Theorem 0subcat 17553

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 17552	. 2 ⊢ (𝐶 ∈ Cat → ∅ ⊆cat (Homf ‘𝐶))
2		ral0 4443	. . 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 2738	. . 3 ⊢ (Homf ‘𝐶) = (Homf ‘𝐶)
5		eqid 2738	. . 3 ⊢ (Id‘𝐶) = (Id‘𝐶)
6		eqid 2738	. . 3 ⊢ (comp‘𝐶) = (comp‘𝐶)
7		id 22	. . 3 ⊢ (𝐶 ∈ Cat → 𝐶 ∈ Cat)
8		f0 6655	. . . . . 6 ⊢ ∅:∅⟶∅
9		ffn 6600	. . . . . 6 ⊢ (∅:∅⟶∅ → ∅ Fn ∅)
10	8, 9	ax-mp 5	. . . . 5 ⊢ ∅ Fn ∅
11		0xp 5685	. . . . . 6 ⊢ (∅ × ∅) = ∅
12	11	fneq2i 6531	. . . . 5 ⊢ (∅ Fn (∅ × ∅) ↔ ∅ Fn ∅)
13	10, 12	mpbir 230	. . . 4 ⊢ ∅ Fn (∅ × ∅)
14	13	a1i 11	. . 3 ⊢ (𝐶 ∈ Cat → ∅ Fn (∅ × ∅))
15	4, 5, 6, 7, 14	issubc2 17551	. 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 710	1 ⊢ (𝐶 ∈ Cat → ∅ ∈ (Subcat‘𝐶))

Syntax hints:   → wi 4   ∧ wa 396   ∈ wcel 2106  ∀wral 3064  ∅c0 4256  ⟨cop 4567   class class class wbr 5074   × cxp 5587   Fn wfn 6428  ⟶wf 6429  ‘cfv 6433  (class class class)co 7275  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-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-ov 7278  df-oprab 7279  df-mpo 7280  df-1st 7831  df-2nd 7832  df-pm 8618  df-ixp 8686  df-homf 17379  df-ssc 17522  df-subc 17524

This theorem is referenced by: (None)