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Theorem catprs 47584
Description: A preorder can be extracted from a category. See catprs2 47585 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 2732 . . . . . 6 (Base‘𝐶) = (Base‘𝐶)
2 eqid 2732 . . . . . 6 (Hom ‘𝐶) = (Hom ‘𝐶)
3 eqid 2732 . . . . . 6 (Id‘𝐶) = (Id‘𝐶)
4 catprs.c . . . . . . 7 (𝜑𝐶 ∈ Cat)
54adantr 481 . . . . . 6 ((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝐶 ∈ Cat)
6 simpr1 1194 . . . . . . 7 ((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝑋𝐵)
7 catprs.b . . . . . . . 8 (𝜑𝐵 = (Base‘𝐶))
87adantr 481 . . . . . . 7 ((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝐵 = (Base‘𝐶))
96, 8eleqtrd 2835 . . . . . 6 ((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝑋 ∈ (Base‘𝐶))
101, 2, 3, 5, 9catidcl 17622 . . . . 5 ((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((Id‘𝐶)‘𝑋) ∈ (𝑋(Hom ‘𝐶)𝑋))
11 catprs.h . . . . . . 7 (𝜑𝐻 = (Hom ‘𝐶))
1211adantr 481 . . . . . 6 ((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝐻 = (Hom ‘𝐶))
1312oveqd 7422 . . . . 5 ((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋𝐻𝑋) = (𝑋(Hom ‘𝐶)𝑋))
1410, 13eleqtrrd 2836 . . . 4 ((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((Id‘𝐶)‘𝑋) ∈ (𝑋𝐻𝑋))
1514ne0d 4334 . . 3 ((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋𝐻𝑋) ≠ ∅)
16 catprs.1 . . . . 5 (𝜑 → ∀𝑥𝐵𝑦𝐵 (𝑥 𝑦 ↔ (𝑥𝐻𝑦) ≠ ∅))
1716adantr 481 . . . 4 ((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ∀𝑥𝐵𝑦𝐵 (𝑥 𝑦 ↔ (𝑥𝐻𝑦) ≠ ∅))
1817, 6, 6catprslem 47583 . . 3 ((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋 𝑋 ↔ (𝑋𝐻𝑋) ≠ ∅))
1915, 18mpbird 256 . 2 ((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝑋 𝑋)
2011ad2antrr 724 . . . . . 6 (((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝑋 𝑌𝑌 𝑍)) → 𝐻 = (Hom ‘𝐶))
2120oveqd 7422 . . . . 5 (((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝑋 𝑌𝑌 𝑍)) → (𝑋𝐻𝑍) = (𝑋(Hom ‘𝐶)𝑍))
227eleq2d 2819 . . . . . . . 8 (𝜑 → (𝑋𝐵𝑋 ∈ (Base‘𝐶)))
237eleq2d 2819 . . . . . . . 8 (𝜑 → (𝑌𝐵𝑌 ∈ (Base‘𝐶)))
247eleq2d 2819 . . . . . . . 8 (𝜑 → (𝑍𝐵𝑍 ∈ (Base‘𝐶)))
2522, 23, 243anbi123d 1436 . . . . . . 7 (𝜑 → ((𝑋𝐵𝑌𝐵𝑍𝐵) ↔ (𝑋 ∈ (Base‘𝐶) ∧ 𝑌 ∈ (Base‘𝐶) ∧ 𝑍 ∈ (Base‘𝐶))))
2625pm5.32i 575 . . . . . 6 ((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ↔ (𝜑 ∧ (𝑋 ∈ (Base‘𝐶) ∧ 𝑌 ∈ (Base‘𝐶) ∧ 𝑍 ∈ (Base‘𝐶))))
27 eqid 2732 . . . . . . 7 (comp‘𝐶) = (comp‘𝐶)
284ad2antrr 724 . . . . . . 7 (((𝜑 ∧ (𝑋 ∈ (Base‘𝐶) ∧ 𝑌 ∈ (Base‘𝐶) ∧ 𝑍 ∈ (Base‘𝐶))) ∧ (𝑋 𝑌𝑌 𝑍)) → 𝐶 ∈ Cat)
29 simplr1 1215 . . . . . . 7 (((𝜑 ∧ (𝑋 ∈ (Base‘𝐶) ∧ 𝑌 ∈ (Base‘𝐶) ∧ 𝑍 ∈ (Base‘𝐶))) ∧ (𝑋 𝑌𝑌 𝑍)) → 𝑋 ∈ (Base‘𝐶))
30 simplr2 1216 . . . . . . 7 (((𝜑 ∧ (𝑋 ∈ (Base‘𝐶) ∧ 𝑌 ∈ (Base‘𝐶) ∧ 𝑍 ∈ (Base‘𝐶))) ∧ (𝑋 𝑌𝑌 𝑍)) → 𝑌 ∈ (Base‘𝐶))
31 simplr3 1217 . . . . . . 7 (((𝜑 ∧ (𝑋 ∈ (Base‘𝐶) ∧ 𝑌 ∈ (Base‘𝐶) ∧ 𝑍 ∈ (Base‘𝐶))) ∧ (𝑋 𝑌𝑌 𝑍)) → 𝑍 ∈ (Base‘𝐶))
3220oveqd 7422 . . . . . . . . 9 (((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝑋 𝑌𝑌 𝑍)) → (𝑋𝐻𝑌) = (𝑋(Hom ‘𝐶)𝑌))
33 simpr2 1195 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝑌𝐵)
3417, 6, 33catprslem 47583 . . . . . . . . . . 11 ((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋 𝑌 ↔ (𝑋𝐻𝑌) ≠ ∅))
3534biimpa 477 . . . . . . . . . 10 (((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ 𝑋 𝑌) → (𝑋𝐻𝑌) ≠ ∅)
3635adantrr 715 . . . . . . . . 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 7422 . . . . . . . . 9 (((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝑋 𝑌𝑌 𝑍)) → (𝑌𝐻𝑍) = (𝑌(Hom ‘𝐶)𝑍))
40 simpr3 1196 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝑍𝐵)
4117, 33, 40catprslem 47583 . . . . . . . . . . 11 ((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑌 𝑍 ↔ (𝑌𝐻𝑍) ≠ ∅))
4241biimpa 477 . . . . . . . . . 10 (((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ 𝑌 𝑍) → (𝑌𝐻𝑍) ≠ ∅)
4342adantrl 714 . . . . . . . . 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 17627 . . . . . 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 47583 . . . . 5 ((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋 𝑍 ↔ (𝑋𝐻𝑍) ≠ ∅))
5049adantr 481 . . . 4 (((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝑋 𝑌𝑌 𝑍)) → (𝑋 𝑍 ↔ (𝑋𝐻𝑍) ≠ ∅))
5148, 50mpbird 256 . . 3 (((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝑋 𝑌𝑌 𝑍)) → 𝑋 𝑍)
5251ex 413 . 2 ((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 𝑌𝑌 𝑍) → 𝑋 𝑍))
5319, 52jca 512 1 ((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋 𝑋 ∧ ((𝑋 𝑌𝑌 𝑍) → 𝑋 𝑍)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  w3a 1087   = wceq 1541  wcel 2106  wne 2940  wral 3061  c0 4321   class class class wbr 5147  cfv 6540  (class class class)co 7405  Basecbs 17140  Hom chom 17204  compcco 17205  Catccat 17604  Idccid 17605
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 2703  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pr 5426
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 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-riota 7361  df-ov 7408  df-cat 17608  df-cid 17609
This theorem is referenced by:  catprs2  47585
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