Theorem 0catg 17609

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 484	. 2 ⊢ ((𝐶 ∈ 𝑉 ∧ ∅ = (Base‘𝐶)) → ∅ = (Base‘𝐶))
2		eqidd 2735	. 2 ⊢ ((𝐶 ∈ 𝑉 ∧ ∅ = (Base‘𝐶)) → (Hom ‘𝐶) = (Hom ‘𝐶))
3		eqidd 2735	. 2 ⊢ ((𝐶 ∈ 𝑉 ∧ ∅ = (Base‘𝐶)) → (comp‘𝐶) = (comp‘𝐶))
4		simpl 482	. 2 ⊢ ((𝐶 ∈ 𝑉 ∧ ∅ = (Base‘𝐶)) → 𝐶 ∈ 𝑉)
5		noel 4288	. . . 4 ⊢ ¬ 𝑥 ∈ ∅
6	5	pm2.21i 119	. . 3 ⊢ (𝑥 ∈ ∅ → ∅ ∈ (𝑥(Hom ‘𝐶)𝑥))
7	6	adantl 481	. 2 ⊢ (((𝐶 ∈ 𝑉 ∧ ∅ = (Base‘𝐶)) ∧ 𝑥 ∈ ∅) → ∅ ∈ (𝑥(Hom ‘𝐶)𝑥))
8		simpr1 1195	. . 3 ⊢ (((𝐶 ∈ 𝑉 ∧ ∅ = (Base‘𝐶)) ∧ (𝑥 ∈ ∅ ∧ 𝑦 ∈ ∅ ∧ 𝑓 ∈ (𝑦(Hom ‘𝐶)𝑥))) → 𝑥 ∈ ∅)
9	5	pm2.21i 119	. . 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 1195	. . 3 ⊢ (((𝐶 ∈ 𝑉 ∧ ∅ = (Base‘𝐶)) ∧ (𝑥 ∈ ∅ ∧ 𝑦 ∈ ∅ ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))) → 𝑥 ∈ ∅)
12	5	pm2.21i 119	. . 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 1207	. . 3 ⊢ (((𝐶 ∈ 𝑉 ∧ ∅ = (Base‘𝐶)) ∧ (𝑥 ∈ ∅ ∧ 𝑦 ∈ ∅ ∧ 𝑧 ∈ ∅) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) → 𝑥 ∈ ∅)
15	5	pm2.21i 119	. . 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 1241	. . 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 119	. . 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 17594	1 ⊢ ((𝐶 ∈ 𝑉 ∧ ∅ = (Base‘𝐶)) → 𝐶 ∈ Cat)

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1541   ∈ wcel 2113  ∅c0 4283  ⟨cop 4584  ‘cfv 6490  (class class class)co 7356  Basecbs 17134  Hom chom 17186  compcco 17187  Catccat 17585

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2706  ax-nul 5249

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2713  df-cleq 2726  df-clel 2809  df-ne 2931  df-ral 3050  df-rex 3059  df-rab 3398  df-v 3440  df-sbc 3739  df-dif 3902  df-un 3904  df-ss 3916  df-nul 4284  df-if 4478  df-sn 4579  df-pr 4581  df-op 4585  df-uni 4862  df-br 5097  df-iota 6446  df-fv 6498  df-ov 7359  df-cat 17589

This theorem is referenced by:  0cat  17610  resccat  49261  0funcg2  49271  0thincg  49645  0fucterm  49730