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Theorem catsubcat 16705
 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 ssid 3771 . . . 4 (Base‘𝐶) ⊆ (Base‘𝐶)
21a1i 11 . . 3 (𝐶 ∈ Cat → (Base‘𝐶) ⊆ (Base‘𝐶))
3 ssid 3771 . . . . 5 (𝑥(Homf𝐶)𝑦) ⊆ (𝑥(Homf𝐶)𝑦)
43a1i 11 . . . 4 ((𝐶 ∈ Cat ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (𝑥(Homf𝐶)𝑦) ⊆ (𝑥(Homf𝐶)𝑦))
54ralrimivva 3119 . . 3 (𝐶 ∈ Cat → ∀𝑥 ∈ (Base‘𝐶)∀𝑦 ∈ (Base‘𝐶)(𝑥(Homf𝐶)𝑦) ⊆ (𝑥(Homf𝐶)𝑦))
6 eqid 2770 . . . . . 6 (Homf𝐶) = (Homf𝐶)
7 eqid 2770 . . . . . 6 (Base‘𝐶) = (Base‘𝐶)
86, 7homffn 16559 . . . . 5 (Homf𝐶) Fn ((Base‘𝐶) × (Base‘𝐶))
98a1i 11 . . . 4 (𝐶 ∈ Cat → (Homf𝐶) Fn ((Base‘𝐶) × (Base‘𝐶)))
10 fvexd 6344 . . . 4 (𝐶 ∈ Cat → (Base‘𝐶) ∈ V)
119, 9, 10isssc 16686 . . 3 (𝐶 ∈ Cat → ((Homf𝐶) ⊆cat (Homf𝐶) ↔ ((Base‘𝐶) ⊆ (Base‘𝐶) ∧ ∀𝑥 ∈ (Base‘𝐶)∀𝑦 ∈ (Base‘𝐶)(𝑥(Homf𝐶)𝑦) ⊆ (𝑥(Homf𝐶)𝑦))))
122, 5, 11mpbir2and 684 . 2 (𝐶 ∈ Cat → (Homf𝐶) ⊆cat (Homf𝐶))
13 eqid 2770 . . . . . 6 (Hom ‘𝐶) = (Hom ‘𝐶)
14 eqid 2770 . . . . . 6 (Id‘𝐶) = (Id‘𝐶)
15 simpl 468 . . . . . 6 ((𝐶 ∈ Cat ∧ 𝑥 ∈ (Base‘𝐶)) → 𝐶 ∈ Cat)
16 simpr 471 . . . . . 6 ((𝐶 ∈ Cat ∧ 𝑥 ∈ (Base‘𝐶)) → 𝑥 ∈ (Base‘𝐶))
177, 13, 14, 15, 16catidcl 16549 . . . . 5 ((𝐶 ∈ Cat ∧ 𝑥 ∈ (Base‘𝐶)) → ((Id‘𝐶)‘𝑥) ∈ (𝑥(Hom ‘𝐶)𝑥))
186, 7, 13, 16, 16homfval 16558 . . . . 5 ((𝐶 ∈ Cat ∧ 𝑥 ∈ (Base‘𝐶)) → (𝑥(Homf𝐶)𝑥) = (𝑥(Hom ‘𝐶)𝑥))
1917, 18eleqtrrd 2852 . . . 4 ((𝐶 ∈ Cat ∧ 𝑥 ∈ (Base‘𝐶)) → ((Id‘𝐶)‘𝑥) ∈ (𝑥(Homf𝐶)𝑥))
20 eqid 2770 . . . . . . . 8 (comp‘𝐶) = (comp‘𝐶)
2115adantr 466 . . . . . . . . 9 (((𝐶 ∈ Cat ∧ 𝑥 ∈ (Base‘𝐶)) ∧ (𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶))) → 𝐶 ∈ Cat)
2221adantr 466 . . . . . . . 8 ((((𝐶 ∈ Cat ∧ 𝑥 ∈ (Base‘𝐶)) ∧ (𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶))) ∧ (𝑓 ∈ (𝑥(Homf𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Homf𝐶)𝑧))) → 𝐶 ∈ Cat)
2316adantr 466 . . . . . . . . 9 (((𝐶 ∈ Cat ∧ 𝑥 ∈ (Base‘𝐶)) ∧ (𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶))) → 𝑥 ∈ (Base‘𝐶))
2423adantr 466 . . . . . . . 8 ((((𝐶 ∈ Cat ∧ 𝑥 ∈ (Base‘𝐶)) ∧ (𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶))) ∧ (𝑓 ∈ (𝑥(Homf𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Homf𝐶)𝑧))) → 𝑥 ∈ (Base‘𝐶))
25 simpl 468 . . . . . . . . . 10 ((𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) → 𝑦 ∈ (Base‘𝐶))
2625adantl 467 . . . . . . . . 9 (((𝐶 ∈ Cat ∧ 𝑥 ∈ (Base‘𝐶)) ∧ (𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶))) → 𝑦 ∈ (Base‘𝐶))
2726adantr 466 . . . . . . . 8 ((((𝐶 ∈ Cat ∧ 𝑥 ∈ (Base‘𝐶)) ∧ (𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶))) ∧ (𝑓 ∈ (𝑥(Homf𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Homf𝐶)𝑧))) → 𝑦 ∈ (Base‘𝐶))
28 simpr 471 . . . . . . . . . 10 ((𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) → 𝑧 ∈ (Base‘𝐶))
2928adantl 467 . . . . . . . . 9 (((𝐶 ∈ Cat ∧ 𝑥 ∈ (Base‘𝐶)) ∧ (𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶))) → 𝑧 ∈ (Base‘𝐶))
3029adantr 466 . . . . . . . 8 ((((𝐶 ∈ Cat ∧ 𝑥 ∈ (Base‘𝐶)) ∧ (𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶))) ∧ (𝑓 ∈ (𝑥(Homf𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Homf𝐶)𝑧))) → 𝑧 ∈ (Base‘𝐶))
316, 7, 13, 23, 26homfval 16558 . . . . . . . . . . . 12 (((𝐶 ∈ Cat ∧ 𝑥 ∈ (Base‘𝐶)) ∧ (𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶))) → (𝑥(Homf𝐶)𝑦) = (𝑥(Hom ‘𝐶)𝑦))
3231eleq2d 2835 . . . . . . . . . . 11 (((𝐶 ∈ Cat ∧ 𝑥 ∈ (Base‘𝐶)) ∧ (𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶))) → (𝑓 ∈ (𝑥(Homf𝐶)𝑦) ↔ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)))
