Theorem prsthinc 45951

Description: Preordered sets as categories. Similar to example 3.3(4.d) of [Adamek] p. 24, but the hom-sets are not pairwise disjoint. One can define a functor from the category of prosets to the category of small thin categories. See catprs 45908 and catprs2 45909 for inducing a preorder from a category. Example 3.26(2) of [Adamek] p. 33 indicates that it induces a bijection from the equivalence class of isomorphic small thin categories to the equivalence class of order-isomorphic preordered sets. (Contributed by Zhi Wang, 18-Sep-2024.)

Hypotheses

Ref	Expression
indthinc.b	⊢ (𝜑 → 𝐵 = (Base‘𝐶))
prsthinc.h	⊢ (𝜑 → ( ≤ × {1o}) = (Hom ‘𝐶))
prsthinc.o	⊢ (𝜑 → ∅ = (comp‘𝐶))
prsthinc.l	⊢ (𝜑 → ≤ = (le‘𝐶))
prsthinc.p	⊢ (𝜑 → 𝐶 ∈ Proset )

Assertion

Ref	Expression
prsthinc	⊢ (𝜑 → (𝐶 ∈ ThinCat ∧ (Id‘𝐶) = (𝑦 ∈ 𝐵 ↦ ∅)))

Distinct variable groups:   𝑦, ≤   𝑦,𝐵   𝑦,𝐶   𝜑,𝑦

Proof of Theorem prsthinc

Dummy variables 𝑓 𝑔 𝑥 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		indthinc.b	. 2 ⊢ (𝜑 → 𝐵 = (Base‘𝐶))
2		prsthinc.h	. 2 ⊢ (𝜑 → ( ≤ × {1o}) = (Hom ‘𝐶))
3		eqidd 2737	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ( ≤ × {1o}) = ( ≤ × {1o}))
4	3	f1omo 45804	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ∃*𝑓 𝑓 ∈ (( ≤ × {1o})‘⟨𝑥, 𝑦⟩))
5		df-ov 7194	. . . . 5 ⊢ (𝑥( ≤ × {1o})𝑦) = (( ≤ × {1o})‘⟨𝑥, 𝑦⟩)
6	5	eleq2i 2822	. . . 4 ⊢ (𝑓 ∈ (𝑥( ≤ × {1o})𝑦) ↔ 𝑓 ∈ (( ≤ × {1o})‘⟨𝑥, 𝑦⟩))
7	6	mobii 2547	. . 3 ⊢ (∃*𝑓 𝑓 ∈ (𝑥( ≤ × {1o})𝑦) ↔ ∃*𝑓 𝑓 ∈ (( ≤ × {1o})‘⟨𝑥, 𝑦⟩))
8	4, 7	sylibr 237	. 2 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ∃*𝑓 𝑓 ∈ (𝑥( ≤ × {1o})𝑦))
9		prsthinc.o	. 2 ⊢ (𝜑 → ∅ = (comp‘𝐶))
10		prsthinc.p	. 2 ⊢ (𝜑 → 𝐶 ∈ Proset )
11		biid 264	. 2 ⊢ (((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑓 ∈ (𝑥( ≤ × {1o})𝑦) ∧ 𝑔 ∈ (𝑦( ≤ × {1o})𝑧))) ↔ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑓 ∈ (𝑥( ≤ × {1o})𝑦) ∧ 𝑔 ∈ (𝑦( ≤ × {1o})𝑧))))
12		0lt1o 8209	. . 3 ⊢ ∅ ∈ 1o
13	1	eleq2d 2816	. . . . . 6 ⊢ (𝜑 → (𝑦 ∈ 𝐵 ↔ 𝑦 ∈ (Base‘𝐶)))
14		eqid 2736	. . . . . . . 8 ⊢ (Base‘𝐶) = (Base‘𝐶)
15		eqid 2736	. . . . . . . 8 ⊢ (le‘𝐶) = (le‘𝐶)
16	14, 15	prsref 17760	. . . . . . 7 ⊢ ((𝐶 ∈ Proset ∧ 𝑦 ∈ (Base‘𝐶)) → 𝑦(le‘𝐶)𝑦)
17	10, 16	sylan 583	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ (Base‘𝐶)) → 𝑦(le‘𝐶)𝑦)
18	13, 17	sylbida 595	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → 𝑦(le‘𝐶)𝑦)
19		prsthinc.l	. . . . . . 7 ⊢ (𝜑 → ≤ = (le‘𝐶))
20	19	breqd 5050	. . . . . 6 ⊢ (𝜑 → (𝑦 ≤ 𝑦 ↔ 𝑦(le‘𝐶)𝑦))
21	20	biimpar 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑦(le‘𝐶)𝑦) → 𝑦 ≤ 𝑦)
