| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | df-risc 37991 | . . 3
⊢ 
≃𝑟 = {〈𝑟, 𝑠〉 ∣ ((𝑟 ∈ RingOps ∧ 𝑠 ∈ RingOps) ∧ ∃𝑓 𝑓 ∈ (𝑟 RingOpsIso 𝑠))} | 
| 2 | 1 | relopabiv 5829 | . 2
⊢ Rel
≃𝑟 | 
| 3 |  | eqid 2736 | . 2
⊢ dom
≃𝑟 = dom ≃𝑟 | 
| 4 |  | vex 3483 | . . . . . . 7
⊢ 𝑟 ∈ V | 
| 5 |  | vex 3483 | . . . . . . 7
⊢ 𝑠 ∈ V | 
| 6 | 4, 5 | isrisc 37993 | . . . . . 6
⊢ (𝑟 ≃𝑟
𝑠 ↔ ((𝑟 ∈ RingOps ∧ 𝑠 ∈ RingOps) ∧
∃𝑓 𝑓 ∈ (𝑟 RingOpsIso 𝑠))) | 
| 7 |  | rngoisocnv 37989 | . . . . . . . . . 10
⊢ ((𝑟 ∈ RingOps ∧ 𝑠 ∈ RingOps ∧ 𝑓 ∈ (𝑟 RingOpsIso 𝑠)) → ◡𝑓 ∈ (𝑠 RingOpsIso 𝑟)) | 
| 8 | 7 | 3expia 1121 | . . . . . . . . 9
⊢ ((𝑟 ∈ RingOps ∧ 𝑠 ∈ RingOps) → (𝑓 ∈ (𝑟 RingOpsIso 𝑠) → ◡𝑓 ∈ (𝑠 RingOpsIso 𝑟))) | 
| 9 |  | risci 37995 | . . . . . . . . . . 11
⊢ ((𝑠 ∈ RingOps ∧ 𝑟 ∈ RingOps ∧ ◡𝑓 ∈ (𝑠 RingOpsIso 𝑟)) → 𝑠 ≃𝑟 𝑟) | 
| 10 | 9 | 3expia 1121 | . . . . . . . . . 10
⊢ ((𝑠 ∈ RingOps ∧ 𝑟 ∈ RingOps) → (◡𝑓 ∈ (𝑠 RingOpsIso 𝑟) → 𝑠 ≃𝑟 𝑟)) | 
| 11 | 10 | ancoms 458 | . . . . . . . . 9
⊢ ((𝑟 ∈ RingOps ∧ 𝑠 ∈ RingOps) → (◡𝑓 ∈ (𝑠 RingOpsIso 𝑟) → 𝑠 ≃𝑟 𝑟)) | 
| 12 | 8, 11 | syld 47 | . . . . . . . 8
⊢ ((𝑟 ∈ RingOps ∧ 𝑠 ∈ RingOps) → (𝑓 ∈ (𝑟 RingOpsIso 𝑠) → 𝑠 ≃𝑟 𝑟)) | 
| 13 | 12 | exlimdv 1932 | . . . . . . 7
⊢ ((𝑟 ∈ RingOps ∧ 𝑠 ∈ RingOps) →
(∃𝑓 𝑓 ∈ (𝑟 RingOpsIso 𝑠) → 𝑠 ≃𝑟 𝑟)) | 
| 14 | 13 | imp 406 | . . . . . 6
⊢ (((𝑟 ∈ RingOps ∧ 𝑠 ∈ RingOps) ∧
∃𝑓 𝑓 ∈ (𝑟 RingOpsIso 𝑠)) → 𝑠 ≃𝑟 𝑟) | 
| 15 | 6, 14 | sylbi 217 | . . . . 5
