Theorem prsthinc 50085

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 49632 and catprs2 49633 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 2763	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ( ≤ × {1o}) = ( ≤ × {1o}))
4	3	f1omo 49514	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ∃*𝑓 𝑓 ∈ (( ≤ × {1o})‘⟨𝑥, 𝑦⟩))
5		df-ov 7399	. . . . 5 ⊢ (𝑥( ≤ × {1o})𝑦) = (( ≤ × {1o})‘⟨𝑥, 𝑦⟩)
6	5	eleq2i 2854	. . . 4 ⊢ (𝑓 ∈ (𝑥( ≤ × {1o})𝑦) ↔ 𝑓 ∈ (( ≤ × {1o})‘⟨𝑥, 𝑦⟩))
7	6	mobii 2575	. . 3 ⊢ (∃*𝑓 𝑓 ∈ (𝑥( ≤ × {1o})𝑦) ↔ ∃*𝑓 𝑓 ∈ (( ≤ × {1o})‘⟨𝑥, 𝑦⟩))
8	4, 7	sylibr 236	. 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 263	. 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 8473	. . 3 ⊢ ∅ ∈ 1o
13	1	eleq2d 2848	. . . . . 6 ⊢ (𝜑 → (𝑦 ∈ 𝐵 ↔ 𝑦 ∈ (Base‘𝐶)))
14		eqid 2762	. . . . . . . 8 ⊢ (Base‘𝐶) = (Base‘𝐶)
15		eqid 2762	. . . . . . . 8 ⊢ (le‘𝐶) = (le‘𝐶)
16	14, 15	prsref 18330	. . . . . . 7 ⊢ ((𝐶 ∈ Proset ∧ 𝑦 ∈ (Base‘𝐶)) → 𝑦(le‘𝐶)𝑦)
17	10, 16	sylan 589	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ (Base‘𝐶)) → 𝑦(le‘𝐶)𝑦)
18	13, 17	sylbida 601	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → 𝑦(le‘𝐶)𝑦)
19		prsthinc.l	. . . . . . 7 ⊢ (𝜑 → ≤ = (le‘𝐶))
20	19	breqd 5111	. . . . . 6 ⊢ (𝜑 → (𝑦 ≤ 𝑦 ↔ 𝑦(le‘𝐶)𝑦))
21	20	biimpar 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑦(le‘𝐶)𝑦) → 𝑦 ≤ 𝑦)
22	18, 21	syldan 600	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → 𝑦 ≤ 𝑦)
23		eqidd 2763	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → ( ≤ × {1o}) = ( ≤ × {1o}))
24		1oex 8447	. . . . . 6 ⊢ 1o ∈ V
25	24	a1i 11	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → 1o ∈ V)
26		1n0 8456	. . . . . 6 ⊢ 1o ≠ ∅
27	26	a1i 11	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → 1o ≠ ∅)
28	23, 25, 27	fvconstr 49483	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → (𝑦 ≤ 𝑦 ↔ (𝑦( ≤ × {1o})𝑦) = 1o))
29	22, 28	mpbid 234	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → (𝑦( ≤ × {1o})𝑦) = 1o)
30	12, 29	eleqtrrid 2869	. 2 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → ∅ ∈ (𝑦( ≤ × {1o})𝑦))
31		0ov 7433	. . . . . 6 ⊢ (⟨𝑥, 𝑦⟩∅𝑧) = ∅
32	31	oveqi 7409	. . . . 5 ⊢ (𝑔(⟨𝑥, 𝑦⟩∅𝑧)𝑓) = (𝑔∅𝑓)
33		0ov 7433	. . . . 5 ⊢ (𝑔∅𝑓) = ∅
34	32, 33	eqtri 2785	. . . 4 ⊢ (𝑔(⟨𝑥, 𝑦⟩∅𝑧)𝑓) = ∅
35	34, 12	eqeltri 2858	. . 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 2848	. . . . . . . . 9 ⊢ (𝜑 → (𝑥 ∈ 𝐵 ↔ 𝑥 ∈ (Base‘𝐶)))
39	1	eleq2d 2848	. . . . . . . . 9 ⊢ (𝜑 → (𝑧 ∈ 𝐵 ↔ 𝑧 ∈ (Base‘𝐶)))
40	38, 13, 39	3anbi123d 1457	. . . . . . . 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 727	. . . . . 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 2763	. . . . . . . 8 ⊢ ((𝜑 ∧ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑓 ∈ (𝑥( ≤ × {1o})𝑦) ∧ 𝑔 ∈ (𝑦( ≤ × {1o})𝑧)))) → ( ≤ × {1o}) = ( ≤ × {1o}))
