Theorem catprs2 47185

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 47186 and catprsc2 47187 for constructions satisfying the hypothesis "catprs.1". See catprs 47184 for a more primitive version. See prsthinc 47227 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 47184	. . 3 ⊢ ((𝜑 ∧ (𝑤 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵 ∧ 𝑢 ∈ 𝐵)) → (𝑤 ≤ 𝑤 ∧ ((𝑤 ≤ 𝑣 ∧ 𝑣 ≤ 𝑢) → 𝑤 ≤ 𝑢)))
6	5	ralrimivvva 3202	. 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 47141	. 2 ⊢ (𝜑 → (𝐶 ∈ Proset ↔ ∀𝑤 ∈ 𝐵 ∀𝑣 ∈ 𝐵 ∀𝑢 ∈ 𝐵 (𝑤 ≤ 𝑤 ∧ ((𝑤 ≤ 𝑣 ∧ 𝑣 ≤ 𝑢) → 𝑤 ≤ 𝑢))))
9	6, 8	mpbird 256	1 ⊢ (𝜑 → 𝐶 ∈ Proset )

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1541   ∈ wcel 2106   ≠ wne 2939  ∀wral 3060  ∅c0 4302   class class class wbr 5125  ‘cfv 6516  (class class class)co 7377  Basecbs 17109  lecple 17169  Hom chom 17173  Catccat 17573   Proset cproset 18211

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702  ax-rep 5262  ax-sep 5276  ax-nul 5283  ax-pr 5404

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rmo 3364  df-reu 3365  df-rab 3419  df-v 3461  df-sbc 3758  df-csb 3874  df-dif 3931  df-un 3933  df-in 3935  df-ss 3945  df-nul 4303  df-if 4507  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4886  df-iun 4976  df-br 5126  df-opab 5188  df-mpt 5209  df-id 5551  df-xp 5659  df-rel 5660  df-cnv 5661  df-co 5662  df-dm 5663  df-rn 5664  df-res 5665  df-ima 5666  df-iota 6468  df-fun 6518  df-fn 6519  df-f 6520  df-f1 6521  df-fo 6522  df-f1o 6523  df-fv 6524  df-riota 7333  df-ov 7380  df-cat 17577  df-cid 17578  df-proset 18213

This theorem is referenced by: (None)