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Mirrors > Home > MPE Home > Th. List > Mathboxes > indthinc | Structured version Visualization version GIF version |
Description: An indiscrete category in which all hom-sets have exactly one morphism is a thin category. Constructed here is an indiscrete category where all morphisms are ∅. This is a special case of prsthinc 48855, where ≤ = (𝐵 × 𝐵). This theorem also implies a functor from the category of sets to the category of small categories. (Contributed by Zhi Wang, 17-Sep-2024.) (Proof shortened by Zhi Wang, 19-Sep-2024.) |
Ref | Expression |
---|---|
indthinc.b | ⊢ (𝜑 → 𝐵 = (Base‘𝐶)) |
indthinc.h | ⊢ (𝜑 → ((𝐵 × 𝐵) × {1o}) = (Hom ‘𝐶)) |
indthinc.o | ⊢ (𝜑 → ∅ = (comp‘𝐶)) |
indthinc.c | ⊢ (𝜑 → 𝐶 ∈ 𝑉) |
Ref | Expression |
---|---|
indthinc | ⊢ (𝜑 → (𝐶 ∈ ThinCat ∧ (Id‘𝐶) = (𝑦 ∈ 𝐵 ↦ ∅))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | indthinc.b | . 2 ⊢ (𝜑 → 𝐵 = (Base‘𝐶)) | |
2 | indthinc.h | . 2 ⊢ (𝜑 → ((𝐵 × 𝐵) × {1o}) = (Hom ‘𝐶)) | |
3 | eqidd 2736 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ((𝐵 × 𝐵) × {1o}) = ((𝐵 × 𝐵) × {1o})) | |
4 | 3 | f1omo 48691 | . . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ∃*𝑓 𝑓 ∈ (((𝐵 × 𝐵) × {1o})‘〈𝑥, 𝑦〉)) |
5 | df-ov 7434 | . . . . 5 ⊢ (𝑥((𝐵 × 𝐵) × {1o})𝑦) = (((𝐵 × 𝐵) × {1o})‘〈𝑥, 𝑦〉) | |
6 | 5 | eleq2i 2831 | . . . 4 ⊢ (𝑓 ∈ (𝑥((𝐵 × 𝐵) × {1o})𝑦) ↔ 𝑓 ∈ (((𝐵 × 𝐵) × {1o})‘〈𝑥, 𝑦〉)) |
7 | 6 | mobii 2546 | . . 3 ⊢ (∃*𝑓 𝑓 ∈ (𝑥((𝐵 × 𝐵) × {1o})𝑦) ↔ ∃*𝑓 𝑓 ∈ (((𝐵 × 𝐵) × {1o})‘〈𝑥, 𝑦〉)) |
8 | 4, 7 | sylibr 234 | . 2 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ∃*𝑓 𝑓 ∈ (𝑥((𝐵 × 𝐵) × {1o})𝑦)) |
9 | indthinc.o | . 2 ⊢ (𝜑 → ∅ = (comp‘𝐶)) | |
10 | indthinc.c | . 2 ⊢ (𝜑 → 𝐶 ∈ 𝑉) | |
11 | biid 261 | . 2 ⊢ (((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑓 ∈ (𝑥((𝐵 × 𝐵) × {1o})𝑦) ∧ 𝑔 ∈ (𝑦((𝐵 × 𝐵) × {1o})𝑧))) ↔ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑓 ∈ (𝑥((𝐵 × 𝐵) × {1o})𝑦) ∧ 𝑔 ∈ (𝑦((𝐵 × 𝐵) × {1o})𝑧)))) | |
12 | id 22 | . . . . 5 ⊢ (𝑦 ∈ 𝐵 → 𝑦 ∈ 𝐵) | |
13 | 12 | ancli 548 | . . . 4 ⊢ (𝑦 ∈ 𝐵 → (𝑦 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) |
14 | 1oex 8515 | . . . . 5 ⊢ 1o ∈ V | |
15 | 14 | ovconst2 7613 | . . . 4 ⊢ ((𝑦 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑦((𝐵 × 𝐵) × {1o})𝑦) = 1o) |
