|   | Mathbox for Zhi Wang | < Previous  
      Next > Nearby theorems | |
| 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 49136, 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 48798 | . . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ∃*𝑓 𝑓 ∈ (((𝐵 × 𝐵) × {1o})‘〈𝑥, 𝑦〉)) | 
| 5 | df-ov 7435 | . . . . 5 ⊢ (𝑥((𝐵 × 𝐵) × {1o})𝑦) = (((𝐵 × 𝐵) × {1o})‘〈𝑥, 𝑦〉) | |
| 6 | 5 | eleq2i 2832 | . . . 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 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 8517 | . . . . 5 ⊢ 1o ∈ V | |
| 15 | 14 | ovconst2 7614 | . . . 4 ⊢ ((𝑦 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑦((𝐵 × 𝐵) × {1o})𝑦) = 1o) | 
| 16 | 0lt1o 8543 | . . . . 5 ⊢ ∅ ∈ 1o | |
| 17 | eleq2 2829 | . . . . 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 7469 | . . . . . . 7 ⊢ (〈𝑥, 𝑦〉∅𝑧) = ∅ | |
| 23 | 22 | oveqi 7445 | . . . . . 6 ⊢ (𝑔(〈𝑥, 𝑦〉∅𝑧)𝑓) = (𝑔∅𝑓) | 
| 24 | 0ov 7469 | . . . . . 6 ⊢ (𝑔∅𝑓) = ∅ | |
| 25 | 23, 24 | eqtri 2764 | . . . . 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 7614 | . . . . 5 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) → (𝑥((𝐵 × 𝐵) × {1o})𝑧) = 1o) | 
| 28 | 27 | 3adant2 1131 | . . . 4 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) → (𝑥((𝐵 × 𝐵) × {1o})𝑧) = 1o) | 
| 29 | 21, 26, 28 | 3eltr4d 2855 | . . 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 49111 | 1 ⊢ (𝜑 → (𝐶 ∈ ThinCat ∧ (Id‘𝐶) = (𝑦 ∈ 𝐵 ↦ ∅))) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1086 = wceq 1539 ∈ wcel 2107 ∃*wmo 2537 ∅c0 4332 {csn 4625 〈cop 4631 ↦ cmpt 5224 × cxp 5682 ‘cfv 6560 (class class class)co 7432 1oc1o 8500 Basecbs 17248 Hom chom 17309 compcco 17310 Idccid 17709 ThinCatcthinc 49091 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-rep 5278 ax-sep 5295 ax-nul 5305 ax-pr 5431 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-ral 3061 df-rex 3070 df-rmo 3379 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-iun 4992 df-br 5143 df-opab 5205 df-mpt 5225 df-id 5577 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-suc 6389 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-riota 7389 df-ov 7435 df-1o 8507 df-cat 17712 df-cid 17713 df-thinc 49092 | 
| This theorem is referenced by: (None) | 
| Copyright terms: Public domain | W3C validator |