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Theorem catprs 48900
Description: A preorder can be extracted from a category. See catprs2 48901 for more details. (Contributed by Zhi Wang, 18-Sep-2024.)
Hypotheses
Ref Expression
catprs.1 (𝜑 → ∀𝑥𝐵𝑦𝐵 (𝑥 𝑦 ↔ (𝑥𝐻𝑦) ≠ ∅))
catprs.b (𝜑𝐵 = (Base‘𝐶))
catprs.h (𝜑𝐻 = (Hom ‘𝐶))
catprs.c (𝜑𝐶 ∈ Cat)
Assertion
Ref Expression
catprs ((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋 𝑋 ∧ ((𝑋 𝑌𝑌 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
StepHypRef Expression
1 eqid 2737 . . . . . 6 (Base‘𝐶) = (Base‘𝐶)
2 eqid 2737 . . . . . 6 (Hom ‘𝐶) = (Hom ‘𝐶)
3 eqid 2737 . . . . . 6 (Id‘𝐶) = (Id‘𝐶)
4 catprs.c . . . . . . 7 (𝜑𝐶 ∈ Cat)
54adantr 480 . . . . . 6 ((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝐶 ∈ Cat)
6 simpr1 1195 . . . . . . 7 ((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝑋𝐵)
7 catprs.b . . . . . . . 8 (𝜑𝐵 = (Base‘𝐶))
87adantr 480 . . . . . . 7 ((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝐵 = (Base‘𝐶))
96, 8eleqtrd 2843 . . . . . 6 ((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝑋 ∈ (Base‘𝐶))
101, 2, 3, 5, 9catidcl 17725 . . . . 5 ((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((Id‘𝐶)‘𝑋) ∈ (𝑋(Hom ‘𝐶)𝑋))
11 catprs.h . . . . . . 7 (𝜑𝐻 = (Hom ‘𝐶))
1211adantr 480 . . . . . 6 ((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝐻 = (Hom ‘𝐶))
1312oveqd 7448 . . . . 5 ((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋𝐻𝑋) = (𝑋(Hom ‘𝐶)𝑋))
1410, 13eleqtrrd 2844 . . . 4 ((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((Id‘𝐶)‘𝑋) ∈ (𝑋𝐻𝑋))
1514ne0d 4342 . . 3 ((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋𝐻𝑋) ≠ ∅)
16 catprs.1 . . . . 5 (𝜑 → ∀𝑥𝐵𝑦𝐵 (𝑥 𝑦 ↔ (𝑥𝐻𝑦) ≠ ∅))
1716adantr 480 . . . 4 ((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ∀𝑥𝐵𝑦𝐵 (𝑥 𝑦 ↔ (𝑥𝐻𝑦) ≠ ∅))
1817, 6, 6catprslem 48899 . . 3 ((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋 𝑋 ↔ (𝑋𝐻𝑋) ≠ ∅))
1915, 18mpbird 257 . 2 ((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝑋 𝑋)
2011ad2antrr 726 . . . . . 6 (((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝑋 𝑌𝑌 𝑍)) → 𝐻 = (Hom ‘𝐶))
2120oveqd 7448 . . . . 5 (((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝑋 𝑌𝑌 𝑍)) → (𝑋𝐻𝑍) = (𝑋(Hom ‘𝐶)𝑍))
227eleq2d 2827 . . . . . . . 8 (𝜑 → (𝑋𝐵𝑋 ∈ (Base‘𝐶)))
237eleq2d 2827 . . . . . . . 8 (𝜑 → (𝑌𝐵𝑌 ∈ (Base‘𝐶)))
247eleq2d 2827 . . . . . . . 8 (𝜑 → (𝑍𝐵𝑍 ∈ (Base‘𝐶)))
2522, 23, 243anbi123d 1438 . . . . . . 7 (𝜑 → ((𝑋𝐵𝑌𝐵𝑍𝐵) ↔ (𝑋 ∈ (Base‘𝐶) ∧ 𝑌 ∈ (Base‘𝐶) ∧ 𝑍 ∈ (Base‘𝐶))))
2625pm5.32i 574 . . . . . 6 ((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ↔ (𝜑 ∧ (𝑋 ∈ (Base‘𝐶) ∧ 𝑌 ∈ (Base‘𝐶) ∧ 𝑍 ∈ (Base‘𝐶))))
27 eqid 2737 . . . . . . 7 (comp‘𝐶) = (comp‘𝐶)
284ad2antrr 726 . . . . . . 7 (((𝜑 ∧ (𝑋 ∈ (Base‘𝐶) ∧ 𝑌 ∈ (Base‘𝐶) ∧ 𝑍 ∈ (Base‘𝐶))) ∧ (𝑋 𝑌𝑌 𝑍)) → 𝐶 ∈ Cat)
29 simplr1 1216 . . . . . . 7 (((𝜑 ∧ (𝑋 ∈ (Base‘𝐶) ∧ 𝑌 ∈ (Base‘𝐶) ∧ 𝑍 ∈ (Base‘𝐶))) ∧ (𝑋 𝑌𝑌 𝑍)) → 𝑋 ∈ (Base‘𝐶))
30 simplr2 1217 . . . . . . 7 (((𝜑 ∧ (𝑋 ∈ (Base‘𝐶) ∧ 𝑌 ∈ (Base‘𝐶) ∧ 𝑍 ∈ (Base‘𝐶))) ∧ (𝑋 𝑌𝑌 𝑍)) → 𝑌 ∈ (Base‘𝐶))