⊢ (𝑟 ≃𝑟
𝑠 → 𝑠 ≃𝑟 𝑟) | 
| 16 |  | vex 3483 | . . . . . . 7
⊢ 𝑡 ∈ V | 
| 17 | 5, 16 | isrisc 37993 | . . . . . 6
⊢ (𝑠 ≃𝑟
𝑡 ↔ ((𝑠 ∈ RingOps ∧ 𝑡 ∈ RingOps) ∧
∃𝑔 𝑔 ∈ (𝑠 RingOpsIso 𝑡))) | 
| 18 |  | exdistrv 1954 | . . . . . . . . . . 11
⊢
(∃𝑓∃𝑔(𝑓 ∈ (𝑟 RingOpsIso 𝑠) ∧ 𝑔 ∈ (𝑠 RingOpsIso 𝑡)) ↔ (∃𝑓 𝑓 ∈ (𝑟 RingOpsIso 𝑠) ∧ ∃𝑔 𝑔 ∈ (𝑠 RingOpsIso 𝑡))) | 
| 19 |  | rngoisoco 37990 | . . . . . . . . . . . . . 14
⊢ (((𝑟 ∈ RingOps ∧ 𝑠 ∈ RingOps ∧ 𝑡 ∈ RingOps) ∧ (𝑓 ∈ (𝑟 RingOpsIso 𝑠) ∧ 𝑔 ∈ (𝑠 RingOpsIso 𝑡))) → (𝑔 ∘ 𝑓) ∈ (𝑟 RingOpsIso 𝑡)) | 
| 20 | 19 | ex 412 | . . . . . . . . . . . . 13
⊢ ((𝑟 ∈ RingOps ∧ 𝑠 ∈ RingOps ∧ 𝑡 ∈ RingOps) → ((𝑓 ∈ (𝑟 RingOpsIso 𝑠) ∧ 𝑔 ∈ (𝑠 RingOpsIso 𝑡)) → (𝑔 ∘ 𝑓) ∈ (𝑟 RingOpsIso 𝑡))) | 
| 21 |  | risci 37995 | . . . . . . . . . . . . . . 15
⊢ ((𝑟 ∈ RingOps ∧ 𝑡 ∈ RingOps ∧ (𝑔 ∘ 𝑓) ∈ (𝑟 RingOpsIso 𝑡)) → 𝑟 ≃𝑟 𝑡) | 
| 22 | 21 | 3expia 1121 | . . . . . . . . . . . . . 14
⊢ ((𝑟 ∈ RingOps ∧ 𝑡 ∈ RingOps) → ((𝑔 ∘ 𝑓) ∈ (𝑟 RingOpsIso 𝑡) → 𝑟 ≃𝑟 𝑡)) | 
| 23 | 22 | 3adant2 1131 | . . . . . . . . . . . . 13
⊢ ((𝑟 ∈ RingOps ∧ 𝑠 ∈ RingOps ∧ 𝑡 ∈ RingOps) → ((𝑔 ∘ 𝑓) ∈ (𝑟 RingOpsIso 𝑡) → 𝑟 ≃𝑟 𝑡)) | 
| 24 | 20, 23 | syld 47 | . . . . . . . . . . . 12
⊢ ((𝑟 ∈ RingOps ∧ 𝑠 ∈ RingOps ∧ 𝑡 ∈ RingOps) → ((𝑓 ∈ (𝑟 RingOpsIso 𝑠) ∧ 𝑔 ∈ (𝑠 RingOpsIso 𝑡)) → 𝑟 ≃𝑟 𝑡)) | 
| 25 | 24 | exlimdvv 1933 | . . . . . . . . . . 11
⊢ ((𝑟 ∈ RingOps ∧ 𝑠 ∈ RingOps ∧ 𝑡 ∈ RingOps) →
(∃𝑓∃𝑔(𝑓 ∈ (𝑟 RingOpsIso 𝑠) ∧ 𝑔 ∈ (𝑠 RingOpsIso 𝑡)) → 𝑟 ≃𝑟 𝑡)) | 