3332biimpcd 239 . . . . . . . . . 10 (𝑓 ∈ (𝑥(Homf𝐶)𝑦) → (((𝐶 ∈ Cat ∧ 𝑥 ∈ (Base‘𝐶)) ∧ (𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶))) → 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)))
3433adantr 466 . . . . . . . . 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)))
3534impcom 394 . . . . . . . 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))
366, 7, 13, 26, 29homfval 16558 . . . . . . . . . . . 12 (((𝐶 ∈ Cat ∧ 𝑥 ∈ (Base‘𝐶)) ∧ (𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶))) → (𝑦(Homf𝐶)𝑧) = (𝑦(Hom ‘𝐶)𝑧))
3736eleq2d 2835 . . . . . . . . . . 11 (((𝐶 ∈ Cat ∧ 𝑥 ∈ (Base‘𝐶)) ∧ (𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶))) → (𝑔 ∈ (𝑦(Homf𝐶)𝑧) ↔ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)))
3837biimpd 219 . . . . . . . . . 10 (((𝐶 ∈ Cat ∧ 𝑥 ∈ (Base‘𝐶)) ∧ (𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶))) → (𝑔 ∈ (𝑦(Homf𝐶)𝑧) → 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)))
3938adantld 474 . . . . . . . . 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)))
4039imp 393 . . . . . . . 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))
417, 13, 20, 22, 24, 27, 30, 35, 40catcocl 16552 . . . . . . 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))
426, 7, 13, 23, 29homfval 16558 . . . . . . . 8 (((𝐶 ∈ Cat ∧ 𝑥 ∈ (Base‘𝐶)) ∧ (𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶))) → (𝑥(Homf𝐶)𝑧) = (𝑥(Hom ‘𝐶)𝑧))
4342adantr 466 . . . . . . 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))
4441, 43eleqtrrd 2852 . . . . . 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))
4544ralrimivva 3119 . . . . 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))
4645ralrimivva 3119 . . . 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))
4719, 46jca 495 . . 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)))
4847ralrimiva 3114 . 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)))
49 id 22 . . 3 (𝐶 ∈ Cat → 𝐶 ∈ Cat)
506, 14, 20, 49, 9issubc2 16702 . 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)))))
5112, 48, 50mpbir2and 684 1 (𝐶 ∈ Cat → (Homf𝐶) ∈ (Subcat‘𝐶))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 382   = wceq 1630   ∈ wcel 2144  ∀wral 3060  Vcvv 3349   ⊆ wss 3721  ⟨cop 4320   class class class wbr 4784   × cxp 5247   Fn wfn 6026  ‘cfv 6031  (class class class)co 6792  Basecbs 16063  Hom chom 16159  compcco 16160  Catccat 16531  Idccid 16532  Homf chomf 16533   ⊆cat cssc 16673  Subcatcsubc 16675 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-rep 4902  ax-sep 4912  ax-nul 4920  ax-pow 4971  ax-pr 5034  ax-un 7095 This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-fal 1636  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-ral 3065  df-rex 3066  df-reu 3067  df-rmo 3068  df-rab 3069  df-v 3351  df-sbc 3586  df-csb 3681  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-pw 4297  df-sn 4315  df-pr 4317  df-op 4321  df-uni 4573  df-iun 4654  df-br 4785  df-opab 4845  df-mpt 4862  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-riota 6753  df-ov 6795  df-oprab 6796  df-mpt2 6797  df-1st 7314  df-2nd 7315  df-pm 8011  df-ixp 8062  df-cat 16535  df-cid 16536  df-homf 16537  df-ssc 16676  df-subc 16678 This theorem is referenced by: (None)
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