16 | 0lt1o 8541 | . . . . 5 ⊢ ∅ ∈ 1o | |
17 | eleq2 2828 | . . . . 5 ⊢ ((𝑦((𝐵 × 𝐵) × {1o})𝑦) = 1o → (∅ ∈ (𝑦((𝐵 × 𝐵) × {1o})𝑦) ↔ ∅ ∈ 1o)) | |
18 | 16, 17 | mpbiri 258 | . . . 4 ⊢ ((𝑦((𝐵 × 𝐵) × {1o})𝑦) = 1o → ∅ ∈ (𝑦((𝐵 × 𝐵) × {1o})𝑦)) |
19 | 13, 15, 18 | 3syl 18 | . . 3 ⊢ (𝑦 ∈ 𝐵 → ∅ ∈ (𝑦((𝐵 × 𝐵) × {1o})𝑦)) |
20 | 19 | adantl 481 | . 2 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → ∅ ∈ (𝑦((𝐵 × 𝐵) × {1o})𝑦)) |
21 | 16 | a1i 11 | . . . 4 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) → ∅ ∈ 1o) |
22 | 0ov 7468 | . . . . . . 7 ⊢ (〈𝑥, 𝑦〉∅𝑧) = ∅ | |
23 | 22 | oveqi 7444 | . . . . . 6 ⊢ (𝑔(〈𝑥, 𝑦〉∅𝑧)𝑓) = (𝑔∅𝑓) |
24 | 0ov 7468 | . . . . . 6 ⊢ (𝑔∅𝑓) = ∅ | |
25 | 23, 24 | eqtri 2763 | . . . . 5 ⊢ (𝑔(〈𝑥, 𝑦〉∅𝑧)𝑓) = ∅ |
26 | 25 | a1i 11 | . . . 4 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) → (𝑔(〈𝑥, 𝑦〉∅𝑧)𝑓) = ∅) |
27 | 14 | ovconst2 7613 | . . . . 5 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) → (𝑥((𝐵 × 𝐵) × {1o})𝑧) = 1o) |
28 | 27 | 3adant2 1130 | . . . 4 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) → (𝑥((𝐵 × 𝐵) × {1o})𝑧) = 1o) |
29 | 21, 26, 28 | 3eltr4d 2854 | . . 3 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) → (𝑔(〈𝑥, 𝑦〉∅𝑧)𝑓) ∈ (𝑥((𝐵 × 𝐵) × {1o})𝑧)) |
30 | 29 | ad2antrl 728 | . 2 ⊢ ((𝜑 ∧ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑓 ∈ (𝑥((𝐵 × 𝐵) × {1o})𝑦) ∧ 𝑔 ∈ (𝑦((𝐵 × 𝐵) × {1o})𝑧)))) → (𝑔(〈𝑥, 𝑦〉∅𝑧)𝑓) ∈ (𝑥((𝐵 × 𝐵) × {1o})𝑧)) |
31 | 1, 2, 8, 9, 10, 11, 20, 30 | isthincd2 48838 | 1 ⊢ (𝜑 → (𝐶 ∈ ThinCat ∧ (Id‘𝐶) = (𝑦 ∈ 𝐵 ↦ ∅))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1086 = wceq 1537 ∈ wcel 2106 ∃*wmo 2536 ∅c0 4339 {csn 4631 〈cop 4637 ↦ cmpt 5231 × cxp 5687 ‘cfv 6563 (class class class)co 7431 1oc1o 8498 Basecbs 17245 Hom chom 17309 compcco 17310 Idccid 17710 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-rep 5285 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-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 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-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 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-riota 7388 df-ov 7434 df-1o 8505 df-cat 17713 df-cid 17714 df-thinc 48820 |
This theorem is referenced by: (None) |
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