31 simplr3 1218 . . . . . . 7 (((𝜑 ∧ (𝑋 ∈ (Base‘𝐶) ∧ 𝑌 ∈ (Base‘𝐶) ∧ 𝑍 ∈ (Base‘𝐶))) ∧ (𝑋 𝑌𝑌 𝑍)) → 𝑍 ∈ (Base‘𝐶))
3220oveqd 7448 . . . . . . . . 9 (((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝑋 𝑌𝑌 𝑍)) → (𝑋𝐻𝑌) = (𝑋(Hom ‘𝐶)𝑌))
33 simpr2 1196 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝑌𝐵)
3417, 6, 33catprslem 48899 . . . . . . . . . . 11 ((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋 𝑌 ↔ (𝑋𝐻𝑌) ≠ ∅))
3534biimpa 476 . . . . . . . . . 10 (((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ 𝑋 𝑌) → (𝑋𝐻𝑌) ≠ ∅)
3635adantrr 717 . . . . . . . . 9 (((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝑋 𝑌𝑌 𝑍)) → (𝑋𝐻𝑌) ≠ ∅)
3732, 36eqnetrrd 3009 . . . . . . . 8 (((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝑋 𝑌𝑌 𝑍)) → (𝑋(Hom ‘𝐶)𝑌) ≠ ∅)
3826, 37sylanbr 582 . . . . . . 7 (((𝜑 ∧ (𝑋 ∈ (Base‘𝐶) ∧ 𝑌 ∈ (Base‘𝐶) ∧ 𝑍 ∈ (Base‘𝐶))) ∧ (𝑋 𝑌𝑌 𝑍)) → (𝑋(Hom ‘𝐶)𝑌) ≠ ∅)
3920oveqd 7448 . . . . . . . . 9 (((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝑋 𝑌𝑌 𝑍)) → (𝑌𝐻𝑍) = (𝑌(Hom ‘𝐶)𝑍))
40 simpr3 1197 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝑍𝐵)
4117, 33, 40catprslem 48899 . . . . . . . . . . 11 ((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑌 𝑍 ↔ (𝑌𝐻𝑍) ≠ ∅))
4241biimpa 476 . . . . . . . . . 10 (((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ 𝑌 𝑍) → (𝑌𝐻𝑍) ≠ ∅)
4342adantrl 716 . . . . . . . . 9 (((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝑋 𝑌𝑌 𝑍)) → (𝑌𝐻𝑍) ≠ ∅)
4439, 43eqnetrrd 3009 . . . . . . . 8 (((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝑋 𝑌𝑌 𝑍)) → (𝑌(Hom ‘𝐶)𝑍) ≠ ∅)
4526, 44sylanbr 582 . . . . . . 7 (((𝜑 ∧ (𝑋 ∈ (Base‘𝐶) ∧ 𝑌 ∈ (Base‘𝐶) ∧ 𝑍 ∈ (Base‘𝐶))) ∧ (𝑋 𝑌𝑌 𝑍)) → (𝑌(Hom ‘𝐶)𝑍) ≠ ∅)
461, 2, 27, 28, 29, 30, 31, 38, 45catcone0 17730 . . . . . 6 (((𝜑 ∧ (𝑋 ∈ (Base‘𝐶) ∧ 𝑌 ∈ (Base‘𝐶) ∧ 𝑍 ∈ (Base‘𝐶))) ∧ (𝑋 𝑌𝑌 𝑍)) → (𝑋(Hom ‘𝐶)𝑍) ≠ ∅)
4726, 46sylanb 581 . . . . 5 (((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝑋 𝑌𝑌 𝑍)) → (𝑋(Hom ‘𝐶)𝑍) ≠ ∅)
4821, 47eqnetrd 3008 . . . 4 (((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝑋 𝑌𝑌 𝑍)) → (𝑋𝐻𝑍) ≠ ∅)
4917, 6, 40catprslem 48899 . . . . 5 ((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋 𝑍 ↔ (𝑋𝐻𝑍) ≠ ∅))
5049adantr 480 . . . 4 (((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝑋 𝑌𝑌 𝑍)) → (𝑋 𝑍 ↔ (𝑋𝐻𝑍) ≠ ∅))
5148, 50mpbird 257 . . 3 (((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝑋 𝑌𝑌 𝑍)) → 𝑋 𝑍)
5251ex 412 . 2 ((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 𝑌𝑌 𝑍) → 𝑋 𝑍))
5319, 52jca 511 1 ((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋 𝑋 ∧ ((𝑋 𝑌𝑌 𝑍) → 𝑋 𝑍)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1087   = wceq 1540  wcel 2108  wne 2940  wral 3061  c0 4333   class class class wbr 5143  cfv 6561  (class class class)co 7431  Basecbs 17247  Hom chom 17308  compcco 17309  Catccat 17707  Idccid 17708
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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-cat 17711  df-cid 17712
This theorem is referenced by:  catprs2  48901
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