Theorem catsubcat 17554

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 3944	. . 3 ⊢ (𝐶 ∈ Cat → (Base‘𝐶) ⊆ (Base‘𝐶))
2		ssidd 3944	. . . 4 ⊢ ((𝐶 ∈ Cat ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (𝑥(Homf ‘𝐶)𝑦) ⊆ (𝑥(Homf ‘𝐶)𝑦))
3	2	ralrimivva 3123	. . 3 ⊢ (𝐶 ∈ Cat → ∀𝑥 ∈ (Base‘𝐶)∀𝑦 ∈ (Base‘𝐶)(𝑥(Homf ‘𝐶)𝑦) ⊆ (𝑥(Homf ‘𝐶)𝑦))
4		eqid 2738	. . . . . 6 ⊢ (Homf ‘𝐶) = (Homf ‘𝐶)
5		eqid 2738	. . . . . 6 ⊢ (Base‘𝐶) = (Base‘𝐶)
6	4, 5	homffn 17402	. . . . 5 ⊢ (Homf ‘𝐶) Fn ((Base‘𝐶) × (Base‘𝐶))
7	6	a1i 11	. . . 4 ⊢ (𝐶 ∈ Cat → (Homf ‘𝐶) Fn ((Base‘𝐶) × (Base‘𝐶)))
8		fvexd 6789	. . . 4 ⊢ (𝐶 ∈ Cat → (Base‘𝐶) ∈ V)
9	7, 7, 8	isssc 17532	. . 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 710	. 2 ⊢ (𝐶 ∈ Cat → (Homf ‘𝐶) ⊆cat (Homf ‘𝐶))
11		eqid 2738	. . . . . 6 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
12		eqid 2738	. . . . . 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 17391	. . . . 5 ⊢ ((𝐶 ∈ Cat ∧ 𝑥 ∈ (Base‘𝐶)) → ((Id‘𝐶)‘𝑥) ∈ (𝑥(Hom ‘𝐶)𝑥))
16	4, 5, 11, 14, 14	homfval 17401	. . . . 5 ⊢ ((𝐶 ∈ Cat ∧ 𝑥 ∈ (Base‘𝐶)) → (𝑥(Homf ‘𝐶)𝑥) = (𝑥(Hom ‘𝐶)𝑥))
17	15, 16	eleqtrrd 2842	. . . 4 ⊢ ((𝐶 ∈ Cat ∧ 𝑥 ∈ (Base‘𝐶)) → ((Id‘𝐶)‘𝑥) ∈ (𝑥(Homf ‘𝐶)𝑥))
18		eqid 2738	. . . . . . . 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 17401	. . . . . . . . . . . 12 ⊢ (((𝐶 ∈ Cat ∧ 𝑥 ∈ (Base‘𝐶)) ∧ (𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶))) → (𝑥(Homf ‘𝐶)𝑦) = (𝑥(Hom ‘𝐶)𝑦))
30	29	eleq2d 2824	. . . . . . . . . . 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 248	. . . . . . . . . 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 17401	. . . . . . . . . . . 12 ⊢ (((𝐶 ∈ Cat ∧ 𝑥 ∈ (Base‘𝐶)) ∧ (𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶))) → (𝑦(Homf ‘𝐶)𝑧) = (𝑦(Hom ‘𝐶)𝑧))
35	34	eleq2d 2824	. . . . . . . . . . 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 228	. . . . . . . . . 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 17394	. . . . . . 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 17401	. . . . . . . 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 2842	. . . . . 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 3123	. . . . 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 3123	. . . 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 512	. . 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 3103	. 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 17551	. 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 710	1 ⊢ (𝐶 ∈ Cat → (Homf ‘𝐶) ∈ (Subcat‘𝐶))

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1539   ∈ wcel 2106  ∀wral 3064  Vcvv 3432   ⊆ wss 3887  ⟨cop 4567   class class class wbr 5074   × cxp 5587   Fn wfn 6428  ‘cfv 6433  (class class class)co 7275  Basecbs 16912  Hom chom 16973  compcco 16974  Catccat 17373  Idccid 17374  Hom_f chomf 17375   ⊆_cat cssc 17519  Subcatcsubc 17521

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-rmo 3071  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-riota 7232  df-ov 7278  df-oprab 7279  df-mpo 7280  df-1st 7831  df-2nd 7832  df-pm 8618  df-ixp 8686  df-cat 17377  df-cid 17378  df-homf 17379  df-ssc 17522  df-subc 17524

This theorem is referenced by: (None)