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Theorem catsubcat 17786
Description: For any category 𝐶, 𝐶 itself is a (full) subcategory of 𝐶, see example 4.3(1.b) in [Adamek] p. 48. (Contributed by AV, 23-Apr-2020.)
Assertion
Ref Expression
catsubcat (𝐶 ∈ Cat → (Homf𝐶) ∈ (Subcat‘𝐶))

Proof of Theorem catsubcat
Dummy variables 𝑓 𝑔 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ssidd 4005 . . 3 (𝐶 ∈ Cat → (Base‘𝐶) ⊆ (Base‘𝐶))
2 ssidd 4005 . . . 4 ((𝐶 ∈ Cat ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (𝑥(Homf𝐶)𝑦) ⊆ (𝑥(Homf𝐶)𝑦))
32ralrimivva 3201 . . 3 (𝐶 ∈ Cat → ∀𝑥 ∈ (Base‘𝐶)∀𝑦 ∈ (Base‘𝐶)(𝑥(Homf𝐶)𝑦) ⊆ (𝑥(Homf𝐶)𝑦))
4 eqid 2733 . . . . . 6 (Homf𝐶) = (Homf𝐶)
5 eqid 2733 . . . . . 6 (Base‘𝐶) = (Base‘𝐶)
64, 5homffn 17634 . . . . 5 (Homf𝐶) Fn ((Base‘𝐶) × (Base‘𝐶))
76a1i 11 . . . 4 (𝐶 ∈ Cat → (Homf𝐶) Fn ((Base‘𝐶) × (Base‘𝐶)))
8 fvexd 6904 . . . 4 (𝐶 ∈ Cat → (Base‘𝐶) ∈ V)
97, 7, 8isssc 17764 . . 3 (𝐶 ∈ Cat → ((Homf𝐶) ⊆cat (Homf𝐶) ↔ ((Base‘𝐶) ⊆ (Base‘𝐶) ∧ ∀𝑥 ∈ (Base‘𝐶)∀𝑦 ∈ (Base‘𝐶)(𝑥(Homf𝐶)𝑦) ⊆ (𝑥(Homf𝐶)𝑦))))
101, 3, 9mpbir2and 712 . 2 (𝐶 ∈ Cat → (Homf𝐶) ⊆cat (Homf𝐶))
11 eqid 2733 . . . . . 6 (Hom ‘𝐶) = (Hom ‘𝐶)
12 eqid 2733 . . . . . 6 (Id‘𝐶) = (Id‘𝐶)
13 simpl 484 . . . . . 6 ((𝐶 ∈ Cat ∧ 𝑥 ∈ (Base‘𝐶)) → 𝐶 ∈ Cat)
14 simpr 486 . . . . . 6 ((𝐶 ∈ Cat ∧ 𝑥 ∈ (Base‘𝐶)) → 𝑥 ∈ (Base‘𝐶))
155, 11, 12, 13, 14catidcl 17623 . . . . 5 ((𝐶 ∈ Cat ∧ 𝑥 ∈ (Base‘𝐶)) → ((Id‘𝐶)‘𝑥) ∈ (𝑥(Hom ‘𝐶)𝑥))
164, 5, 11, 14, 14homfval 17633 . . . . 5 ((𝐶 ∈ Cat ∧ 𝑥 ∈ (Base‘𝐶)) → (𝑥(Homf𝐶)𝑥) = (𝑥(Hom ‘𝐶)𝑥))
1715, 16eleqtrrd 2837 . . . 4 ((𝐶 ∈ Cat ∧ 𝑥 ∈ (Base‘𝐶)) → ((Id‘𝐶)‘𝑥) ∈ (𝑥(Homf𝐶)𝑥))
18 eqid 2733 . . . . . . . 8 (comp‘𝐶) = (comp‘𝐶)
1913adantr 482 . . . . . . . . 9 (((𝐶 ∈ Cat ∧ 𝑥 ∈ (Base‘𝐶)) ∧ (𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶))) → 𝐶 ∈ Cat)
2019adantr 482 . . . . . . . 8 ((((𝐶 ∈ Cat ∧ 𝑥 ∈ (Base‘𝐶)) ∧ (𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶))) ∧ (𝑓 ∈ (𝑥(Homf𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Homf𝐶)𝑧))) → 𝐶 ∈ Cat)
2114adantr 482 . . . . . . . . 9 (((𝐶 ∈ Cat ∧ 𝑥 ∈ (Base‘𝐶)) ∧ (𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶))) → 𝑥 ∈ (Base‘𝐶))
2221adantr 482 . . . . . . . 8 ((((𝐶 ∈ Cat ∧ 𝑥 ∈ (Base‘𝐶)) ∧ (𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶))) ∧ (𝑓 ∈ (𝑥(Homf𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Homf𝐶)𝑧))) → 𝑥 ∈ (Base‘𝐶))
23 simpl 484 . . . . . . . . . 10 ((𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) → 𝑦 ∈ (Base‘𝐶))
2423adantl 483 . . . . . . . . 9 (((𝐶 ∈ Cat ∧ 𝑥 ∈ (Base‘𝐶)) ∧ (𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶))) → 𝑦 ∈ (Base‘𝐶))
2524adantr 482 . . . . . . . 8 ((((𝐶 ∈ Cat ∧ 𝑥 ∈ (Base‘𝐶)) ∧ (𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶))) ∧ (𝑓 ∈ (𝑥(Homf𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Homf𝐶)𝑧))) → 𝑦 ∈ (Base‘𝐶))
