Theorem 0subcat 17796

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 17795	. 2 ⊢ (𝐶 ∈ Cat → ∅ ⊆cat (Homf ‘𝐶))
2		ral0 4439	. . 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 2737	. . 3 ⊢ (Homf ‘𝐶) = (Homf ‘𝐶)
5		eqid 2737	. . 3 ⊢ (Id‘𝐶) = (Id‘𝐶)
6		eqid 2737	. . 3 ⊢ (comp‘𝐶) = (comp‘𝐶)
7		id 22	. . 3 ⊢ (𝐶 ∈ Cat → 𝐶 ∈ Cat)
8		f0 6715	. . . . . 6 ⊢ ∅:∅⟶∅
9		ffn 6662	. . . . . 6 ⊢ (∅:∅⟶∅ → ∅ Fn ∅)
10	8, 9	ax-mp 5	. . . . 5 ⊢ ∅ Fn ∅
11		0xp 5723	. . . . . 6 ⊢ (∅ × ∅) = ∅
12	11	fneq2i 6590	. . . . 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 17794	. 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 714	1 ⊢ (𝐶 ∈ Cat → ∅ ∈ (Subcat‘𝐶))

Syntax hints:   → wi 4   ∧ wa 395   ∈ wcel 2114  ∀wral 3052  ∅c0 4274  ⟨cop 4574   class class class wbr 5086   × cxp 5622   Fn wfn 6487  ⟶wf 6488  ‘cfv 6492  (class class class)co 7360  compcco 17223  Catccat 17621  Idccid 17622  Hom_f chomf 17623   ⊆_cat cssc 17765  Subcatcsubc 17767

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5302  ax-pr 5370  ax-un 7682

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-ov 7363  df-oprab 7364  df-mpo 7365  df-1st 7935  df-2nd 7936  df-pm 8769  df-ixp 8839  df-homf 17627  df-ssc 17768  df-subc 17770

This theorem is referenced by: (None)