Theorem 0subcat 17469

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 17468	. 2 ⊢ (𝐶 ∈ Cat → ∅ ⊆cat (Homf ‘𝐶))
2		ral0 4440	. . 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 6639	. . . . . 6 ⊢ ∅:∅⟶∅
9		ffn 6584	. . . . . 6 ⊢ (∅:∅⟶∅ → ∅ Fn ∅)
10	8, 9	ax-mp 5	. . . . 5 ⊢ ∅ Fn ∅
11		0xp 5675	. . . . . 6 ⊢ (∅ × ∅) = ∅
12	11	fneq2i 6515	. . . . 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 17467	. 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 709	1 ⊢ (𝐶 ∈ Cat → ∅ ∈ (Subcat‘𝐶))

Syntax hints:   → wi 4   ∧ wa 395   ∈ wcel 2108  ∀wral 3063  ∅c0 4253  ⟨cop 4564   class class class wbr 5070   × cxp 5578   Fn wfn 6413  ⟶wf 6414  ‘cfv 6418  (class class class)co 7255  compcco 16900  Catccat 17290  Idccid 17291  Hom_f chomf 17292   ⊆_cat cssc 17436  Subcatcsubc 17438

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-reu 3070  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-ov 7258  df-oprab 7259  df-mpo 7260  df-1st 7804  df-2nd 7805  df-pm 8576  df-ixp 8644  df-homf 17296  df-ssc 17439  df-subc 17441

This theorem is referenced by: (None)