26 simpr 486 . . . . . . . . . 10 ((𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) → 𝑧 ∈ (Base‘𝐶))
2726adantl 483 . . . . . . . . 9 (((𝐶 ∈ Cat ∧ 𝑥 ∈ (Base‘𝐶)) ∧ (𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶))) → 𝑧 ∈ (Base‘𝐶))
2827adantr 482 . . . . . . . 8 ((((𝐶 ∈ Cat ∧ 𝑥 ∈ (Base‘𝐶)) ∧ (𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶))) ∧ (𝑓 ∈ (𝑥(Homf𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Homf𝐶)𝑧))) → 𝑧 ∈ (Base‘𝐶))
294, 5, 11, 21, 24homfval 17633 . . . . . . . . . . . 12 (((𝐶 ∈ Cat ∧ 𝑥 ∈ (Base‘𝐶)) ∧ (𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶))) → (𝑥(Homf𝐶)𝑦) = (𝑥(Hom ‘𝐶)𝑦))
3029eleq2d 2820 . . . . . . . . . . 11 (((𝐶 ∈ Cat ∧ 𝑥 ∈ (Base‘𝐶)) ∧ (𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶))) → (𝑓 ∈ (𝑥(Homf𝐶)𝑦) ↔ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)))
3130biimpcd 248 . . . . . . . . . 10 (𝑓 ∈ (𝑥(Homf𝐶)𝑦) → (((𝐶 ∈ Cat ∧ 𝑥 ∈ (Base‘𝐶)) ∧ (𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶))) → 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)))
3231adantr 482 . . . . . . . . 9 ((𝑓 ∈ (𝑥(Homf𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Homf𝐶)𝑧)) → (((𝐶 ∈ Cat ∧ 𝑥 ∈ (Base‘𝐶)) ∧ (𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶))) → 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)))
3332impcom 409 . . . . . . . 8 ((((𝐶 ∈ Cat ∧ 𝑥 ∈ (Base‘𝐶)) ∧ (𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶))) ∧ (𝑓 ∈ (𝑥(Homf𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Homf𝐶)𝑧))) → 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))
344, 5, 11, 24, 27homfval 17633 . . . . . . . . . . . 12 (((𝐶 ∈ Cat ∧ 𝑥 ∈ (Base‘𝐶)) ∧ (𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶))) → (𝑦(Homf𝐶)𝑧) = (𝑦(Hom ‘𝐶)𝑧))
3534eleq2d 2820 . . . . . . . . . . 11 (((𝐶 ∈ Cat ∧ 𝑥 ∈ (Base‘𝐶)) ∧ (𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶))) → (𝑔 ∈ (𝑦(Homf𝐶)𝑧) ↔ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)))
3635biimpd 228 . . . . . . . . . 10 (((𝐶 ∈ Cat ∧ 𝑥 ∈ (Base‘𝐶)) ∧ (𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶))) → (𝑔 ∈ (𝑦(Homf𝐶)𝑧) → 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)))
3736adantld 492 . . . . . . . . 9 (((𝐶 ∈ Cat ∧ 𝑥 ∈ (Base‘𝐶)) ∧ (𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶))) → ((𝑓 ∈ (𝑥(Homf𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Homf𝐶)𝑧)) → 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)))
3837imp 408 . . . . . . . 8 ((((𝐶 ∈ Cat ∧ 𝑥 ∈ (Base‘𝐶)) ∧ (𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶))) ∧ (𝑓 ∈ (𝑥(Homf𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Homf𝐶)𝑧))) → 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))
395, 11, 18, 20, 22, 25, 28, 33, 38catcocl 17626 . . . . . . 7 ((((𝐶 ∈ Cat ∧ 𝑥 ∈ (Base‘𝐶)) ∧ (𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶))) ∧ (𝑓 ∈ (𝑥(Homf𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Homf𝐶)𝑧))) → (𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥(Hom ‘𝐶)𝑧))
