Theorem 0catg 17766

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 490	. 2 ⊢ ((𝐶 ∈ 𝑉 ∧ ∅ = (Base‘𝐶)) → ∅ = (Base‘𝐶))
2		eqidd 2766	. 2 ⊢ ((𝐶 ∈ 𝑉 ∧ ∅ = (Base‘𝐶)) → (Hom ‘𝐶) = (Hom ‘𝐶))
3		eqidd 2766	. 2 ⊢ ((𝐶 ∈ 𝑉 ∧ ∅ = (Base‘𝐶)) → (comp‘𝐶) = (comp‘𝐶))
4		simpl 488	. 2 ⊢ ((𝐶 ∈ 𝑉 ∧ ∅ = (Base‘𝐶)) → 𝐶 ∈ 𝑉)
5		noel 4291	. . . 4 ⊢ ¬ 𝑥 ∈ ∅
6	5	pm2.21i 120	. . 3 ⊢ (𝑥 ∈ ∅ → ∅ ∈ (𝑥(Hom ‘𝐶)𝑥))
7	6	adantl 487	. 2 ⊢ (((𝐶 ∈ 𝑉 ∧ ∅ = (Base‘𝐶)) ∧ 𝑥 ∈ ∅) → ∅ ∈ (𝑥(Hom ‘𝐶)𝑥))
8		simpr1 1213	. . 3 ⊢ (((𝐶 ∈ 𝑉 ∧ ∅ = (Base‘𝐶)) ∧ (𝑥 ∈ ∅ ∧ 𝑦 ∈ ∅ ∧ 𝑓 ∈ (𝑦(Hom ‘𝐶)𝑥))) → 𝑥 ∈ ∅)
9	5	pm2.21i 120	. . 3 ⊢ (𝑥 ∈ ∅ → (∅(⟨𝑦, 𝑥⟩(comp‘𝐶)𝑥)𝑓) = 𝑓)
10	8, 9	syl 18	. 2 ⊢ (((𝐶 ∈ 𝑉 ∧ ∅ = (Base‘𝐶)) ∧ (𝑥 ∈ ∅ ∧ 𝑦 ∈ ∅ ∧ 𝑓 ∈ (𝑦(Hom ‘𝐶)𝑥))) → (∅(⟨𝑦, 𝑥⟩(comp‘𝐶)𝑥)𝑓) = 𝑓)
11		simpr1 1213	. . 3 ⊢ (((𝐶 ∈ 𝑉 ∧ ∅ = (Base‘𝐶)) ∧ (𝑥 ∈ ∅ ∧ 𝑦 ∈ ∅ ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))) → 𝑥 ∈ ∅)
12	5	pm2.21i 120	. . 3 ⊢ (𝑥 ∈ ∅ → (𝑓(⟨𝑥, 𝑥⟩(comp‘𝐶)𝑦)∅) = 𝑓)
13	11, 12	syl 18	. 2 ⊢ (((𝐶 ∈ 𝑉 ∧ ∅ = (Base‘𝐶)) ∧ (𝑥 ∈ ∅ ∧ 𝑦 ∈ ∅ ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))) → (𝑓(⟨𝑥, 𝑥⟩(comp‘𝐶)𝑦)∅) = 𝑓)
14		simp21 1225	. . 3 ⊢ (((𝐶 ∈ 𝑉 ∧ ∅ = (Base‘𝐶)) ∧ (𝑥 ∈ ∅ ∧ 𝑦 ∈ ∅ ∧ 𝑧 ∈ ∅) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) → 𝑥 ∈ ∅)
15	5	pm2.21i 120	. . 3 ⊢ (𝑥 ∈ ∅ → (𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥(Hom ‘𝐶)𝑧))
16	14, 15	syl 18	. 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 1259	. . 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 120	. . 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 18	. 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 17751	1 ⊢ ((𝐶 ∈ 𝑉 ∧ ∅ = (Base‘𝐶)) → 𝐶 ∈ Cat)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2146  ∅c0 4286  ⟨cop 4597  ‘cfv 6540  (class class class)co 7419  Basecbs 17291  Hom chom 17343  compcco 17344  Catccat 17742

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2737  ax-nul 5271

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2744  df-cleq 2757  df-clel 2840  df-ne 2961  df-ral 3082  df-rex 3092  df-rab 3419  df-v 3459  df-sbc 3747  df-dif 3909  df-un 3911  df-ss 3923  df-nul 4287  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-br 5112  df-iota 6496  df-fv 6548  df-ov 7422  df-cat 17746

This theorem is used by:  0cat  17767  resccat  49909  0funcg2  49919  0thincg  50293  0fucterm  50378