Theorem catsubcat 17111

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 → (Homf ‘𝐶) ∈ (Subcat‘𝐶))

Proof of Theorem catsubcat

Dummy variables 𝑓 𝑔 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ssidd 3992	. . 3 ⊢ (𝐶 ∈ Cat → (Base‘𝐶) ⊆ (Base‘𝐶))
2		ssidd 3992	. . . 4 ⊢ ((𝐶 ∈ Cat ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (𝑥(Homf ‘𝐶)𝑦) ⊆ (𝑥(Homf ‘𝐶)𝑦))
3	2	ralrimivva 3193	. . 3 ⊢ (𝐶 ∈ Cat → ∀𝑥 ∈ (Base‘𝐶)∀𝑦 ∈ (Base‘𝐶)(𝑥(Homf ‘𝐶)𝑦) ⊆ (𝑥(Homf ‘𝐶)𝑦))
4		eqid 2823	. . . . . 6 ⊢ (Homf ‘𝐶) = (Homf ‘𝐶)
5		eqid 2823	. . . . . 6 ⊢ (Base‘𝐶) = (Base‘𝐶)
6	4, 5	homffn 16965	. . . . 5 ⊢ (Homf ‘𝐶) Fn ((Base‘𝐶) × (Base‘𝐶))
7	6	a1i 11	. . . 4 ⊢ (𝐶 ∈ Cat → (Homf ‘𝐶) Fn ((Base‘𝐶) × (Base‘𝐶)))
8		fvexd 6687	. . . 4 ⊢ (𝐶 ∈ Cat → (Base‘𝐶) ∈ V)
9	7, 7, 8	isssc 17092	. . 3 ⊢ (𝐶 ∈ Cat → ((Homf ‘𝐶) ⊆cat (Homf ‘𝐶) ↔ ((Base‘𝐶) ⊆ (Base‘𝐶) ∧ ∀𝑥 ∈ (Base‘𝐶)∀𝑦 ∈ (Base‘𝐶)(𝑥(Homf ‘𝐶)𝑦) ⊆ (𝑥(Homf ‘𝐶)𝑦))))
10	1, 3, 9	mpbir2and 711	. 2 ⊢ (𝐶 ∈ Cat → (Homf ‘𝐶) ⊆cat (Homf ‘𝐶))
11		eqid 2823	. . . . . 6 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
12		eqid 2823	. . . . . 6 ⊢ (Id‘𝐶) = (Id‘𝐶)
13		simpl 485	. . . . . 6 ⊢ ((𝐶 ∈ Cat ∧ 𝑥 ∈ (Base‘𝐶)) → 𝐶 ∈ Cat)
14		simpr 487	. . . . . 6 ⊢ ((𝐶 ∈ Cat ∧ 𝑥 ∈ (Base‘𝐶)) → 𝑥 ∈ (Base‘𝐶))
15	5, 11, 12, 13, 14	catidcl 16955	. . . . 5 ⊢ ((𝐶 ∈ Cat ∧ 𝑥 ∈ (Base‘𝐶)) → ((Id‘𝐶)‘𝑥) ∈ (𝑥(Hom ‘𝐶)𝑥))
16	4, 5, 11, 14, 14	homfval 16964	. . . . 5 ⊢ ((𝐶 ∈ Cat ∧ 𝑥 ∈ (Base‘𝐶)) → (𝑥(Homf ‘𝐶)𝑥) = (𝑥(Hom ‘𝐶)𝑥))
17	15, 16	eleqtrrd 2918	. . . 4 ⊢ ((𝐶 ∈ Cat ∧ 𝑥 ∈ (Base‘𝐶)) → ((Id‘𝐶)‘𝑥) ∈ (𝑥(Homf ‘𝐶)𝑥))
18		eqid 2823	. . . . . . . 8 ⊢ (comp‘𝐶) = (comp‘𝐶)
19	13	adantr 483	. . . . . . . . 9 ⊢ (((𝐶 ∈ Cat ∧ 𝑥 ∈ (Base‘𝐶)) ∧ (𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶))) → 𝐶 ∈ Cat)
20	19	adantr 483	. . . . . . . 8 ⊢ ((((𝐶 ∈ Cat ∧ 𝑥 ∈ (Base‘𝐶)) ∧ (𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶))) ∧ (𝑓 ∈ (𝑥(Homf ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Homf ‘𝐶)𝑧))) → 𝐶 ∈ Cat)
21	14	adantr 483	. . . . . . . . 9 ⊢ (((𝐶 ∈ Cat ∧ 𝑥 ∈ (Base‘𝐶)) ∧ (𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶))) → 𝑥 ∈ (Base‘𝐶))
22	21	adantr 483	. . . . . . . 8 ⊢ ((((𝐶 ∈ Cat ∧ 𝑥 ∈ (Base‘𝐶)) ∧ (𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶))) ∧ (𝑓 ∈ (𝑥(Homf ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Homf ‘𝐶)𝑧))) → 𝑥 ∈ (Base‘𝐶))
23		simpl 485	. . . . . . . . . 10 ⊢ ((𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) → 𝑦 ∈ (Base‘𝐶))
24	23	adantl 484	. . . . . . . . 9 ⊢ (((𝐶 ∈ Cat ∧ 𝑥 ∈ (Base‘𝐶)) ∧ (𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶))) → 𝑦 ∈ (Base‘𝐶))
