Theorem prsthinc 49954

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 49501 and catprs2 49502 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 2738	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ( ≤ × {1o}) = ( ≤ × {1o}))
4	3	f1omo 49383	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ∃*𝑓 𝑓 ∈ (( ≤ × {1o})‘⟨𝑥, 𝑦⟩))
5		df-ov 7364	. . . . 5 ⊢ (𝑥( ≤ × {1o})𝑦) = (( ≤ × {1o})‘⟨𝑥, 𝑦⟩)
6	5	eleq2i 2829	. . . 4 ⊢ (𝑓 ∈ (𝑥( ≤ × {1o})𝑦) ↔ 𝑓 ∈ (( ≤ × {1o})‘⟨𝑥, 𝑦⟩))
7	6	mobii 2549	. . 3 ⊢ (∃*𝑓 𝑓 ∈ (𝑥( ≤ × {1o})𝑦) ↔ ∃*𝑓 𝑓 ∈ (( ≤ × {1o})‘⟨𝑥, 𝑦⟩))
8	4, 7	sylibr 234	. 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 261	. 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 8433	. . 3 ⊢ ∅ ∈ 1o
13	1	eleq2d 2823	. . . . . 6 ⊢ (𝜑 → (𝑦 ∈ 𝐵 ↔ 𝑦 ∈ (Base‘𝐶)))
14		eqid 2737	. . . . . . . 8 ⊢ (Base‘𝐶) = (Base‘𝐶)
15		eqid 2737	. . . . . . . 8 ⊢ (le‘𝐶) = (le‘𝐶)
16	14, 15	prsref 18258	. . . . . . 7 ⊢ ((𝐶 ∈ Proset ∧ 𝑦 ∈ (Base‘𝐶)) → 𝑦(le‘𝐶)𝑦)
17	10, 16	sylan 581	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ (Base‘𝐶)) → 𝑦(le‘𝐶)𝑦)
18	13, 17	sylbida 593	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → 𝑦(le‘𝐶)𝑦)
19		prsthinc.l	. . . . . . 7 ⊢ (𝜑 → ≤ = (le‘𝐶))
20	19	breqd 5097	. . . . . 6 ⊢ (𝜑 → (𝑦 ≤ 𝑦 ↔ 𝑦(le‘𝐶)𝑦))
21	20	biimpar 477	. . . . 5 ⊢ ((𝜑 ∧ 𝑦(le‘𝐶)𝑦) → 𝑦 ≤ 𝑦)
22	18, 21	syldan 592	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → 𝑦 ≤ 𝑦)
23		eqidd 2738	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → ( ≤ × {1o}) = ( ≤ × {1o}))
24		1oex 8409	. . . . . 6 ⊢ 1o ∈ V
25	24	a1i 11	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → 1o ∈ V)
26		1n0 8417	. . . . . 6 ⊢ 1o ≠ ∅
27	26	a1i 11	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → 1o ≠ ∅)
28	23, 25, 27	fvconstr 49352	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → (𝑦 ≤ 𝑦 ↔ (𝑦( ≤ × {1o})𝑦) = 1o))
29	22, 28	mpbid 232	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → (𝑦( ≤ × {1o})𝑦) = 1o)
30	12, 29	eleqtrrid 2844	. 2 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → ∅ ∈ (𝑦( ≤ × {1o})𝑦))
31		0ov 7398	. . . . . 6 ⊢ (⟨𝑥, 𝑦⟩∅𝑧) = ∅
32	31	oveqi 7374	. . . . 5 ⊢ (𝑔(⟨𝑥, 𝑦⟩∅𝑧)𝑓) = (𝑔∅𝑓)
33		0ov 7398	. . . . 5 ⊢ (𝑔∅𝑓) = ∅
34	32, 33	eqtri 2760	. . . 4 ⊢ (𝑔(⟨𝑥, 𝑦⟩∅𝑧)𝑓) = ∅
35	34, 12	eqeltri 2833	. . 3 ⊢ (𝑔(⟨𝑥, 𝑦⟩∅𝑧)𝑓) ∈ 1o
36		simpl 482	. . . . 5 ⊢ ((𝜑 ∧ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑓 ∈ (𝑥( ≤ × {1o})𝑦) ∧ 𝑔 ∈ (𝑦( ≤ × {1o})𝑧)))) → 𝜑)
37	10	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑓 ∈ (𝑥( ≤ × {1o})𝑦) ∧ 𝑔 ∈ (𝑦( ≤ × {1o})𝑧)))) → 𝐶 ∈ Proset )