44		simprrl 790	. . . . . . . 8 ⊢ ((𝜑 ∧ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑓 ∈ (𝑥( ≤ × {1o})𝑦) ∧ 𝑔 ∈ (𝑦( ≤ × {1o})𝑧)))) → 𝑓 ∈ (𝑥( ≤ × {1o})𝑦))
45	43, 44	fvconstr2 49485	. . . . . . 7 ⊢ ((𝜑 ∧ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑓 ∈ (𝑥( ≤ × {1o})𝑦) ∧ 𝑔 ∈ (𝑦( ≤ × {1o})𝑧)))) → 𝑥 ≤ 𝑦)
46	19	breqd 5111	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ≤ 𝑦 ↔ 𝑥(le‘𝐶)𝑦))
47	46	biimpd 231	. . . . . . 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 791	. . . . . . . 8 ⊢ ((𝜑 ∧ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑓 ∈ (𝑥( ≤ × {1o})𝑦) ∧ 𝑔 ∈ (𝑦( ≤ × {1o})𝑧)))) → 𝑔 ∈ (𝑦( ≤ × {1o})𝑧))
50	43, 49	fvconstr2 49485	. . . . . . 7 ⊢ ((𝜑 ∧ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑓 ∈ (𝑥( ≤ × {1o})𝑦) ∧ 𝑔 ∈ (𝑦( ≤ × {1o})𝑧)))) → 𝑦 ≤ 𝑧)
51	19	breqd 5111	. . . . . . . 8 ⊢ (𝜑 → (𝑦 ≤ 𝑧 ↔ 𝑦(le‘𝐶)𝑧))
52	51	biimpd 231	. . . . . . 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 18331	. . . . . 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 1393	. . . . 5 ⊢ ((𝜑 ∧ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑓 ∈ (𝑥( ≤ × {1o})𝑦) ∧ 𝑔 ∈ (𝑦( ≤ × {1o})𝑧)))) → 𝑥(le‘𝐶)𝑧)
56	19	breqd 5111	. . . . . 6 ⊢ (𝜑 → (𝑥 ≤ 𝑧 ↔ 𝑥(le‘𝐶)𝑧))
57	56	biimprd 250	. . . . 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 49483	. . . 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 234	. . 3 ⊢ ((𝜑 ∧ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑓 ∈ (𝑥( ≤ × {1o})𝑦) ∧ 𝑔 ∈ (𝑦( ≤ × {1o})𝑧)))) → (𝑥( ≤ × {1o})𝑧) = 1o)
63	35, 62	eleqtrrid 2869	. 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 50058	1 ⊢ (𝜑 → (𝐶 ∈ ThinCat ∧ (Id‘𝐶) = (𝑦 ∈ 𝐵 ↦ ∅)))

Syntax hints:   → wi 4   ∧ wa 399   ∧ w3a 1098   = wceq 1560   ∈ wcel 2142  ∃*wmo 2564   ≠ wne 2957  Vcvv 3454  ∅c0 4285  {csn 4582  ⟨cop 4588   class class class wbr 5100   ↦ cmpt 5181   × cxp 5645  ‘cfv 6521  (class class class)co 7396  1_oc1o 8430  Basecbs 17245  lecple 17293  Hom chom 17297  compcco 17298  Idccid 17697   Proset cproset 18324  ThinCatcthinc 50038

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-rep 5227  ax-sep 5246  ax-nul 5256  ax-pr 5390

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-nf 1804  df-sb 2091  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3077  df-rex 3087  df-rmo 3367  df-reu 3368  df-rab 3415  df-v 3456  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4481  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4951  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5542  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-suc 6352  df-iota 6477  df-fun 6523  df-fn 6524  df-f 6525  df-f1 6526  df-fo 6527  df-f1o 6528  df-fv 6529  df-riota 7353  df-ov 7399  df-1o 8437  df-cat 17700  df-cid 17701  df-proset 18326  df-thinc 50039

This theorem is referenced by:  prstcthin  50182