25	24	adantr 483	. . . . . . . 8 ⊢ ((((𝐶 ∈ Cat ∧ 𝑥 ∈ (Base‘𝐶)) ∧ (𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶))) ∧ (𝑓 ∈ (𝑥(Homf ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Homf ‘𝐶)𝑧))) → 𝑦 ∈ (Base‘𝐶))
26		simpr 487	. . . . . . . . . 10 ⊢ ((𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) → 𝑧 ∈ (Base‘𝐶))
27	26	adantl 484	. . . . . . . . 9 ⊢ (((𝐶 ∈ Cat ∧ 𝑥 ∈ (Base‘𝐶)) ∧ (𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶))) → 𝑧 ∈ (Base‘𝐶))
28	27	adantr 483	. . . . . . . 8 ⊢ ((((𝐶 ∈ Cat ∧ 𝑥 ∈ (Base‘𝐶)) ∧ (𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶))) ∧ (𝑓 ∈ (𝑥(Homf ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Homf ‘𝐶)𝑧))) → 𝑧 ∈ (Base‘𝐶))
29	4, 5, 11, 21, 24	homfval 16964	. . . . . . . . . . . 12 ⊢ (((𝐶 ∈ Cat ∧ 𝑥 ∈ (Base‘𝐶)) ∧ (𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶))) → (𝑥(Homf ‘𝐶)𝑦) = (𝑥(Hom ‘𝐶)𝑦))
30	29	eleq2d 2900	. . . . . . . . . . 11 ⊢ (((𝐶 ∈ Cat ∧ 𝑥 ∈ (Base‘𝐶)) ∧ (𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶))) → (𝑓 ∈ (𝑥(Homf ‘𝐶)𝑦) ↔ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)))
31	30	biimpcd 251	. . . . . . . . . 10 ⊢ (𝑓 ∈ (𝑥(Homf ‘𝐶)𝑦) → (((𝐶 ∈ Cat ∧ 𝑥 ∈ (Base‘𝐶)) ∧ (𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶))) → 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)))
32	31	adantr 483	. . . . . . . . 9 ⊢ ((𝑓 ∈ (𝑥(Homf ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Homf ‘𝐶)𝑧)) → (((𝐶 ∈ Cat ∧ 𝑥 ∈ (Base‘𝐶)) ∧ (𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶))) → 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)))
33	32	impcom 410	. . . . . . . 8 ⊢ ((((𝐶 ∈ Cat ∧ 𝑥 ∈ (Base‘𝐶)) ∧ (𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶))) ∧ (𝑓 ∈ (𝑥(Homf ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Homf ‘𝐶)𝑧))) → 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))
34	4, 5, 11, 24, 27	homfval 16964	. . . . . . . . . . . 12 ⊢ (((𝐶 ∈ Cat ∧ 𝑥 ∈ (Base‘𝐶)) ∧ (𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶))) → (𝑦(Homf ‘𝐶)𝑧) = (𝑦(Hom ‘𝐶)𝑧))
35	34	eleq2d 2900	. . . . . . . . . . 11 ⊢ (((𝐶 ∈ Cat ∧ 𝑥 ∈ (Base‘𝐶)) ∧ (𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶))) → (𝑔 ∈ (𝑦(Homf ‘𝐶)𝑧) ↔ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)))
36	35	biimpd 231	. . . . . . . . . 10 ⊢ (((𝐶 ∈ Cat ∧ 𝑥 ∈ (Base‘𝐶)) ∧ (𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶))) → (𝑔 ∈ (𝑦(Homf ‘𝐶)𝑧) → 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)))
37	36	adantld 493	. . . . . . . . 9 ⊢ (((𝐶 ∈ Cat ∧ 𝑥 ∈ (Base‘𝐶)) ∧ (𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶))) → ((𝑓 ∈ (𝑥(Homf ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Homf ‘𝐶)𝑧)) → 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)))
