Theorem 0subcat 17856

Description: For any category 𝐶, the empty set is a (full) subcategory of 𝐶, see example 4.3(1.a) in [Adamek] p. 48. (Contributed by AV, 23-Apr-2020.)

Assertion

Ref	Expression
0subcat	⊢ (𝐶 ∈ Cat → ∅ ∈ (Subcat‘𝐶))

Proof of Theorem 0subcat

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

Step	Hyp	Ref	Expression
1		0ssc 17855	. 2 ⊢ (𝐶 ∈ Cat → ∅ ⊆cat (Homf ‘𝐶))
2		ral0 4493	. . 3 ⊢ ∀𝑥 ∈ ∅ (((Id‘𝐶)‘𝑥) ∈ (𝑥∅𝑥) ∧ ∀𝑦 ∈ ∅ ∀𝑧 ∈ ∅ ∀𝑓 ∈ (𝑥∅𝑦)∀𝑔 ∈ (𝑦∅𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥∅𝑧))
3	2	a1i 11	. 2 ⊢ (𝐶 ∈ Cat → ∀𝑥 ∈ ∅ (((Id‘𝐶)‘𝑥) ∈ (𝑥∅𝑥) ∧ ∀𝑦 ∈ ∅ ∀𝑧 ∈ ∅ ∀𝑓 ∈ (𝑥∅𝑦)∀𝑔 ∈ (𝑦∅𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥∅𝑧)))
4		eqid 2736	. . 3 ⊢ (Homf ‘𝐶) = (Homf ‘𝐶)
5		eqid 2736	. . 3 ⊢ (Id‘𝐶) = (Id‘𝐶)
6		eqid 2736	. . 3 ⊢ (comp‘𝐶) = (comp‘𝐶)
7		id 22	. . 3 ⊢ (𝐶 ∈ Cat → 𝐶 ∈ Cat)
8		f0 6764	. . . . . 6 ⊢ ∅:∅⟶∅
9		ffn 6711	. . . . . 6 ⊢ (∅:∅⟶∅ → ∅ Fn ∅)
10	8, 9	ax-mp 5	. . . . 5 ⊢ ∅ Fn ∅
11		0xp 5758	. . . . . 6 ⊢ (∅ × ∅) = ∅
12	11	fneq2i 6641	. . . . 5 ⊢ (∅ Fn (∅ × ∅) ↔ ∅ Fn ∅)
13	10, 12	mpbir 231	. . . 4 ⊢ ∅ Fn (∅ × ∅)
14	13	a1i 11	. . 3 ⊢ (𝐶 ∈ Cat → ∅ Fn (∅ × ∅))
15	4, 5, 6, 7, 14	issubc2 17854	. 2 ⊢ (𝐶 ∈ Cat → (∅ ∈ (Subcat‘𝐶) ↔ (∅ ⊆cat (Homf ‘𝐶) ∧ ∀𝑥 ∈ ∅ (((Id‘𝐶)‘𝑥) ∈ (𝑥∅𝑥) ∧ ∀𝑦 ∈ ∅ ∀𝑧 ∈ ∅ ∀𝑓 ∈ (𝑥∅𝑦)∀𝑔 ∈ (𝑦∅𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥∅𝑧)))))
16	1, 3, 15	mpbir2and 713	1 ⊢ (𝐶 ∈ Cat → ∅ ∈ (Subcat‘𝐶))

Syntax hints:   → wi 4   ∧ wa 395   ∈ wcel 2109  ∀wral 3052  ∅c0 4313  ⟨cop 4612   class class class wbr 5124   × cxp 5657   Fn wfn 6531  ⟶wf 6532  ‘cfv 6536  (class class class)co 7410  compcco 17288  Catccat 17681  Idccid 17682  Hom_f chomf 17683   ⊆_cat cssc 17825  Subcatcsubc 17827

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2708  ax-rep 5254  ax-sep 5271  ax-nul 5281  ax-pow 5340  ax-pr 5407  ax-un 7734

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-reu 3365  df-rab 3421  df-v 3466  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-iun 4974  df-br 5125  df-opab 5187  df-mpt 5207  df-id 5553  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-iota 6489  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-ov 7413  df-oprab 7414  df-mpo 7415  df-1st 7993  df-2nd 7994  df-pm 8848  df-ixp 8917  df-homf 17687  df-ssc 17828  df-subc 17830

This theorem is referenced by: (None)