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Theorem catprs 47679
Description: A preorder can be extracted from a category. See catprs2 47680 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 2733 . . . . . 6 (Base‘𝐶) = (Base‘𝐶)
2 eqid 2733 . . . . . 6 (Hom ‘𝐶) = (Hom ‘𝐶)
3 eqid 2733 . . . . . 6 (Id‘𝐶) = (Id‘𝐶)
4 catprs.c . . . . . . 7 (𝜑𝐶 ∈ Cat)
54adantr 482 . . . . . 6 ((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝐶 ∈ Cat)
6 simpr1 1195 . . . . . . 7 ((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝑋𝐵)
7 catprs.b . . . . . . . 8 (𝜑𝐵 = (Base‘𝐶))
87adantr 482 . . . . . . 7 ((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝐵 = (Base‘𝐶))
96, 8eleqtrd 2836 . . . . . 6 ((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝑋 ∈ (Base‘𝐶))
101, 2, 3, 5, 9catidcl 17626 . . . . 5 ((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((Id‘𝐶)‘𝑋) ∈ (𝑋(Hom ‘𝐶)𝑋))
11 catprs.h . . . . . . 7 (𝜑𝐻 = (Hom ‘𝐶))
1211adantr 482 . . . . . 6 ((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝐻 = (Hom ‘𝐶))
1312oveqd 7426 . . . . 5 ((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋𝐻𝑋) = (𝑋(Hom ‘𝐶)𝑋))
1410, 13eleqtrrd 2837 . . . 4 ((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((Id‘𝐶)‘𝑋) ∈ (𝑋𝐻𝑋))
1514ne0d 4336 . . 3 ((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋𝐻𝑋) ≠ ∅)
16 catprs.1 . . . . 5 (𝜑 → ∀𝑥𝐵𝑦𝐵 (𝑥 𝑦 ↔ (𝑥𝐻𝑦) ≠ ∅))
1716adantr 482 . . . 4 ((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ∀𝑥𝐵𝑦𝐵 (𝑥 𝑦 ↔ (𝑥𝐻𝑦) ≠ ∅))
1817, 6, 6catprslem 47678 . . 3 ((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋 𝑋 ↔ (𝑋𝐻𝑋) ≠ ∅))
1915, 18mpbird 257 . 2 ((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝑋 𝑋)
2011ad2antrr 725 . . . . . 6 (((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝑋 𝑌𝑌 𝑍)) → 𝐻 = (Hom ‘𝐶))
2120oveqd 7426 . . . . 5 (((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝑋 𝑌𝑌 𝑍)) → (𝑋𝐻𝑍) = (𝑋(Hom ‘𝐶)𝑍))
227eleq2d 2820 . . . . . . . 8 (𝜑 → (𝑋𝐵𝑋 ∈ (Base‘𝐶)))
237eleq2d 2820 . . . . . . . 8 (𝜑 → (𝑌𝐵𝑌 ∈ (Base‘𝐶)))
247eleq2d 2820 . . . . . . . 8 (𝜑 → (𝑍𝐵𝑍 ∈ (Base‘𝐶)))
2522, 23, 243anbi123d 1437 . . . . . . 7 (𝜑 → ((𝑋𝐵𝑌𝐵𝑍𝐵) ↔ (𝑋 ∈ (Base‘𝐶) ∧ 𝑌 ∈ (Base‘𝐶) ∧ 𝑍 ∈ (Base‘𝐶))))
2625pm5.32i 576 . . . . . 6 ((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ↔ (𝜑 ∧ (𝑋 ∈ (Base‘𝐶) ∧ 𝑌 ∈ (Base‘𝐶) ∧ 𝑍 ∈ (Base‘𝐶))))
27 eqid 2733 . . . . . . 7 (comp‘𝐶) = (comp‘𝐶)
284ad2antrr 725 . . . . . . 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 7426 . . . . . . . . 9 (((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝑋 𝑌𝑌 𝑍)) → (𝑋𝐻𝑌) = (𝑋(Hom ‘𝐶)𝑌))
33 simpr2 1196 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝑌𝐵)
