Theorem catprs 48800

Description: A preorder can be extracted from a category. See catprs2 48801 for more details. (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)

Assertion

Ref	Expression
catprs	⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (𝑋 ≤ 𝑋 ∧ ((𝑋 ≤ 𝑌 ∧ 𝑌 ≤ 𝑍) → 𝑋 ≤ 𝑍)))

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

Allowed substitution hints:   𝜑(𝑥,𝑦)   𝐶(𝑥,𝑦)   𝑋(𝑥,𝑦)   𝑌(𝑥,𝑦)   𝑍(𝑥,𝑦)

Proof of Theorem catprs

Step	Hyp	Ref	Expression
1		eqid 2735	. . . . . 6 ⊢ (Base‘𝐶) = (Base‘𝐶)
2		eqid 2735	. . . . . 6 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
3		eqid 2735	. . . . . 6 ⊢ (Id‘𝐶) = (Id‘𝐶)
4		catprs.c	. . . . . . 7 ⊢ (𝜑 → 𝐶 ∈ Cat)
5	4	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → 𝐶 ∈ Cat)
6		simpr1 1193	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → 𝑋 ∈ 𝐵)
7		catprs.b	. . . . . . . 8 ⊢ (𝜑 → 𝐵 = (Base‘𝐶))
8	7	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → 𝐵 = (Base‘𝐶))
9	6, 8	eleqtrd 2841	. . . . . 6 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → 𝑋 ∈ (Base‘𝐶))
10	1, 2, 3, 5, 9	catidcl 17727	. . . . 5 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((Id‘𝐶)‘𝑋) ∈ (𝑋(Hom ‘𝐶)𝑋))
11		catprs.h	. . . . . . 7 ⊢ (𝜑 → 𝐻 = (Hom ‘𝐶))
12	11	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → 𝐻 = (Hom ‘𝐶))
13	12	oveqd 7448	. . . . 5 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (𝑋𝐻𝑋) = (𝑋(Hom ‘𝐶)𝑋))
14	10, 13	eleqtrrd 2842	. . . 4 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((Id‘𝐶)‘𝑋) ∈ (𝑋𝐻𝑋))
15	14	ne0d 4348	. . 3 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (𝑋𝐻𝑋) ≠ ∅)
16		catprs.1	. . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 ≤ 𝑦 ↔ (𝑥𝐻𝑦) ≠ ∅))
17	16	adantr 480	. . . 4 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 ≤ 𝑦 ↔ (𝑥𝐻𝑦) ≠ ∅))
18	17, 6, 6	catprslem 48799	. . 3 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (𝑋 ≤ 𝑋 ↔ (𝑋𝐻𝑋) ≠ ∅))
19	15, 18	mpbird 257	. 2 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → 𝑋 ≤ 𝑋)
20	11	ad2antrr 726	. . . . . 6 ⊢ (((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) ∧ (𝑋 ≤ 𝑌 ∧ 𝑌 ≤ 𝑍)) → 𝐻 = (Hom ‘𝐶))
21	20	oveqd 7448	. . . . 5 ⊢ (((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) ∧ (𝑋 ≤ 𝑌 ∧ 𝑌 ≤ 𝑍)) → (𝑋𝐻𝑍) = (𝑋(Hom ‘𝐶)𝑍))
22	7	eleq2d 2825	. . . . . . . 8 ⊢ (𝜑 → (𝑋 ∈ 𝐵 ↔ 𝑋 ∈ (Base‘𝐶)))
23	7	eleq2d 2825	. . . . . . . 8 ⊢ (𝜑 → (𝑌 ∈ 𝐵 ↔ 𝑌 ∈ (Base‘𝐶)))
24	7	eleq2d 2825	. . . . . . . 8 ⊢ (𝜑 → (𝑍 ∈ 𝐵 ↔ 𝑍 ∈ (Base‘𝐶)))
25	22, 23, 24	3anbi123d 1435	. . . . . . 7 ⊢ (𝜑 → ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ↔ (𝑋 ∈ (Base‘𝐶) ∧ 𝑌 ∈ (Base‘𝐶) ∧ 𝑍 ∈ (Base‘𝐶))))
26	25	pm5.32i 574	. . . . . 6 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) ↔ (𝜑 ∧ (𝑋 ∈ (Base‘𝐶) ∧ 𝑌 ∈ (Base‘𝐶) ∧ 𝑍 ∈ (Base‘𝐶))))
27		eqid 2735	. . . . . . 7 ⊢ (comp‘𝐶) = (comp‘𝐶)
28	4	ad2antrr 726	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑋 ∈ (Base‘𝐶) ∧ 𝑌 ∈ (Base‘𝐶) ∧ 𝑍 ∈ (Base‘𝐶))) ∧ (𝑋 ≤ 𝑌 ∧ 𝑌 ≤ 𝑍)) → 𝐶 ∈ Cat)
29		simplr1 1214	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑋 ∈ (Base‘𝐶) ∧ 𝑌 ∈ (Base‘𝐶) ∧ 𝑍 ∈ (Base‘𝐶))) ∧ (𝑋 ≤ 𝑌 ∧ 𝑌 ≤ 𝑍)) → 𝑋 ∈ (Base‘𝐶))
30		simplr2 1215	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑋 ∈ (Base‘𝐶) ∧ 𝑌 ∈ (Base‘𝐶) ∧ 𝑍 ∈ (Base‘𝐶))) ∧ (𝑋 ≤ 𝑌 ∧ 𝑌 ≤ 𝑍)) → 𝑌 ∈ (Base‘𝐶))
