Theorem catsubcat 17804

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 3945	. . 3 ⊢ (𝐶 ∈ Cat → (Base‘𝐶) ⊆ (Base‘𝐶))
2		ssidd 3945	. . . 4 ⊢ ((𝐶 ∈ Cat ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (𝑥(Homf ‘𝐶)𝑦) ⊆ (𝑥(Homf ‘𝐶)𝑦))
3	2	ralrimivva 3183	. . 3 ⊢ (𝐶 ∈ Cat → ∀𝑥 ∈ (Base‘𝐶)∀𝑦 ∈ (Base‘𝐶)(𝑥(Homf ‘𝐶)𝑦) ⊆ (𝑥(Homf ‘𝐶)𝑦))
4		eqid 2740	. . . . . 6 ⊢ (Homf ‘𝐶) = (Homf ‘𝐶)
5		eqid 2740	. . . . . 6 ⊢ (Base‘𝐶) = (Base‘𝐶)
6	4, 5	homffn 17657	. . . . 5 ⊢ (Homf ‘𝐶) Fn ((Base‘𝐶) × (Base‘𝐶))
7	6	a1i 11	. . . 4 ⊢ (𝐶 ∈ Cat → (Homf ‘𝐶) Fn ((Base‘𝐶) × (Base‘𝐶)))
8		fvexd 6849	. . . 4 ⊢ (𝐶 ∈ Cat → (Base‘𝐶) ∈ V)
9	7, 7, 8	isssc 17785	. . 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 719	. 2 ⊢ (𝐶 ∈ Cat → (Homf ‘𝐶) ⊆cat (Homf ‘𝐶))
11		eqid 2740	. . . . . 6 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
12		eqid 2740	. . . . . 6 ⊢ (Id‘𝐶) = (Id‘𝐶)
13		simpl 483	. . . . . 6 ⊢ ((𝐶 ∈ Cat ∧ 𝑥 ∈ (Base‘𝐶)) → 𝐶 ∈ Cat)
14		simpr 485	. . . . . 6 ⊢ ((𝐶 ∈ Cat ∧ 𝑥 ∈ (Base‘𝐶)) → 𝑥 ∈ (Base‘𝐶))
15	5, 11, 12, 13, 14	catidcl 17646	. . . . 5 ⊢ ((𝐶 ∈ Cat ∧ 𝑥 ∈ (Base‘𝐶)) → ((Id‘𝐶)‘𝑥) ∈ (𝑥(Hom ‘𝐶)𝑥))
16	4, 5, 11, 14, 14	homfval 17656	. . . . 5 ⊢ ((𝐶 ∈ Cat ∧ 𝑥 ∈ (Base‘𝐶)) → (𝑥(Homf ‘𝐶)𝑥) = (𝑥(Hom ‘𝐶)𝑥))
17	15, 16	eleqtrrd 2843	. . . 4 ⊢ ((𝐶 ∈ Cat ∧ 𝑥 ∈ (Base‘𝐶)) → ((Id‘𝐶)‘𝑥) ∈ (𝑥(Homf ‘𝐶)𝑥))
18		eqid 2740	. . . . . . . 8 ⊢ (comp‘𝐶) = (comp‘𝐶)
19	13	adantr 481	. . . . . . . . 9 ⊢ (((𝐶 ∈ Cat ∧ 𝑥 ∈ (Base‘𝐶)) ∧ (𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶))) → 𝐶 ∈ Cat)
20	19	adantr 481	. . . . . . . 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 481	. . . . . . . . 9 ⊢ (((𝐶 ∈ Cat ∧ 𝑥 ∈ (Base‘𝐶)) ∧ (𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶))) → 𝑥 ∈ (Base‘𝐶))
22	21	adantr 481	. . . . . . . 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 483	. . . . . . . . . 10 ⊢ ((𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) → 𝑦 ∈ (Base‘𝐶))
24	23	adantl 482	. . . . . . . . 9 ⊢ (((𝐶 ∈ Cat ∧ 𝑥 ∈ (Base‘𝐶)) ∧ (𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶))) → 𝑦 ∈ (Base‘𝐶))
25	24	adantr 481	. . . . . . . 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 485	. . . . . . . . . 10 ⊢ ((𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) → 𝑧 ∈ (Base‘𝐶))
27	26	adantl 482	. . . . . . . . 9 ⊢ (((𝐶 ∈ Cat ∧ 𝑥 ∈ (Base‘𝐶)) ∧ (𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶))) → 𝑧 ∈ (Base‘𝐶))
28	27	adantr 481	. . . . . . . 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 17656	. . . . . . . . . . . 12 ⊢ (((𝐶 ∈ Cat ∧ 𝑥 ∈ (Base‘𝐶)) ∧ (𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶))) → (𝑥(Homf ‘𝐶)𝑦) = (𝑥(Hom ‘𝐶)𝑦))
30	29	eleq2d 2826	. . . . . . . . . . 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 250	. . . . . . . . . 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 481	. . . . . . . . 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 408	. . . . . . . 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 17656	. . . . . . . . . . . 12 ⊢ (((𝐶 ∈ Cat ∧ 𝑥 ∈ (Base‘𝐶)) ∧ (𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶))) → (𝑦(Homf ‘𝐶)𝑧) = (𝑦(Hom ‘𝐶)𝑧))
35	34	eleq2d 2826	. . . . . . . . . . 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 230	. . . . . . . . . 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 491	. . . . . . . . 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 407	. . . . . . . 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 17649	. . . . . . 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 17656	. . . . . . . 8 ⊢ (((𝐶 ∈ Cat ∧ 𝑥 ∈ (Base‘𝐶)) ∧ (𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶))) → (𝑥(Homf ‘𝐶)𝑧) = (𝑥(Hom ‘𝐶)𝑧))
41	40	adantr 481	. . . . . . 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 2843	. . . . . 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 3183	. . . . 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 3183	. . . 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 516	. . 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 3132	. 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 17801	. 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 719	1 ⊢ (𝐶 ∈ Cat → (Homf ‘𝐶) ∈ (Subcat‘𝐶))

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1547   ∈ wcel 2119  ∀wral 3054  Vcvv 3432   ⊆ wss 3890  ⟨cop 4568   class class class wbr 5079   × cxp 5623   Fn wfn 6487  ‘cfv 6492  (class class class)co 7363  Basecbs 17177  Hom chom 17229  compcco 17230  Catccat 17628  Idccid 17629  Hom_f chomf 17630   ⊆_cat cssc 17772  Subcatcsubc 17774

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2712  ax-rep 5206  ax-sep 5225  ax-nul 5235  ax-pow 5301  ax-pr 5369  ax-un 7685

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2719  df-cleq 2732  df-clel 2815  df-nfc 2889  df-ne 2936  df-ral 3055  df-rex 3065  df-rmo 3345  df-reu 3346  df-rab 3393  df-v 3434  df-sbc 3731  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4269  df-if 4462  df-pw 4538  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4846  df-iun 4930  df-br 5080  df-opab 5142  df-mpt 5161  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-riota 7320  df-ov 7366  df-oprab 7367  df-mpo 7368  df-1st 7938  df-2nd 7939  df-pm 8773  df-ixp 8843  df-cat 17632  df-cid 17633  df-homf 17634  df-ssc 17775  df-subc 17777

This theorem is referenced by: (None)