Theorem catsubcat 17855

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 3959	. . 3 ⊢ (𝐶 ∈ Cat → (Base‘𝐶) ⊆ (Base‘𝐶))
2		ssidd 3959	. . . 4 ⊢ ((𝐶 ∈ Cat ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (𝑥(Homf ‘𝐶)𝑦) ⊆ (𝑥(Homf ‘𝐶)𝑦))
3	2	ralrimivva 3204	. . 3 ⊢ (𝐶 ∈ Cat → ∀𝑥 ∈ (Base‘𝐶)∀𝑦 ∈ (Base‘𝐶)(𝑥(Homf ‘𝐶)𝑦) ⊆ (𝑥(Homf ‘𝐶)𝑦))
4		eqid 2761	. . . . . 6 ⊢ (Homf ‘𝐶) = (Homf ‘𝐶)
5		eqid 2761	. . . . . 6 ⊢ (Base‘𝐶) = (Base‘𝐶)
6	4, 5	homffn 17708	. . . . 5 ⊢ (Homf ‘𝐶) Fn ((Base‘𝐶) × (Base‘𝐶))
7	6	a1i 11	. . . 4 ⊢ (𝐶 ∈ Cat → (Homf ‘𝐶) Fn ((Base‘𝐶) × (Base‘𝐶)))
8		fvexd 6878	. . . 4 ⊢ (𝐶 ∈ Cat → (Base‘𝐶) ∈ V)
9	7, 7, 8	isssc 17836	. . 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 723	. 2 ⊢ (𝐶 ∈ Cat → (Homf ‘𝐶) ⊆cat (Homf ‘𝐶))
11		eqid 2761	. . . . . 6 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
12		eqid 2761	. . . . . 6 ⊢ (Id‘𝐶) = (Id‘𝐶)
13		simpl 486	. . . . . 6 ⊢ ((𝐶 ∈ Cat ∧ 𝑥 ∈ (Base‘𝐶)) → 𝐶 ∈ Cat)
14		simpr 488	. . . . . 6 ⊢ ((𝐶 ∈ Cat ∧ 𝑥 ∈ (Base‘𝐶)) → 𝑥 ∈ (Base‘𝐶))
15	5, 11, 12, 13, 14	catidcl 17697	. . . . 5 ⊢ ((𝐶 ∈ Cat ∧ 𝑥 ∈ (Base‘𝐶)) → ((Id‘𝐶)‘𝑥) ∈ (𝑥(Hom ‘𝐶)𝑥))
16	4, 5, 11, 14, 14	homfval 17707	. . . . 5 ⊢ ((𝐶 ∈ Cat ∧ 𝑥 ∈ (Base‘𝐶)) → (𝑥(Homf ‘𝐶)𝑥) = (𝑥(Hom ‘𝐶)𝑥))
17	15, 16	eleqtrrd 2864	. . . 4 ⊢ ((𝐶 ∈ Cat ∧ 𝑥 ∈ (Base‘𝐶)) → ((Id‘𝐶)‘𝑥) ∈ (𝑥(Homf ‘𝐶)𝑥))
18		eqid 2761	. . . . . . . 8 ⊢ (comp‘𝐶) = (comp‘𝐶)
19	13	adantr 484	. . . . . . . . 9 ⊢ (((𝐶 ∈ Cat ∧ 𝑥 ∈ (Base‘𝐶)) ∧ (𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶))) → 𝐶 ∈ Cat)
20	19	adantr 484	. . . . . . . 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 484	. . . . . . . . 9 ⊢ (((𝐶 ∈ Cat ∧ 𝑥 ∈ (Base‘𝐶)) ∧ (𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶))) → 𝑥 ∈ (Base‘𝐶))
22	21	adantr 484	. . . . . . . 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 486	. . . . . . . . . 10 ⊢ ((𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) → 𝑦 ∈ (Base‘𝐶))
24	23	adantl 485	. . . . . . . . 9 ⊢ (((𝐶 ∈ Cat ∧ 𝑥 ∈ (Base‘𝐶)) ∧ (𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶))) → 𝑦 ∈ (Base‘𝐶))
25	24	adantr 484	. . . . . . . 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 488	. . . . . . . . . 10 ⊢ ((𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) → 𝑧 ∈ (Base‘𝐶))
27	26	adantl 485	. . . . . . . . 9 ⊢ (((𝐶 ∈ Cat ∧ 𝑥 ∈ (Base‘𝐶)) ∧ (𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶))) → 𝑧 ∈ (Base‘𝐶))
28	27	adantr 484	. . . . . . . 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 17707	. . . . . . . . . . . 12 ⊢ (((𝐶 ∈ Cat ∧ 𝑥 ∈ (Base‘𝐶)) ∧ (𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶))) → (𝑥(Homf ‘𝐶)𝑦) = (𝑥(Hom ‘𝐶)𝑦))
30	29	eleq2d 2847	. . . . . . . . . . 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 484	. . . . . . . . 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 411	. . . . . . . 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 17707	. . . . . . . . . . . 12 ⊢ (((𝐶 ∈ Cat ∧ 𝑥 ∈ (Base‘𝐶)) ∧ (𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶))) → (𝑦(Homf ‘𝐶)𝑧) = (𝑦(Hom ‘𝐶)𝑧))
35	34	eleq2d 2847	. . . . . . . . . . 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 494	. . . . . . . . 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 410	. . . . . . . 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 17700	. . . . . . 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 17707	. . . . . . . 8 ⊢ (((𝐶 ∈ Cat ∧ 𝑥 ∈ (Base‘𝐶)) ∧ (𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶))) → (𝑥(Homf ‘𝐶)𝑧) = (𝑥(Hom ‘𝐶)𝑧))
41	40	adantr 484	. . . . . . 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 2864	. . . . . 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 3204	. . . . 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 3204	. . . 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 519	. . 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 3153	. 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 17852	. 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 723	1 ⊢ (𝐶 ∈ Cat → (Homf ‘𝐶) ∈ (Subcat‘𝐶))

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1559   ∈ wcel 2141  ∀wral 3075  Vcvv 3453   ⊆ wss 3904  ⟨cop 4587   class class class wbr 5099   × cxp 5643   Fn wfn 6512  ‘cfv 6517  (class class class)co 7392  Basecbs 17228  Hom chom 17280  compcco 17281  Catccat 17679  Idccid 17680  Hom_f chomf 17681   ⊆_cat cssc 17823  Subcatcsubc 17825

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-rep 5226  ax-sep 5245  ax-nul 5255  ax-pow 5321  ax-pr 5389  ax-un 7714

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3076  df-rex 3086  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3455  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-iun 4950  df-br 5100  df-opab 5162  df-mpt 5181  df-id 5540  df-xp 5651  df-rel 5652  df-cnv 5653  df-co 5654  df-dm 5655  df-rn 5656  df-res 5657  df-ima 5658  df-iota 6473  df-fun 6519  df-fn 6520  df-f 6521  df-f1 6522  df-fo 6523  df-f1o 6524  df-fv 6525  df-riota 7349  df-ov 7395  df-oprab 7396  df-mpo 7397  df-1st 7966  df-2nd 7967  df-pm 8806  df-ixp 8876  df-cat 17683  df-cid 17684  df-homf 17685  df-ssc 17826  df-subc 17828

This theorem is referenced by: (None)