| 26 | 18, 25 | biimtrrid 243 | . . . . . . . . . 10
⊢ ((𝑟 ∈ RingOps ∧ 𝑠 ∈ RingOps ∧ 𝑡 ∈ RingOps) →
((∃𝑓 𝑓 ∈ (𝑟 RingOpsIso 𝑠) ∧ ∃𝑔 𝑔 ∈ (𝑠 RingOpsIso 𝑡)) → 𝑟 ≃𝑟 𝑡)) | 
| 27 | 26 | 3expb 1120 | . . . . . . . . 9
⊢ ((𝑟 ∈ RingOps ∧ (𝑠 ∈ RingOps ∧ 𝑡 ∈ RingOps)) →
((∃𝑓 𝑓 ∈ (𝑟 RingOpsIso 𝑠) ∧ ∃𝑔 𝑔 ∈ (𝑠 RingOpsIso 𝑡)) → 𝑟 ≃𝑟 𝑡)) | 
| 28 | 27 | adantlr 715 | . . . . . . . 8
⊢ (((𝑟 ∈ RingOps ∧ 𝑠 ∈ RingOps) ∧ (𝑠 ∈ RingOps ∧ 𝑡 ∈ RingOps)) →
((∃𝑓 𝑓 ∈ (𝑟 RingOpsIso 𝑠) ∧ ∃𝑔 𝑔 ∈ (𝑠 RingOpsIso 𝑡)) → 𝑟 ≃𝑟 𝑡)) | 
| 29 | 28 | imp 406 | . . . . . . 7
⊢ ((((𝑟 ∈ RingOps ∧ 𝑠 ∈ RingOps) ∧ (𝑠 ∈ RingOps ∧ 𝑡 ∈ RingOps)) ∧
(∃𝑓 𝑓 ∈ (𝑟 RingOpsIso 𝑠) ∧ ∃𝑔 𝑔 ∈ (𝑠 RingOpsIso 𝑡))) → 𝑟 ≃𝑟 𝑡) | 
| 30 | 29 | an4s 660 | . . . . . 6
⊢ ((((𝑟 ∈ RingOps ∧ 𝑠 ∈ RingOps) ∧
∃𝑓 𝑓 ∈ (𝑟 RingOpsIso 𝑠)) ∧ ((𝑠 ∈ RingOps ∧ 𝑡 ∈ RingOps) ∧ ∃𝑔 𝑔 ∈ (𝑠 RingOpsIso 𝑡))) → 𝑟 ≃𝑟 𝑡) | 
| 31 | 6, 17, 30 | syl2anb 598 | . . . . 5
⊢ ((𝑟 ≃𝑟
𝑠 ∧ 𝑠 ≃𝑟 𝑡) → 𝑟 ≃𝑟 𝑡) | 
| 32 | 15, 31 | pm3.2i 470 | . . . 4
⊢ ((𝑟 ≃𝑟
𝑠 → 𝑠 ≃𝑟 𝑟) ∧ ((𝑟 ≃𝑟 𝑠 ∧ 𝑠 ≃𝑟 𝑡) → 𝑟 ≃𝑟 𝑡)) | 
| 33 | 32 | ax-gen 1794 | . . 3
⊢
∀𝑡((𝑟 ≃𝑟
𝑠 → 𝑠 ≃𝑟 𝑟) ∧ ((𝑟 ≃𝑟 𝑠 ∧ 𝑠 ≃𝑟 𝑡) → 𝑟 ≃𝑟 𝑡)) | 
| 34 | 33 | gen2 1795 | . 2
⊢
∀𝑟∀𝑠∀𝑡((𝑟 ≃𝑟 𝑠 → 𝑠 ≃𝑟 𝑟) ∧ ((𝑟 ≃𝑟 𝑠 ∧ 𝑠 ≃𝑟 𝑡) → 𝑟 ≃𝑟 𝑡)) | 
| 35 |  | dfer2 8747 | . 2
⊢ (
≃𝑟 Er dom ≃𝑟 ↔ (Rel
≃𝑟 ∧ dom ≃𝑟 = dom
≃𝑟 ∧ ∀𝑟∀𝑠∀𝑡((𝑟 ≃𝑟 𝑠 → 𝑠 ≃𝑟 𝑟) ∧ ((𝑟 ≃𝑟 𝑠 ∧ 𝑠 ≃𝑟 𝑡) → 𝑟 ≃𝑟 𝑡)))) | 
| 36 | 2, 3, 34, 35 | mpbir3an 1341 | 1
⊢ 
≃𝑟 Er dom ≃𝑟 |