31		simplr3 1216	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑋 ∈ (Base‘𝐶) ∧ 𝑌 ∈ (Base‘𝐶) ∧ 𝑍 ∈ (Base‘𝐶))) ∧ (𝑋 ≤ 𝑌 ∧ 𝑌 ≤ 𝑍)) → 𝑍 ∈ (Base‘𝐶))
32	20	oveqd 7448	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) ∧ (𝑋 ≤ 𝑌 ∧ 𝑌 ≤ 𝑍)) → (𝑋𝐻𝑌) = (𝑋(Hom ‘𝐶)𝑌))
33		simpr2 1194	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → 𝑌 ∈ 𝐵)
34	17, 6, 33	catprslem 48799	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (𝑋 ≤ 𝑌 ↔ (𝑋𝐻𝑌) ≠ ∅))
35	34	biimpa 476	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) ∧ 𝑋 ≤ 𝑌) → (𝑋𝐻𝑌) ≠ ∅)
36	35	adantrr 717	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) ∧ (𝑋 ≤ 𝑌 ∧ 𝑌 ≤ 𝑍)) → (𝑋𝐻𝑌) ≠ ∅)
37	32, 36	eqnetrrd 3007	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) ∧ (𝑋 ≤ 𝑌 ∧ 𝑌 ≤ 𝑍)) → (𝑋(Hom ‘𝐶)𝑌) ≠ ∅)
38	26, 37	sylanbr 582	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑋 ∈ (Base‘𝐶) ∧ 𝑌 ∈ (Base‘𝐶) ∧ 𝑍 ∈ (Base‘𝐶))) ∧ (𝑋 ≤ 𝑌 ∧ 𝑌 ≤ 𝑍)) → (𝑋(Hom ‘𝐶)𝑌) ≠ ∅)
39	20	oveqd 7448	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) ∧ (𝑋 ≤ 𝑌 ∧ 𝑌 ≤ 𝑍)) → (𝑌𝐻𝑍) = (𝑌(Hom ‘𝐶)𝑍))
40		simpr3 1195	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → 𝑍 ∈ 𝐵)
41	17, 33, 40	catprslem 48799	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (𝑌 ≤ 𝑍 ↔ (𝑌𝐻𝑍) ≠ ∅))
42	41	biimpa 476	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) ∧ 𝑌 ≤ 𝑍) → (𝑌𝐻𝑍) ≠ ∅)
43	42	adantrl 716	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) ∧ (𝑋 ≤ 𝑌 ∧ 𝑌 ≤ 𝑍)) → (𝑌𝐻𝑍) ≠ ∅)
44	39, 43	eqnetrrd 3007	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) ∧ (𝑋 ≤ 𝑌 ∧ 𝑌 ≤ 𝑍)) → (𝑌(Hom ‘𝐶)𝑍) ≠ ∅)
45	26, 44	sylanbr 582	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑋 ∈ (Base‘𝐶) ∧ 𝑌 ∈ (Base‘𝐶) ∧ 𝑍 ∈ (Base‘𝐶))) ∧ (𝑋 ≤ 𝑌 ∧ 𝑌 ≤ 𝑍)) → (𝑌(Hom ‘𝐶)𝑍) ≠ ∅)
46	1, 2, 27, 28, 29, 30, 31, 38, 45	catcone0 17732	. . . . . 6 ⊢ (((𝜑 ∧ (𝑋 ∈ (Base‘𝐶) ∧ 𝑌 ∈ (Base‘𝐶) ∧ 𝑍 ∈ (Base‘𝐶))) ∧ (𝑋 ≤ 𝑌 ∧ 𝑌 ≤ 𝑍)) → (𝑋(Hom ‘𝐶)𝑍) ≠ ∅)
47	26, 46	sylanb 581	. . . . 5 ⊢ (((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) ∧ (𝑋 ≤ 𝑌 ∧ 𝑌 ≤ 𝑍)) → (𝑋(Hom ‘𝐶)𝑍) ≠ ∅)
48	21, 47	eqnetrd 3006	. . . 4 ⊢ (((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) ∧ (𝑋 ≤ 𝑌 ∧ 𝑌 ≤ 𝑍)) → (𝑋𝐻𝑍) ≠ ∅)
49	17, 6, 40	catprslem 48799	. . . . 5 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (𝑋 ≤ 𝑍 ↔ (𝑋𝐻𝑍) ≠ ∅))
50	49	adantr 480	. . . 4 ⊢ (((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) ∧ (𝑋 ≤ 𝑌 ∧ 𝑌 ≤ 𝑍)) → (𝑋 ≤ 𝑍 ↔ (𝑋𝐻𝑍) ≠ ∅))
51	48, 50	mpbird 257	. . 3 ⊢ (((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) ∧ (𝑋 ≤ 𝑌 ∧ 𝑌 ≤ 𝑍)) → 𝑋 ≤ 𝑍)
52	51	ex 412	. 2 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((𝑋 ≤ 𝑌 ∧ 𝑌 ≤ 𝑍) → 𝑋 ≤ 𝑍))
53	19, 52	jca 511	1 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (𝑋 ≤ 𝑋 ∧ ((𝑋 ≤ 𝑌 ∧ 𝑌 ≤ 𝑍) → 𝑋 ≤ 𝑍)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1537   ∈ wcel 2106   ≠ wne 2938  ∀wral 3059  ∅c0 4339   class class class wbr 5148  ‘cfv 6563  (class class class)co 7431  Basecbs 17245  Hom chom 17309  compcco 17310  Catccat 17709  Idccid 17710

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pr 5438

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-cat 17713  df-cid 17714

This theorem is referenced by:  catprs2  48801