404, 5, 11, 21, 27homfval 17633 . . . . . . . 8 (((𝐶 ∈ Cat ∧ 𝑥 ∈ (Base‘𝐶)) ∧ (𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶))) → (𝑥(Homf𝐶)𝑧) = (𝑥(Hom ‘𝐶)𝑧))
4140adantr 482 . . . . . . 7 ((((𝐶 ∈ Cat ∧ 𝑥 ∈ (Base‘𝐶)) ∧ (𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶))) ∧ (𝑓 ∈ (𝑥(Homf𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Homf𝐶)𝑧))) → (𝑥(Homf𝐶)𝑧) = (𝑥(Hom ‘𝐶)𝑧))
4239, 41eleqtrrd 2837 . . . . . 6 ((((𝐶 ∈ Cat ∧ 𝑥 ∈ (Base‘𝐶)) ∧ (𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶))) ∧ (𝑓 ∈ (𝑥(Homf𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Homf𝐶)𝑧))) → (𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥(Homf𝐶)𝑧))
4342ralrimivva 3201 . . . . 5 (((𝐶 ∈ Cat ∧ 𝑥 ∈ (Base‘𝐶)) ∧ (𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶))) → ∀𝑓 ∈ (𝑥(Homf𝐶)𝑦)∀𝑔 ∈ (𝑦(Homf𝐶)𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥(Homf𝐶)𝑧))
4443ralrimivva 3201 . . . 4 ((𝐶 ∈ Cat ∧ 𝑥 ∈ (Base‘𝐶)) → ∀𝑦 ∈ (Base‘𝐶)∀𝑧 ∈ (Base‘𝐶)∀𝑓 ∈ (𝑥(Homf𝐶)𝑦)∀𝑔 ∈ (𝑦(Homf𝐶)𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥(Homf𝐶)𝑧))
4517, 44jca 513 . . 3 ((𝐶 ∈ Cat ∧ 𝑥 ∈ (Base‘𝐶)) → (((Id‘𝐶)‘𝑥) ∈ (𝑥(Homf𝐶)𝑥) ∧ ∀𝑦 ∈ (Base‘𝐶)∀𝑧 ∈ (Base‘𝐶)∀𝑓 ∈ (𝑥(Homf𝐶)𝑦)∀𝑔 ∈ (𝑦(Homf𝐶)𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥(Homf𝐶)𝑧)))
4645ralrimiva 3147 . 2 (𝐶 ∈ Cat → ∀𝑥 ∈ (Base‘𝐶)(((Id‘𝐶)‘𝑥) ∈ (𝑥(Homf𝐶)𝑥) ∧ ∀𝑦 ∈ (Base‘𝐶)∀𝑧 ∈ (Base‘𝐶)∀𝑓 ∈ (𝑥(Homf𝐶)𝑦)∀𝑔 ∈ (𝑦(Homf𝐶)𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥(Homf𝐶)𝑧)))
47 id 22 . . 3 (𝐶 ∈ Cat → 𝐶 ∈ Cat)
484, 12, 18, 47, 7issubc2 17783 . 2 (𝐶 ∈ Cat → ((Homf𝐶) ∈ (Subcat‘𝐶) ↔ ((Homf𝐶) ⊆cat (Homf𝐶) ∧ ∀𝑥 ∈ (Base‘𝐶)(((Id‘𝐶)‘𝑥) ∈ (𝑥(Homf𝐶)𝑥) ∧ ∀𝑦 ∈ (Base‘𝐶)∀𝑧 ∈ (Base‘𝐶)∀𝑓 ∈ (𝑥(Homf𝐶)𝑦)∀𝑔 ∈ (𝑦(Homf𝐶)𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥(Homf𝐶)𝑧)))))
4910, 46, 48mpbir2and 712 1 (𝐶 ∈ Cat → (Homf𝐶) ∈ (Subcat‘𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397   = wceq 1542  wcel 2107  wral 3062  Vcvv 3475  wss 3948  cop 4634   class class class wbr 5148   × cxp 5674   Fn wfn 6536  cfv 6541  (class class class)co 7406  Basecbs 17141  Hom chom 17205  compcco 17206  Catccat 17605  Idccid 17606  Homf chomf 17607  cat cssc 17751  Subcatcsubc 17753
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 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7722
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 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6493  df-fun 6543  df-fn 6544  df-f 6545  df-f1 6546  df-fo 6547  df-f1o 6548  df-fv 6549  df-riota 7362  df-ov 7409  df-oprab 7410  df-mpo 7411  df-1st 7972  df-2nd 7973  df-pm 8820  df-ixp 8889  df-cat 17609  df-cid 17610  df-homf 17611  df-ssc 17754  df-subc 17756
This theorem is referenced by: (None)
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