Theorem 0catg 16701

Description: Any structure with an empty set of objects is a category. (Contributed by Mario Carneiro, 3-Jan-2017.)

Assertion

Ref	Expression
0catg	⊢ ((𝐶 ∈ 𝑉 ∧ ∅ = (Base‘𝐶)) → 𝐶 ∈ Cat)

Proof of Theorem 0catg

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

Step	Hyp	Ref	Expression
1		simpr 479	. 2 ⊢ ((𝐶 ∈ 𝑉 ∧ ∅ = (Base‘𝐶)) → ∅ = (Base‘𝐶))
2		eqidd 2827	. 2 ⊢ ((𝐶 ∈ 𝑉 ∧ ∅ = (Base‘𝐶)) → (Hom ‘𝐶) = (Hom ‘𝐶))
3		eqidd 2827	. 2 ⊢ ((𝐶 ∈ 𝑉 ∧ ∅ = (Base‘𝐶)) → (comp‘𝐶) = (comp‘𝐶))
4		simpl 476	. 2 ⊢ ((𝐶 ∈ 𝑉 ∧ ∅ = (Base‘𝐶)) → 𝐶 ∈ 𝑉)
5		noel 4149	. . . 4 ⊢ ¬ 𝑥 ∈ ∅
6	5	pm2.21i 117	. . 3 ⊢ (𝑥 ∈ ∅ → ∅ ∈ (𝑥(Hom ‘𝐶)𝑥))
7	6	adantl 475	. 2 ⊢ (((𝐶 ∈ 𝑉 ∧ ∅ = (Base‘𝐶)) ∧ 𝑥 ∈ ∅) → ∅ ∈ (𝑥(Hom ‘𝐶)𝑥))
8		simpr1 1254	. . 3 ⊢ (((𝐶 ∈ 𝑉 ∧ ∅ = (Base‘𝐶)) ∧ (𝑥 ∈ ∅ ∧ 𝑦 ∈ ∅ ∧ 𝑓 ∈ (𝑦(Hom ‘𝐶)𝑥))) → 𝑥 ∈ ∅)
9	5	pm2.21i 117	. . 3 ⊢ (𝑥 ∈ ∅ → (∅(⟨𝑦, 𝑥⟩(comp‘𝐶)𝑥)𝑓) = 𝑓)
10	8, 9	syl 17	. 2 ⊢ (((𝐶 ∈ 𝑉 ∧ ∅ = (Base‘𝐶)) ∧ (𝑥 ∈ ∅ ∧ 𝑦 ∈ ∅ ∧ 𝑓 ∈ (𝑦(Hom ‘𝐶)𝑥))) → (∅(⟨𝑦, 𝑥⟩(comp‘𝐶)𝑥)𝑓) = 𝑓)
11		simpr1 1254	. . 3 ⊢ (((𝐶 ∈ 𝑉 ∧ ∅ = (Base‘𝐶)) ∧ (𝑥 ∈ ∅ ∧ 𝑦 ∈ ∅ ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))) → 𝑥 ∈ ∅)
12	5	pm2.21i 117	. . 3 ⊢ (𝑥 ∈ ∅ → (𝑓(⟨𝑥, 𝑥⟩(comp‘𝐶)𝑦)∅) = 𝑓)
13	11, 12	syl 17	. 2 ⊢ (((𝐶 ∈ 𝑉 ∧ ∅ = (Base‘𝐶)) ∧ (𝑥 ∈ ∅ ∧ 𝑦 ∈ ∅ ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))) → (𝑓(⟨𝑥, 𝑥⟩(comp‘𝐶)𝑦)∅) = 𝑓)
14		simp21 1269	. . 3 ⊢ (((𝐶 ∈ 𝑉 ∧ ∅ = (Base‘𝐶)) ∧ (𝑥 ∈ ∅ ∧ 𝑦 ∈ ∅ ∧ 𝑧 ∈ ∅) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) → 𝑥 ∈ ∅)
15	5	pm2.21i 117	. . 3 ⊢ (𝑥 ∈ ∅ → (𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥(Hom ‘𝐶)𝑧))
16	14, 15	syl 17	. 2 ⊢ (((𝐶 ∈ 𝑉 ∧ ∅ = (Base‘𝐶)) ∧ (𝑥 ∈ ∅ ∧ 𝑦 ∈ ∅ ∧ 𝑧 ∈ ∅) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) → (𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥(Hom ‘𝐶)𝑧))
17		simp2ll 1327	. . 3 ⊢ (((𝐶 ∈ 𝑉 ∧ ∅ = (Base‘𝐶)) ∧ ((𝑥 ∈ ∅ ∧ 𝑦 ∈ ∅) ∧ (𝑧 ∈ ∅ ∧ 𝑤 ∈ ∅)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤))) → 𝑥 ∈ ∅)
18	5	pm2.21i 117	. . 3 ⊢ (𝑥 ∈ ∅ → ((ℎ(⟨𝑦, 𝑧⟩(comp‘𝐶)𝑤)𝑔)(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑤)𝑓) = (ℎ(⟨𝑥, 𝑧⟩(comp‘𝐶)𝑤)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓)))
19	17, 18	syl 17	. 2 ⊢ (((𝐶 ∈ 𝑉 ∧ ∅ = (Base‘𝐶)) ∧ ((𝑥 ∈ ∅ ∧ 𝑦 ∈ ∅) ∧ (𝑧 ∈ ∅ ∧ 𝑤 ∈ ∅)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤))) → ((ℎ(⟨𝑦, 𝑧⟩(comp‘𝐶)𝑤)𝑔)(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑤)𝑓) = (ℎ(⟨𝑥, 𝑧⟩(comp‘𝐶)𝑤)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓)))
20	1, 2, 3, 4, 7, 10, 13, 16, 19	iscatd 16687	1 ⊢ ((𝐶 ∈ 𝑉 ∧ ∅ = (Base‘𝐶)) → 𝐶 ∈ Cat)

Syntax hints:   → wi 4   ∧ wa 386   ∧ w3a 1113   = wceq 1658   ∈ wcel 2166  ∅c0 4145  ⟨cop 4404  ‘cfv 6124  (class class class)co 6906  Basecbs 16223  Hom chom 16317  compcco 16318  Catccat 16678

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2804  ax-nul 5014

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2606  df-eu 2641  df-clab 2813  df-cleq 2819  df-clel 2822  df-nfc 2959  df-ral 3123  df-rex 3124  df-rab 3127  df-v 3417  df-sbc 3664  df-dif 3802  df-un 3804  df-in 3806  df-ss 3813  df-nul 4146  df-if 4308  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4660  df-br 4875  df-iota 6087  df-fv 6132  df-ov 6909  df-cat 16682

This theorem is referenced by:  0cat  16702