| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | irngval.r | . . 3
⊢ (𝜑 → 𝑅 ∈ Ring) | 
| 2 | 1 | elexd 3504 | . 2
⊢ (𝜑 → 𝑅 ∈ V) | 
| 3 |  | irngval.b | . . . . 5
⊢ 𝐵 = (Base‘𝑅) | 
| 4 | 3 | fvexi 6920 | . . . 4
⊢ 𝐵 ∈ V | 
| 5 | 4 | a1i 11 | . . 3
⊢ (𝜑 → 𝐵 ∈ V) | 
| 6 |  | irngval.s | . . 3
⊢ (𝜑 → 𝑆 ⊆ 𝐵) | 
| 7 | 5, 6 | ssexd 5324 | . 2
⊢ (𝜑 → 𝑆 ∈ V) | 
| 8 |  | fvexd 6921 | . . 3
⊢ (𝜑 →
(Monic1p‘𝑈) ∈ V) | 
| 9 |  | fvex 6919 | . . . . . 6
⊢ (𝑂‘𝑓) ∈ V | 
| 10 | 9 | cnvex 7947 | . . . . 5
⊢ ◡(𝑂‘𝑓) ∈ V | 
| 11 | 10 | imaex 7936 | . . . 4
⊢ (◡(𝑂‘𝑓) “ { 0 }) ∈
V | 
| 12 | 11 | rgenw 3065 | . . 3
⊢
∀𝑓 ∈
(Monic1p‘𝑈)(◡(𝑂‘𝑓) “ { 0 }) ∈
V | 
| 13 |  | iunexg 7988 | . . 3
⊢
(((Monic1p‘𝑈) ∈ V ∧ ∀𝑓 ∈ (Monic1p‘𝑈)(◡(𝑂‘𝑓) “ { 0 }) ∈ V) →
∪ 𝑓 ∈ (Monic1p‘𝑈)(◡(𝑂‘𝑓) “ { 0 }) ∈
V) | 
| 14 | 8, 12, 13 | sylancl 586 | . 2
⊢ (𝜑 → ∪ 𝑓 ∈ (Monic1p‘𝑈)(◡(𝑂‘𝑓) “ { 0 }) ∈
V) | 
| 15 |  | oveq12 7440 | . . . . . 6
⊢ ((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) → (𝑟 ↾s 𝑠) = (𝑅 ↾s 𝑆)) | 
| 16 |  | irngval.u | . . . . . 6
⊢ 𝑈 = (𝑅 ↾s 𝑆) | 
| 17 | 15, 16 | eqtr4di 2795 | . . . . 5
⊢ ((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) → (𝑟 ↾s 𝑠) = 𝑈) | 
| 18 | 17 | fveq2d 6910 | . . . 4
⊢ ((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) → (Monic1p‘(𝑟 ↾s 𝑠)) =
(Monic1p‘𝑈)) | 
| 19 |  | oveq12 7440 | . . . . . . . 8
⊢ ((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) → (𝑟 evalSub1 𝑠) = (𝑅 evalSub1 𝑆)) | 
| 20 |  | irngval.o | . . . . . . . 8
⊢ 𝑂 = (𝑅 evalSub1 𝑆) | 
| 21 | 19, 20 | eqtr4di 2795 | . . . . . . 7
⊢ ((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) → (𝑟 evalSub1 𝑠) = 𝑂) | 
| 22 | 21 | fveq1d 6908 | . . . . . 6
⊢ ((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) → ((𝑟 evalSub1 𝑠)‘𝑓) = (𝑂‘𝑓)) | 
| 23 | 22 | cnveqd 5886 | . . . . 5
⊢ ((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) → ◡((𝑟 evalSub1 𝑠)‘𝑓) = ◡(𝑂‘𝑓)) | 
| 24 |  | simpl 482 | . . . . . . . 8
⊢ ((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) → 𝑟 = 𝑅) | 
| 25 | 24 | fveq2d 6910 | . . . . . . 7
⊢ ((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) → (0g‘𝑟) = (0g‘𝑅)) | 
| 26 |  | irngval.0 | . . . . . . 7
⊢  0 =
(0g‘𝑅) | 
| 27 | 25, 26 | eqtr4di 2795 | . . . . . 6
⊢ ((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) → (0g‘𝑟) = 0 ) | 
| 28 | 27 | sneqd 4638 | . . . . 5
⊢ ((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) → {(0g‘𝑟)} = { 0 }) | 
| 29 | 23, 28 | imaeq12d 6079 | . . . 4
⊢ ((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) → (◡((𝑟 evalSub1 𝑠)‘𝑓) “ {(0g‘𝑟)}) = (◡(𝑂‘𝑓) “ { 0 })) | 
| 30 | 18, 29 | iuneq12d 5021 | . . 3
⊢ ((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) → ∪
𝑓 ∈
(Monic1p‘(𝑟 ↾s 𝑠))(◡((𝑟 evalSub1 𝑠)‘𝑓) “ {(0g‘𝑟)}) = ∪ 𝑓 ∈ (Monic1p‘𝑈)(◡(𝑂‘𝑓) “ { 0 })) | 
| 31 |  | df-irng 33734 | . . 3
⊢  IntgRing
= (𝑟 ∈ V, 𝑠 ∈ V ↦ ∪ 𝑓 ∈ (Monic1p‘(𝑟 ↾s 𝑠))(◡((𝑟 evalSub1 𝑠)‘𝑓) “ {(0g‘𝑟)})) | 
| 32 | 30, 31 | ovmpoga 7587 | . 2
⊢ ((𝑅 ∈ V ∧ 𝑆 ∈ V ∧ ∪ 𝑓 ∈ (Monic1p‘𝑈)(◡(𝑂‘𝑓) “ { 0 }) ∈ V) → (𝑅 IntgRing 𝑆) = ∪
𝑓 ∈
(Monic1p‘𝑈)(◡(𝑂‘𝑓) “ { 0 })) | 
| 33 | 2, 7, 14, 32 | syl3anc 1373 | 1
⊢ (𝜑 → (𝑅 IntgRing 𝑆) = ∪
𝑓 ∈
(Monic1p‘𝑈)(◡(𝑂‘𝑓) “ { 0 })) |