22	18, 21	syldan 594	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → 𝑦 ≤ 𝑦)
23		eqidd 2737	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → ( ≤ × {1o}) = ( ≤ × {1o}))
24		1oex 8193	. . . . . 6 ⊢ 1o ∈ V
25	24	a1i 11	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → 1o ∈ V)
26		1n0 8199	. . . . . 6 ⊢ 1o ≠ ∅
27	26	a1i 11	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → 1o ≠ ∅)
28	23, 25, 27	fvconstr 45799	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → (𝑦 ≤ 𝑦 ↔ (𝑦( ≤ × {1o})𝑦) = 1o))
29	22, 28	mpbid 235	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → (𝑦( ≤ × {1o})𝑦) = 1o)
30	12, 29	eleqtrrid 2838	. 2 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → ∅ ∈ (𝑦( ≤ × {1o})𝑦))
31		0ov 7228	. . . . . 6 ⊢ (⟨𝑥, 𝑦⟩∅𝑧) = ∅
32	31	oveqi 7204	. . . . 5 ⊢ (𝑔(⟨𝑥, 𝑦⟩∅𝑧)𝑓) = (𝑔∅𝑓)
33		0ov 7228	. . . . 5 ⊢ (𝑔∅𝑓) = ∅
34	32, 33	eqtri 2759	. . . 4 ⊢ (𝑔(⟨𝑥, 𝑦⟩∅𝑧)𝑓) = ∅
35	34, 12	eqeltri 2827	. . 3 ⊢ (𝑔(⟨𝑥, 𝑦⟩∅𝑧)𝑓) ∈ 1o
36		simpl 486	. . . . 5 ⊢ ((𝜑 ∧ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑓 ∈ (𝑥( ≤ × {1o})𝑦) ∧ 𝑔 ∈ (𝑦( ≤ × {1o})𝑧)))) → 𝜑)
37	10	adantr 484	. . . . . 6 ⊢ ((𝜑 ∧ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑓 ∈ (𝑥( ≤ × {1o})𝑦) ∧ 𝑔 ∈ (𝑦( ≤ × {1o})𝑧)))) → 𝐶 ∈ Proset )
38	1	eleq2d 2816	. . . . . . . . 9 ⊢ (𝜑 → (𝑥 ∈ 𝐵 ↔ 𝑥 ∈ (Base‘𝐶)))
39	1	eleq2d 2816	. . . . . . . . 9 ⊢ (𝜑 → (𝑧 ∈ 𝐵 ↔ 𝑧 ∈ (Base‘𝐶)))
40	38, 13, 39	3anbi123d 1438	. . . . . . . 8 ⊢ (𝜑 → ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ↔ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶))))
41	40	biimpa 480	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)))
42	41	adantrr 717	. . . . . 6 ⊢ ((𝜑 ∧ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑓 ∈ (𝑥( ≤ × {1o})𝑦) ∧ 𝑔 ∈ (𝑦( ≤ × {1o})𝑧)))) → (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)))
43		eqidd 2737	. . . . . . . 8 ⊢ ((𝜑 ∧ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑓 ∈ (𝑥( ≤ × {1o})𝑦) ∧ 𝑔 ∈ (𝑦( ≤ × {1o})𝑧)))) → ( ≤ × {1o}) = ( ≤ × {1o}))
44		simprrl 781	. . . . . . . 8 ⊢ ((𝜑 ∧ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑓 ∈ (𝑥( ≤ × {1o})𝑦) ∧ 𝑔 ∈ (𝑦( ≤ × {1o})𝑧)))) → 𝑓 ∈ (𝑥( ≤ × {1o})𝑦))
45	43, 44	fvconstr2 45801	. . . . . . 7 ⊢ ((𝜑 ∧ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑓 ∈ (𝑥( ≤ × {1o})𝑦) ∧ 𝑔 ∈ (𝑦( ≤ × {1o})𝑧)))) → 𝑥 ≤ 𝑦)
46	19	breqd 5050	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ≤ 𝑦 ↔ 𝑥(le‘𝐶)𝑦))
47	46	biimpd 232	. . . . . . 7 ⊢ (𝜑 → (𝑥 ≤ 𝑦 → 𝑥(le‘𝐶)𝑦))
48	36, 45, 47	sylc 65	. . . . . 6 ⊢ ((𝜑 ∧ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑓 ∈ (𝑥( ≤ × {1o})𝑦) ∧ 𝑔 ∈ (𝑦( ≤ × {1o})𝑧)))) → 𝑥(le‘𝐶)𝑦)