38	37	imp 409	. . . . . . . 8 ⊢ ((((𝐶 ∈ Cat ∧ 𝑥 ∈ (Base‘𝐶)) ∧ (𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶))) ∧ (𝑓 ∈ (𝑥(Homf ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Homf ‘𝐶)𝑧))) → 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))
39	5, 11, 18, 20, 22, 25, 28, 33, 38	catcocl 16958	. . . . . . 7 ⊢ ((((𝐶 ∈ Cat ∧ 𝑥 ∈ (Base‘𝐶)) ∧ (𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶))) ∧ (𝑓 ∈ (𝑥(Homf ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Homf ‘𝐶)𝑧))) → (𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥(Hom ‘𝐶)𝑧))
40	4, 5, 11, 21, 27	homfval 16964	. . . . . . . 8 ⊢ (((𝐶 ∈ Cat ∧ 𝑥 ∈ (Base‘𝐶)) ∧ (𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶))) → (𝑥(Homf ‘𝐶)𝑧) = (𝑥(Hom ‘𝐶)𝑧))
41	40	adantr 483	. . . . . . 7 ⊢ ((((𝐶 ∈ Cat ∧ 𝑥 ∈ (Base‘𝐶)) ∧ (𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶))) ∧ (𝑓 ∈ (𝑥(Homf ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Homf ‘𝐶)𝑧))) → (𝑥(Homf ‘𝐶)𝑧) = (𝑥(Hom ‘𝐶)𝑧))
42	39, 41	eleqtrrd 2918	. . . . . 6 ⊢ ((((𝐶 ∈ Cat ∧ 𝑥 ∈ (Base‘𝐶)) ∧ (𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶))) ∧ (𝑓 ∈ (𝑥(Homf ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Homf ‘𝐶)𝑧))) → (𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥(Homf ‘𝐶)𝑧))
43	42	ralrimivva 3193	. . . . 5 ⊢ (((𝐶 ∈ Cat ∧ 𝑥 ∈ (Base‘𝐶)) ∧ (𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶))) → ∀𝑓 ∈ (𝑥(Homf ‘𝐶)𝑦)∀𝑔 ∈ (𝑦(Homf ‘𝐶)𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥(Homf ‘𝐶)𝑧))
44	43	ralrimivva 3193	. . . 4 ⊢ ((𝐶 ∈ Cat ∧ 𝑥 ∈ (Base‘𝐶)) → ∀𝑦 ∈ (Base‘𝐶)∀𝑧 ∈ (Base‘𝐶)∀𝑓 ∈ (𝑥(Homf ‘𝐶)𝑦)∀𝑔 ∈ (𝑦(Homf ‘𝐶)𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥(Homf ‘𝐶)𝑧))
45	17, 44	jca 514	. . 3 ⊢ ((𝐶 ∈ Cat ∧ 𝑥 ∈ (Base‘𝐶)) → (((Id‘𝐶)‘𝑥) ∈ (𝑥(Homf ‘𝐶)𝑥) ∧ ∀𝑦 ∈ (Base‘𝐶)∀𝑧 ∈ (Base‘𝐶)∀𝑓 ∈ (𝑥(Homf ‘𝐶)𝑦)∀𝑔 ∈ (𝑦(Homf ‘𝐶)𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥(Homf ‘𝐶)𝑧)))
46	45	ralrimiva 3184	. 2 ⊢ (𝐶 ∈ Cat → ∀𝑥 ∈ (Base‘𝐶)(((Id‘𝐶)‘𝑥) ∈ (𝑥(Homf ‘𝐶)𝑥) ∧ ∀𝑦 ∈ (Base‘𝐶)∀𝑧 ∈ (Base‘𝐶)∀𝑓 ∈ (𝑥(Homf ‘𝐶)𝑦)∀𝑔 ∈ (𝑦(Homf ‘𝐶)𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥(Homf ‘𝐶)𝑧)))
47		id 22	. . 3 ⊢ (𝐶 ∈ Cat → 𝐶 ∈ Cat)
48	4, 12, 18, 47, 7	issubc2 17108	. 2 ⊢ (𝐶 ∈ Cat → ((Homf ‘𝐶) ∈ (Subcat‘𝐶) ↔ ((Homf ‘𝐶) ⊆cat (Homf ‘𝐶) ∧ ∀𝑥 ∈ (Base‘𝐶)(((Id‘𝐶)‘𝑥) ∈ (𝑥(Homf ‘𝐶)𝑥) ∧ ∀𝑦 ∈ (Base‘𝐶)∀𝑧 ∈ (Base‘𝐶)∀𝑓 ∈ (𝑥(Homf ‘𝐶)𝑦)∀𝑔 ∈ (𝑦(Homf ‘𝐶)𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥(Homf ‘𝐶)𝑧)))))
49	10, 46, 48	mpbir2and 711	1 ⊢ (𝐶 ∈ Cat → (Homf ‘𝐶) ∈ (Subcat‘𝐶))

