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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 45951, 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 2737 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ((𝐵 × 𝐵) × {1o}) = ((𝐵 × 𝐵) × {1o})) | |
4 | 3 | f1omo 45804 | . . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ∃*𝑓 𝑓 ∈ (((𝐵 × 𝐵) × {1o})‘〈𝑥, 𝑦〉)) |
5 | df-ov 7194 | . . . . 5 ⊢ (𝑥((𝐵 × 𝐵) × {1o})𝑦) = (((𝐵 × 𝐵) × {1o})‘〈𝑥, 𝑦〉) | |
6 | 5 | eleq2i 2822 | . . . 4 ⊢ (𝑓 ∈ (𝑥((𝐵 × 𝐵) × {1o})𝑦) ↔ 𝑓 ∈ (((𝐵 × 𝐵) × {1o})‘〈𝑥, 𝑦〉)) |
7 | 6 | mobii 2547 | . . 3 ⊢ (∃*𝑓 𝑓 ∈ (𝑥((𝐵 × 𝐵) × {1o})𝑦) ↔ ∃*𝑓 𝑓 ∈ (((𝐵 × 𝐵) × {1o})‘〈𝑥, 𝑦〉)) |
8 | 4, 7 | sylibr 237 | . 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 264 | . 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 552 | . . . 4 ⊢ (𝑦 ∈ 𝐵 → (𝑦 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) |
14 | 1oex 8193 | . . . . 5 ⊢ 1o ∈ V | |
15 | 14 | ovconst2 7366 | . . . 4 ⊢ ((𝑦 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑦((𝐵 × 𝐵) × {1o})𝑦) = 1o) |
16 | 0lt1o 8209 | . . . . 5 ⊢ ∅ ∈ 1o | |
17 | eleq2 2819 | . . . . 5 ⊢ ((𝑦((𝐵 × 𝐵) × {1o})𝑦) = 1o → (∅ ∈ (𝑦((𝐵 × 𝐵) × {1o})𝑦) ↔ ∅ ∈ 1o)) | |
18 | 16, 17 | mpbiri 261 | . . . 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 485 | . 2 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → ∅ ∈ (𝑦((𝐵 × 𝐵) × {1o})𝑦)) |
21 | 16 | a1i 11 | . . . 4 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) → ∅ ∈ 1o) |
22 | 0ov 7228 | . . . . . . 7 ⊢ (〈𝑥, 𝑦〉∅𝑧) = ∅ | |
23 | 22 | oveqi 7204 | . . . . . 6 ⊢ (𝑔(〈𝑥, 𝑦〉∅𝑧)𝑓) = (𝑔∅𝑓) |
24 | 0ov 7228 | . . . . . 6 ⊢ (𝑔∅𝑓) = ∅ | |
25 | 23, 24 | eqtri 2759 | . . . . 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 7366 | . . . . 5 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) → (𝑥((𝐵 × 𝐵) × {1o})𝑧) = 1o) |
28 | 27 | 3adant2 1133 | . . . 4 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) → (𝑥((𝐵 × 𝐵) × {1o})𝑧) = 1o) |
29 | 21, 26, 28 | 3eltr4d 2846 | . . 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 45935 | 1 ⊢ (𝜑 → (𝐶 ∈ ThinCat ∧ (Id‘𝐶) = (𝑦 ∈ 𝐵 ↦ ∅))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∧ w3a 1089 = wceq 1543 ∈ wcel 2112 ∃*wmo 2537 ∅c0 4223 {csn 4527 〈cop 4533 ↦ cmpt 5120 × cxp 5534 ‘cfv 6358 (class class class)co 7191 1oc1o 8173 Basecbs 16666 Hom chom 16760 compcco 16761 Idccid 17122 ThinCatcthinc 45916 |
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 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-rep 5164 ax-sep 5177 ax-nul 5184 ax-pr 5307 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-ral 3056 df-rex 3057 df-reu 3058 df-rmo 3059 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-if 4426 df-sn 4528 df-pr 4530 df-op 4534 df-uni 4806 df-iun 4892 df-br 5040 df-opab 5102 df-mpt 5121 df-id 5440 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-suc 6197 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7148 df-ov 7194 df-1o 8180 df-cat 17125 df-cid 17126 df-thinc 45917 |
This theorem is referenced by: (None) |
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