.h | ⊢ 𝐻 = (Hom ‘𝐶) |
Ref | Expression |
---|---|
isthinc2 | ⊢ (𝐶 ∈ ThinCat ↔ (𝐶 ∈ Cat ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥𝐻𝑦) ≼ 1o)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | isthinc.b | . . 3 ⊢ 𝐵 = (Base‘𝐶) | |
2 | isthinc.h | . . 3 ⊢ 𝐻 = (Hom ‘𝐶) | |
3 | 1, 2 | isthinc 48377 | . 2 ⊢ (𝐶 ∈ ThinCat ↔ (𝐶 ∈ Cat ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∃*𝑓 𝑓 ∈ (𝑥𝐻𝑦))) |
4 | modom2 9271 | . . . 4 ⊢ (∃*𝑓 𝑓 ∈ (𝑥𝐻𝑦) ↔ (𝑥𝐻𝑦) ≼ 1o) | |
5 | 4 | 2ralbii 3118 | . . 3 ⊢ (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∃*𝑓 𝑓 ∈ (𝑥𝐻𝑦) ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥𝐻𝑦) ≼ 1o) |
6 | 5 | anbi2i 621 | . 2 ⊢ ((𝐶 ∈ Cat ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∃*𝑓 𝑓 ∈ (𝑥𝐻𝑦)) ↔ (𝐶 ∈ Cat ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥𝐻𝑦) ≼ 1o)) |
7 | 3, 6 | bitri 274 | 1 ⊢ (𝐶 ∈ ThinCat ↔ (𝐶 ∈ Cat ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (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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