3417, 6, 33catprslem 47678 . . . . . . . . . . 11 ((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋 𝑌 ↔ (𝑋𝐻𝑌) ≠ ∅))
3534biimpa 478 . . . . . . . . . 10 (((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ 𝑋 𝑌) → (𝑋𝐻𝑌) ≠ ∅)
3635adantrr 716 . . . . . . . . 9 (((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝑋 𝑌𝑌 𝑍)) → (𝑋𝐻𝑌) ≠ ∅)
3732, 36eqnetrrd 3010 . . . . . . . 8 (((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝑋 𝑌𝑌 𝑍)) → (𝑋(Hom ‘𝐶)𝑌) ≠ ∅)
3826, 37sylanbr 583 . . . . . . 7 (((𝜑 ∧ (𝑋 ∈ (Base‘𝐶) ∧ 𝑌 ∈ (Base‘𝐶) ∧ 𝑍 ∈ (Base‘𝐶))) ∧ (𝑋 𝑌𝑌 𝑍)) → (𝑋(Hom ‘𝐶)𝑌) ≠ ∅)
3920oveqd 7426 . . . . . . . . 9 (((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝑋 𝑌𝑌 𝑍)) → (𝑌𝐻𝑍) = (𝑌(Hom ‘𝐶)𝑍))
40 simpr3 1197 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝑍𝐵)
4117, 33, 40catprslem 47678 . . . . . . . . . . 11 ((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑌 𝑍 ↔ (𝑌𝐻𝑍) ≠ ∅))
4241biimpa 478 . . . . . . . . . 10 (((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ 𝑌 𝑍) → (𝑌𝐻𝑍) ≠ ∅)
4342adantrl 715 . . . . . . . . 9 (((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝑋 𝑌𝑌 𝑍)) → (𝑌𝐻𝑍) ≠ ∅)
4439, 43eqnetrrd 3010 . . . . . . . 8 (((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝑋 𝑌𝑌 𝑍)) → (𝑌(Hom ‘𝐶)𝑍) ≠ ∅)
4526, 44sylanbr 583 . . . . . . 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 17631 . . . . . 6 (((𝜑 ∧ (𝑋 ∈ (Base‘𝐶) ∧ 𝑌 ∈ (Base‘𝐶) ∧ 𝑍 ∈ (Base‘𝐶))) ∧ (𝑋 𝑌𝑌 𝑍)) → (𝑋(Hom ‘𝐶)𝑍) ≠ ∅)
4726, 46sylanb 582 . . . . 5 (((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝑋 𝑌𝑌 𝑍)) → (𝑋(Hom ‘𝐶)𝑍) ≠ ∅)
4821, 47eqnetrd 3009 . . . 4 (((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝑋 𝑌𝑌 𝑍)) → (𝑋𝐻𝑍) ≠ ∅)
4917, 6, 40catprslem 47678 . . . . 5 ((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋 𝑍 ↔ (𝑋𝐻𝑍) ≠ ∅))
5049adantr 482 . . . 4 (((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝑋 𝑌𝑌 𝑍)) → (𝑋 𝑍 ↔ (𝑋𝐻𝑍) ≠ ∅))
5148, 50mpbird 257 . . 3 (((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝑋 𝑌𝑌 𝑍)) → 𝑋 𝑍)
5251ex 414 . 2 ((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 𝑌𝑌 𝑍) → 𝑋 𝑍))
5319, 52jca 513 1 ((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋 𝑋 ∧ ((𝑋 𝑌𝑌 𝑍) → 𝑋 𝑍)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397  w3a 1088   = wceq 1542  wcel 2107  wne 2941  wral 3062  c0 4323   class class class wbr 5149  cfv 6544  (class class class)co 7409  Basecbs 17144  Hom chom 17208  compcco 17209  Catccat 17608  Idccid 17609
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pr 5428
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-riota 7365  df-ov 7412  df-cat 17612  df-cid 17613
This theorem is referenced by:  catprs2  47680
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