38	1	eleq2d 2823	. . . . . . . . 9 ⊢ (𝜑 → (𝑥 ∈ 𝐵 ↔ 𝑥 ∈ (Base‘𝐶)))
39	1	eleq2d 2823	. . . . . . . . 9 ⊢ (𝜑 → (𝑧 ∈ 𝐵 ↔ 𝑧 ∈ (Base‘𝐶)))
40	38, 13, 39	3anbi123d 1439	. . . . . . . 8 ⊢ (𝜑 → ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ↔ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶))))
41	40	biimpa 476	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)))
42	41	adantrr 718	. . . . . 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 2738	. . . . . . . 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 49354	. . . . . . 7 ⊢ ((𝜑 ∧ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑓 ∈ (𝑥( ≤ × {1o})𝑦) ∧ 𝑔 ∈ (𝑦( ≤ × {1o})𝑧)))) → 𝑥 ≤ 𝑦)
46	19	breqd 5097	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ≤ 𝑦 ↔ 𝑥(le‘𝐶)𝑦))
47	46	biimpd 229	. . . . . . 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 49354	. . . . . . 7 ⊢ ((𝜑 ∧ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑓 ∈ (𝑥( ≤ × {1o})𝑦) ∧ 𝑔 ∈ (𝑦( ≤ × {1o})𝑧)))) → 𝑦 ≤ 𝑧)
51	19	breqd 5097	. . . . . . . 8 ⊢ (𝜑 → (𝑦 ≤ 𝑧 ↔ 𝑦(le‘𝐶)𝑧))
52	51	biimpd 229	. . . . . . 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 18259	. . . . . 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 1377	. . . . 5 ⊢ ((𝜑 ∧ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑓 ∈ (𝑥( ≤ × {1o})𝑦) ∧ 𝑔 ∈ (𝑦( ≤ × {1o})𝑧)))) → 𝑥(le‘𝐶)𝑧)
56	19	breqd 5097	. . . . . 6 ⊢ (𝜑 → (𝑥 ≤ 𝑧 ↔ 𝑥(le‘𝐶)𝑧))
57	56	biimprd 248	. . . . 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 49352	. . . 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 232	. . 3 ⊢ ((𝜑 ∧ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑓 ∈ (𝑥( ≤ × {1o})𝑦) ∧ 𝑔 ∈ (𝑦( ≤ × {1o})𝑧)))) → (𝑥( ≤ × {1o})𝑧) = 1o)
63	35, 62	eleqtrrid 2844	. 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 49927	1 ⊢ (𝜑 → (𝐶 ∈ ThinCat ∧ (Id‘𝐶) = (𝑦 ∈ 𝐵 ↦ ∅)))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114  ∃*wmo 2538   ≠ wne 2933  Vcvv 3430  ∅c0 4274  {csn 4568  ⟨cop 4574   class class class wbr 5086   ↦ cmpt 5167   × cxp 5623  ‘cfv 6493  (class class class)co 7361  1_oc1o 8392  Basecbs 17173  lecple 17221  Hom chom 17225  compcco 17226  Idccid 17625   Proset cproset 18252  ThinCatcthinc 49907

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5213  ax-sep 5232  ax-nul 5242  ax-pr 5371

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rmo 3343  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-suc 6324  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-riota 7318  df-ov 7364  df-1o 8399  df-cat 17628  df-cid 17629  df-proset 18254  df-thinc 49908

This theorem is referenced by:  prstcthin  50051