Syntax hints:   → wi 4   ∧ wa 398   = wceq 1537   ∈ wcel 2114  ∀wral 3140  Vcvv 3496   ⊆ wss 3938  ⟨cop 4575   class class class wbr 5068   × cxp 5555   Fn wfn 6352  ‘cfv 6357  (class class class)co 7158  Basecbs 16485  Hom chom 16578  compcco 16579  Catccat 16937  Idccid 16938  Hom_f chomf 16939   ⊆_cat cssc 17079  Subcatcsubc 17081

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-rep 5192  ax-sep 5205  ax-nul 5212  ax-pow 5268  ax-pr 5332  ax-un 7463

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-fal 1550  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-ral 3145  df-rex 3146  df-reu 3147  df-rmo 3148  df-rab 3149  df-v 3498  df-sbc 3775  df-csb 3886  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-pw 4543  df-sn 4570  df-pr 4572  df-op 4576  df-uni 4841  df-iun 4923  df-br 5069  df-opab 5131  df-mpt 5149  df-id 5462  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-rn 5568  df-res 5569  df-ima 5570  df-iota 6316  df-fun 6359  df-fn 6360  df-f 6361  df-f1 6362  df-fo 6363  df-f1o 6364  df-fv 6365  df-riota 7116  df-ov 7161  df-oprab 7162  df-mpo 7163  df-1st 7691  df-2nd 7692  df-pm 8411  df-ixp 8464  df-cat 16941  df-cid 16942  df-homf 16943  df-ssc 17082  df-subc 17084

This theorem is referenced by: (None)