Theorem catprs2 48989

Description: A category equipped with the induced preorder, where an object 𝑥 is defined to be "less than or equal to" 𝑦 iff there is a morphism from 𝑥 to 𝑦, is a preordered set, or a proset. The category might not be thin. See catprsc 48990 and catprsc2 48991 for constructions satisfying the hypothesis "catprs.1". See catprs 48988 for a more primitive version. See prsthinc 49433 for constructing a thin category from a proset. (Contributed by Zhi Wang, 18-Sep-2024.)

Hypotheses

Ref	Expression
catprs.1	⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 ≤ 𝑦 ↔ (𝑥𝐻𝑦) ≠ ∅))
catprs.b	⊢ (𝜑 → 𝐵 = (Base‘𝐶))
catprs.h	⊢ (𝜑 → 𝐻 = (Hom ‘𝐶))
catprs.c	⊢ (𝜑 → 𝐶 ∈ Cat)
catprs2.l	⊢ (𝜑 → ≤ = (le‘𝐶))

Assertion

Ref	Expression
catprs2	⊢ (𝜑 → 𝐶 ∈ Proset )

Distinct variable groups:   𝑥, ≤ ,𝑦   𝑥,𝐵,𝑦   𝑥,𝐻,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑦)   𝐶(𝑥,𝑦)

Proof of Theorem catprs2

Dummy variables 𝑤 𝑢 𝑣 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		catprs.1	. . . 4 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 ≤ 𝑦 ↔ (𝑥𝐻𝑦) ≠ ∅))
2		catprs.b	. . . 4 ⊢ (𝜑 → 𝐵 = (Base‘𝐶))
3		catprs.h	. . . 4 ⊢ (𝜑 → 𝐻 = (Hom ‘𝐶))
4		catprs.c	. . . 4 ⊢ (𝜑 → 𝐶 ∈ Cat)
5	1, 2, 3, 4	catprs 48988	. . 3 ⊢ ((𝜑 ∧ (𝑤 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵 ∧ 𝑢 ∈ 𝐵)) → (𝑤 ≤ 𝑤 ∧ ((𝑤 ≤ 𝑣 ∧ 𝑣 ≤ 𝑢) → 𝑤 ≤ 𝑢)))
6	5	ralrimivvva 3184	. 2 ⊢ (𝜑 → ∀𝑤 ∈ 𝐵 ∀𝑣 ∈ 𝐵 ∀𝑢 ∈ 𝐵 (𝑤 ≤ 𝑤 ∧ ((𝑤 ≤ 𝑣 ∧ 𝑣 ≤ 𝑢) → 𝑤 ≤ 𝑢)))
7		catprs2.l	. . 3 ⊢ (𝜑 → ≤ = (le‘𝐶))
8	2, 7, 4	isprsd 48931	. 2 ⊢ (𝜑 → (𝐶 ∈ Proset ↔ ∀𝑤 ∈ 𝐵 ∀𝑣 ∈ 𝐵 ∀𝑢 ∈ 𝐵 (𝑤 ≤ 𝑤 ∧ ((𝑤 ≤ 𝑣 ∧ 𝑣 ≤ 𝑢) → 𝑤 ≤ 𝑢))))
9	6, 8	mpbird 257	1 ⊢ (𝜑 → 𝐶 ∈ Proset )

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109   ≠ wne 2926  ∀wral 3045  ∅c0 4298   class class class wbr 5109  ‘cfv 6513  (class class class)co 7389  Basecbs 17185  lecple 17233  Hom chom 17237  Catccat 17631   Proset cproset 18259

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-rep 5236  ax-sep 5253  ax-nul 5263  ax-pr 5389

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rmo 3356  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3756  df-csb 3865  df-dif 3919  df-un 3921  df-in 3923  df-ss 3933  df-nul 4299  df-if 4491  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4874  df-iun 4959  df-br 5110  df-opab 5172  df-mpt 5191  df-id 5535  df-xp 5646  df-rel 5647  df-cnv 5648  df-co 5649  df-dm 5650  df-rn 5651  df-res 5652  df-ima 5653  df-iota 6466  df-fun 6515  df-fn 6516  df-f 6517  df-f1 6518  df-fo 6519  df-f1o 6520  df-fv 6521  df-riota 7346  df-ov 7392  df-cat 17635  df-cid 17636  df-proset 18261

This theorem is referenced by: (None)