| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | relopabv 5830 | . . . . . 6
⊢ Rel
{〈𝑥, 𝑦〉 ∣ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ ((Iso‘𝐶)‘〈𝑥, 𝑦〉) ≠ ∅)} | 
| 2 | 1 | a1i 11 | . . . . 5
⊢ (𝐶 ∈ Cat → Rel
{〈𝑥, 𝑦〉 ∣ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ ((Iso‘𝐶)‘〈𝑥, 𝑦〉) ≠ ∅)}) | 
| 3 |  | fveq2 6905 | . . . . . . . . 9
⊢ (𝑓 = 〈𝑥, 𝑦〉 → ((Iso‘𝐶)‘𝑓) = ((Iso‘𝐶)‘〈𝑥, 𝑦〉)) | 
| 4 | 3 | neeq1d 2999 | . . . . . . . 8
⊢ (𝑓 = 〈𝑥, 𝑦〉 → (((Iso‘𝐶)‘𝑓) ≠ ∅ ↔ ((Iso‘𝐶)‘〈𝑥, 𝑦〉) ≠ ∅)) | 
| 5 | 4 | rabxp 5732 | . . . . . . 7
⊢ {𝑓 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∣ ((Iso‘𝐶)‘𝑓) ≠ ∅} = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ ((Iso‘𝐶)‘〈𝑥, 𝑦〉) ≠ ∅)} | 
| 6 | 5 | a1i 11 | . . . . . 6
⊢ (𝐶 ∈ Cat → {𝑓 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∣ ((Iso‘𝐶)‘𝑓) ≠ ∅} = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ ((Iso‘𝐶)‘〈𝑥, 𝑦〉) ≠ ∅)}) | 
| 7 | 6 | releqd 5787 | . . . . 5
⊢ (𝐶 ∈ Cat → (Rel {𝑓 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∣ ((Iso‘𝐶)‘𝑓) ≠ ∅} ↔ Rel {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ ((Iso‘𝐶)‘〈𝑥, 𝑦〉) ≠ ∅)})) | 
| 8 | 2, 7 | mpbird 257 | . . . 4
⊢ (𝐶 ∈ Cat → Rel {𝑓 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∣ ((Iso‘𝐶)‘𝑓) ≠ ∅}) | 
| 9 |  | isofn 17820 | . . . . . 6
⊢ (𝐶 ∈ Cat →
(Iso‘𝐶) Fn
((Base‘𝐶) ×
(Base‘𝐶))) | 
| 10 |  | fvex 6918 | . . . . . . 7
⊢
(Base‘𝐶)
∈ V | 
| 11 |  | sqxpexg 7776 | . . . . . . 7
⊢
((Base‘𝐶)
∈ V → ((Base‘𝐶) × (Base‘𝐶)) ∈ V) | 
| 12 | 10, 11 | mp1i 13 | . . . . . 6
⊢ (𝐶 ∈ Cat →
((Base‘𝐶) ×
(Base‘𝐶)) ∈
V) | 
| 13 |  | 0ex 5306 | . . . . . . 7
⊢ ∅
∈ V | 
| 14 | 13 | a1i 11 | . . . . . 6
⊢ (𝐶 ∈ Cat → ∅
∈ V) | 
| 15 |  | suppvalfn 8194 | . . . . . 6
⊢
(((Iso‘𝐶) Fn
((Base‘𝐶) ×
(Base‘𝐶)) ∧
((Base‘𝐶) ×
(Base‘𝐶)) ∈ V
∧ ∅ ∈ V) → ((Iso‘𝐶) supp ∅) = {𝑓 ∈ ((Base‘𝐶) × (Base‘𝐶)) ∣ ((Iso‘𝐶)‘𝑓) ≠ ∅}) | 
| 16 | 9, 12, 14, 15 | syl3anc 1372 | . . . . 5
⊢ (𝐶 ∈ Cat →
((Iso‘𝐶) supp
∅) = {𝑓 ∈
((Base‘𝐶) ×
(Base‘𝐶)) ∣
((Iso‘𝐶)‘𝑓) ≠
∅}) | 
| 17 | 16 | releqd 5787 | . . . 4
⊢ (𝐶 ∈ Cat → (Rel
((Iso‘𝐶) supp
∅) ↔ Rel {𝑓
∈ ((Base‘𝐶)
× (Base‘𝐶))
∣ ((Iso‘𝐶)‘𝑓) ≠ ∅})) | 
| 18 | 8, 17 | mpbird 257 | . . 3
⊢ (𝐶 ∈ Cat → Rel
((Iso‘𝐶) supp
∅)) | 
| 19 |  | cicfval 17842 | . . . 4
⊢ (𝐶 ∈ Cat → (
≃𝑐 ‘𝐶) = ((Iso‘𝐶) supp ∅)) | 
| 20 | 19 | releqd 5787 | . . 3
⊢ (𝐶 ∈ Cat → (Rel (
≃𝑐 ‘𝐶) ↔ Rel ((Iso‘𝐶) supp ∅))) | 
| 21 | 18, 20 | mpbird 257 | . 2
⊢ (𝐶 ∈ Cat → Rel (
≃𝑐 ‘𝐶)) | 
| 22 |  | cicsym 17849 | . 2
⊢ ((𝐶 ∈ Cat ∧ 𝑥( ≃𝑐
‘𝐶)𝑦) → 𝑦( ≃𝑐 ‘𝐶)𝑥) | 
| 23 |  | cictr 17850 | . . 3
⊢ ((𝐶 ∈ Cat ∧ 𝑥( ≃𝑐
‘𝐶)𝑦 ∧ 𝑦( ≃𝑐 ‘𝐶)𝑧) → 𝑥( ≃𝑐 ‘𝐶)𝑧) | 
| 24 | 23 | 3expb 1120 | . 2
⊢ ((𝐶 ∈ Cat ∧ (𝑥( ≃𝑐
‘𝐶)𝑦 ∧ 𝑦( ≃𝑐 ‘𝐶)𝑧)) → 𝑥( ≃𝑐 ‘𝐶)𝑧) | 
| 25 |  | cicref 17846 | . . 3
⊢ ((𝐶 ∈ Cat ∧ 𝑥 ∈ (Base‘𝐶)) → 𝑥( ≃𝑐 ‘𝐶)𝑥) | 
| 26 |  | ciclcl 17847 | . . 3
⊢ ((𝐶 ∈ Cat ∧ 𝑥( ≃𝑐
‘𝐶)𝑥) → 𝑥 ∈ (Base‘𝐶)) | 
| 27 | 25, 26 | impbida 800 | . 2
⊢ (𝐶 ∈ Cat → (𝑥 ∈ (Base‘𝐶) ↔ 𝑥( ≃𝑐 ‘𝐶)𝑥)) | 
| 28 | 21, 22, 24, 27 | iserd 8772 | 1
⊢ (𝐶 ∈ Cat → (
≃𝑐 ‘𝐶) Er (Base‘𝐶)) |