Theorem catsubcat 18019

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 3931	. . 3 ⊢ (𝐶 ∈ Cat → (Base‘𝐶) ⊆ (Base‘𝐶))
2		ssidd 3931	. . . 4 ⊢ ((𝐶 ∈ Cat ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (𝑥(Homf ‘𝐶)𝑦) ⊆ (𝑥(Homf ‘𝐶)𝑦))
3	2	ralrimivva 3205	. . 3 ⊢ (𝐶 ∈ Cat → ∀𝑥 ∈ (Base‘𝐶)∀𝑦 ∈ (Base‘𝐶)(𝑥(Homf ‘𝐶)𝑦) ⊆ (𝑥(Homf ‘𝐶)𝑦))
4		eqid 2760	. . . . . 6 ⊢ (Homf ‘𝐶) = (Homf ‘𝐶)
5		eqid 2760	. . . . . 6 ⊢ (Base‘𝐶) = (Base‘𝐶)
6	4, 5	homffn 17872	. . . . 5 ⊢ (Homf ‘𝐶) Fn ((Base‘𝐶) × (Base‘𝐶))
7	6	a1i 11	. . . 4 ⊢ (𝐶 ∈ Cat → (Homf ‘𝐶) Fn ((Base‘𝐶) × (Base‘𝐶)))
8		fvexd 6900	. . . 4 ⊢ (𝐶 ∈ Cat → (Base‘𝐶) ∈ V)
9	7, 7, 8	isssc 18000	. . 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 726	. 2 ⊢ (𝐶 ∈ Cat → (Homf ‘𝐶) ⊆cat (Homf ‘𝐶))
11		eqid 2760	. . . . . 6 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
12		eqid 2760	. . . . . 6 ⊢ (Id‘𝐶) = (Id‘𝐶)
13		simpl 488	. . . . . 6 ⊢ ((𝐶 ∈ Cat ∧ 𝑥 ∈ (Base‘𝐶)) → 𝐶 ∈ Cat)
14		simpr 490	. . . . . 6 ⊢ ((𝐶 ∈ Cat ∧ 𝑥 ∈ (Base‘𝐶)) → 𝑥 ∈ (Base‘𝐶))
15	5, 11, 12, 13, 14	catidcl 17861	. . . . 5 ⊢ ((𝐶 ∈ Cat ∧ 𝑥 ∈ (Base‘𝐶)) → ((Id‘𝐶)‘𝑥) ∈ (𝑥(Hom ‘𝐶)𝑥))
16	4, 5, 11, 14, 14	homfval 17871	. . . . 5 ⊢ ((𝐶 ∈ Cat ∧ 𝑥 ∈ (Base‘𝐶)) → (𝑥(Homf ‘𝐶)𝑥) = (𝑥(Hom ‘𝐶)𝑥))
17	15, 16	eleqtrrd 2863	. . . 4 ⊢ ((𝐶 ∈ Cat ∧ 𝑥 ∈ (Base‘𝐶)) → ((Id‘𝐶)‘𝑥) ∈ (𝑥(Homf ‘𝐶)𝑥))
18		eqid 2760	. . . . . . . 8 ⊢ (comp‘𝐶) = (comp‘𝐶)
19	13	adantr 486	. . . . . . . . 9 ⊢ (((𝐶 ∈ Cat ∧ 𝑥 ∈ (Base‘𝐶)) ∧ (𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶))) → 𝐶 ∈ Cat)
20	19	adantr 486	. . . . . . . 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 486	. . . . . . . . 9 ⊢ (((𝐶 ∈ Cat ∧ 𝑥 ∈ (Base‘𝐶)) ∧ (𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶))) → 𝑥 ∈ (Base‘𝐶))
22	21	adantr 486	. . . . . . . 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 488	. . . . . . . . . 10 ⊢ ((𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) → 𝑦 ∈ (Base‘𝐶))
24	23	adantl 487	. . . . . . . . 9 ⊢ (((𝐶 ∈ Cat ∧ 𝑥 ∈ (Base‘𝐶)) ∧ (𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶))) → 𝑦 ∈ (Base‘𝐶))
25	24	adantr 486	. . . . . . . 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 490	. . . . . . . . . 10 ⊢ ((𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) → 𝑧 ∈ (Base‘𝐶))
27	26	adantl 487	. . . . . . . . 9 ⊢ (((𝐶 ∈ Cat ∧ 𝑥 ∈ (Base‘𝐶)) ∧ (𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶))) → 𝑧 ∈ (Base‘𝐶))
28	27	adantr 486	. . . . . . . 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 17871	. . . . . . . . . . . 12 ⊢ (((𝐶 ∈ Cat ∧ 𝑥 ∈ (Base‘𝐶)) ∧ (𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶))) → (𝑥(Homf ‘𝐶)𝑦) = (𝑥(Hom ‘𝐶)𝑦))
30	29	eleq2d 2846	. . . . . . . . . . 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 252	. . . . . . . . . 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 486	. . . . . . . . 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 413	. . . . . . . 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 17871	. . . . . . . . . . . 12 ⊢ (((𝐶 ∈ Cat ∧ 𝑥 ∈ (Base‘𝐶)) ∧ (𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶))) → (𝑦(Homf ‘𝐶)𝑧) = (𝑦(Hom ‘𝐶)𝑧))
35	34	eleq2d 2846	. . . . . . . . . . 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 232	. . . . . . . . . 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 496	. . . . . . . . 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 412	. . . . . . . 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 17864	. . . . . . 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 17871	. . . . . . . 8 ⊢ (((𝐶 ∈ Cat ∧ 𝑥 ∈ (Base‘𝐶)) ∧ (𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶))) → (𝑥(Homf ‘𝐶)𝑧) = (𝑥(Hom ‘𝐶)𝑧))
41	40	adantr 486	. . . . . . 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 2863	. . . . . 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 3205	. . . . 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 3205	. . . 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 521	. . 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 3154	. 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 23	. . 3 ⊢ (𝐶 ∈ Cat → 𝐶 ∈ Cat)
48	4, 12, 18, 47, 7	issubc2 18016	. 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 726	1 ⊢ (𝐶 ∈ Cat → (Homf ‘𝐶) ∈ (Subcat‘𝐶))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1569   ∈ wcel 2144  ∀wral 3076  Vcvv 3450   ⊆ wss 3895  ⟨cop 4834   class class class wbr 5102   × cxp 5648   Fn wfn 6533  ‘cfv 6538  (class class class)co 7420  Basecbs 17392  Hom chom 17444  compcco 17445  Catccat 17843  Idccid 17844  Hom_f chomf 17845   ⊆_cat cssc 17987  Subcatcsubc 17989

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-riota 7377  df-ov 7423  df-oprab 7424  df-mpo 7425  df-1st 8001  df-2nd 8002  df-pm 8853  df-ixp 8929  df-cat 17847  df-cid 17848  df-homf 17849  df-ssc 17990  df-subc 17992

This theorem is used by: (None)