49		simprrr 782	. . . . . . . 8 ⊢ ((𝜑 ∧ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑓 ∈ (𝑥( ≤ × {1o})𝑦) ∧ 𝑔 ∈ (𝑦( ≤ × {1o})𝑧)))) → 𝑔 ∈ (𝑦( ≤ × {1o})𝑧))
50	43, 49	fvconstr2 45801	. . . . . . 7 ⊢ ((𝜑 ∧ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑓 ∈ (𝑥( ≤ × {1o})𝑦) ∧ 𝑔 ∈ (𝑦( ≤ × {1o})𝑧)))) → 𝑦 ≤ 𝑧)
51	19	breqd 5050	. . . . . . . 8 ⊢ (𝜑 → (𝑦 ≤ 𝑧 ↔ 𝑦(le‘𝐶)𝑧))
52	51	biimpd 232	. . . . . . 7 ⊢ (𝜑 → (𝑦 ≤ 𝑧 → 𝑦(le‘𝐶)𝑧))
53	36, 50, 52	sylc 65	. . . . . 6 ⊢ ((𝜑 ∧ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑓 ∈ (𝑥( ≤ × {1o})𝑦) ∧ 𝑔 ∈ (𝑦( ≤ × {1o})𝑧)))) → 𝑦(le‘𝐶)𝑧)
54	14, 15	prstr 17761	. . . . . 6 ⊢ ((𝐶 ∈ Proset ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑥(le‘𝐶)𝑦 ∧ 𝑦(le‘𝐶)𝑧)) → 𝑥(le‘𝐶)𝑧)
55	37, 42, 48, 53, 54	syl112anc 1376	. . . . 5 ⊢ ((𝜑 ∧ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑓 ∈ (𝑥( ≤ × {1o})𝑦) ∧ 𝑔 ∈ (𝑦( ≤ × {1o})𝑧)))) → 𝑥(le‘𝐶)𝑧)
56	19	breqd 5050	. . . . . 6 ⊢ (𝜑 → (𝑥 ≤ 𝑧 ↔ 𝑥(le‘𝐶)𝑧))
57	56	biimprd 251	. . . . 5 ⊢ (𝜑 → (𝑥(le‘𝐶)𝑧 → 𝑥 ≤ 𝑧))
58	36, 55, 57	sylc 65	. . . 4 ⊢ ((𝜑 ∧ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑓 ∈ (𝑥( ≤ × {1o})𝑦) ∧ 𝑔 ∈ (𝑦( ≤ × {1o})𝑧)))) → 𝑥 ≤ 𝑧)
59	24	a1i 11	. . . . 5 ⊢ ((𝜑 ∧ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑓 ∈ (𝑥( ≤ × {1o})𝑦) ∧ 𝑔 ∈ (𝑦( ≤ × {1o})𝑧)))) → 1o ∈ V)
60	26	a1i 11	. . . . 5 ⊢ ((𝜑 ∧ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑓 ∈ (𝑥( ≤ × {1o})𝑦) ∧ 𝑔 ∈ (𝑦( ≤ × {1o})𝑧)))) → 1o ≠ ∅)
61	43, 59, 60	fvconstr 45799	. . . 4 ⊢ ((𝜑 ∧ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑓 ∈ (𝑥( ≤ × {1o})𝑦) ∧ 𝑔 ∈ (𝑦( ≤ × {1o})𝑧)))) → (𝑥 ≤ 𝑧 ↔ (𝑥( ≤ × {1o})𝑧) = 1o))
62	58, 61	mpbid 235	. . 3 ⊢ ((𝜑 ∧ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑓 ∈ (𝑥( ≤ × {1o})𝑦) ∧ 𝑔 ∈ (𝑦( ≤ × {1o})𝑧)))) → (𝑥( ≤ × {1o})𝑧) = 1o)
63	35, 62	eleqtrrid 2838	. 2 ⊢ ((𝜑 ∧ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑓 ∈ (𝑥( ≤ × {1o})𝑦) ∧ 𝑔 ∈ (𝑦( ≤ × {1o})𝑧)))) → (𝑔(⟨𝑥, 𝑦⟩∅𝑧)𝑓) ∈ (𝑥( ≤ × {1o})𝑧))
64	1, 2, 8, 9, 10, 11, 30, 63	isthincd2 45935	1 ⊢ (𝜑 → (𝐶 ∈ ThinCat ∧ (Id‘𝐶) = (𝑦 ∈ 𝐵 ↦ ∅)))

Syntax hints:   → wi 4   ∧ wa 399   ∧ w3a 1089   = wceq 1543   ∈ wcel 2112  ∃*wmo 2537   ≠ wne 2932  Vcvv 3398  ∅c0 4223  {csn 4527  ⟨cop 4533   class class class wbr 5039   ↦ cmpt 5120   × cxp 5534  ‘cfv 6358  (class class class)co 7191  1_oc1o 8173  Basecbs 16666  lecple 16756  Hom chom 16760  compcco 16761  Idccid 17122   Proset cproset 17754  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-proset 17756  df-thinc 45917

This theorem is referenced by:  prstcthin  45971