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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 48821 | . 2 ⊢ (𝐶 ∈ ThinCat ↔ (𝐶 ∈ Cat ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∃*𝑓 𝑓 ∈ (𝑥𝐻𝑦))) |
4 | modom2 9279 | . . . 4 ⊢ (∃*𝑓 𝑓 ∈ (𝑥𝐻𝑦) ↔ (𝑥𝐻𝑦) ≼ 1o) | |
5 | 4 | 2ralbii 3126 | . . 3 ⊢ (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∃*𝑓 𝑓 ∈ (𝑥𝐻𝑦) ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥𝐻𝑦) ≼ 1o) |
6 | 5 | anbi2i 623 | . 2 ⊢ ((𝐶 ∈ Cat ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∃*𝑓 𝑓 ∈ (𝑥𝐻𝑦)) ↔ (𝐶 ∈ Cat ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥𝐻𝑦) ≼ 1o)) |
7 | 3, 6 | bitri 275 | 1 ⊢ (𝐶 ∈ ThinCat ↔ (𝐶 ∈ Cat ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥𝐻𝑦) ≼ 1o)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 206 ∧ wa 395 = wceq 1537 ∈ wcel 2106 ∃*wmo 2536 ∀wral 3059 class class class wbr 5148 ‘cfv 6563 (class class class)co 7431 1oc1o 8498 ≼ cdom 8982 Basecbs 17245 Hom chom 17309 Catccat 17709 ThinCatcthinc 48819 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-ne 2939 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-ov 7434 df-1o 8505 df-en 8985 df-dom 8986 df-sdom 8987 df-thinc 48820 |
This theorem is referenced by: (None) |
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