| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | fmptcof.1 | . . . . 5
⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝑅 ∈ 𝐵) | 
| 2 |  | nfcsb1v 3923 | . . . . . . 7
⊢
Ⅎ𝑥⦋𝑧 / 𝑥⦌𝑅 | 
| 3 | 2 | nfel1 2922 | . . . . . 6
⊢
Ⅎ𝑥⦋𝑧 / 𝑥⦌𝑅 ∈ 𝐵 | 
| 4 |  | csbeq1a 3913 | . . . . . . 7
⊢ (𝑥 = 𝑧 → 𝑅 = ⦋𝑧 / 𝑥⦌𝑅) | 
| 5 | 4 | eleq1d 2826 | . . . . . 6
⊢ (𝑥 = 𝑧 → (𝑅 ∈ 𝐵 ↔ ⦋𝑧 / 𝑥⦌𝑅 ∈ 𝐵)) | 
| 6 | 3, 5 | rspc 3610 | . . . . 5
⊢ (𝑧 ∈ 𝐴 → (∀𝑥 ∈ 𝐴 𝑅 ∈ 𝐵 → ⦋𝑧 / 𝑥⦌𝑅 ∈ 𝐵)) | 
| 7 | 1, 6 | mpan9 506 | . . . 4
⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → ⦋𝑧 / 𝑥⦌𝑅 ∈ 𝐵) | 
| 8 |  | fmptcof.2 | . . . . 5
⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝑅)) | 
| 9 |  | nfcv 2905 | . . . . . 6
⊢
Ⅎ𝑧𝑅 | 
| 10 | 9, 2, 4 | cbvmpt 5253 | . . . . 5
⊢ (𝑥 ∈ 𝐴 ↦ 𝑅) = (𝑧 ∈ 𝐴 ↦ ⦋𝑧 / 𝑥⦌𝑅) | 
| 11 | 8, 10 | eqtrdi 2793 | . . . 4
⊢ (𝜑 → 𝐹 = (𝑧 ∈ 𝐴 ↦ ⦋𝑧 / 𝑥⦌𝑅)) | 
| 12 |  | fmptcof.3 | . . . . 5
⊢ (𝜑 → 𝐺 = (𝑦 ∈ 𝐵 ↦ 𝑆)) | 
| 13 |  | nfcv 2905 | . . . . . 6
⊢
Ⅎ𝑤𝑆 | 
| 14 |  | nfcsb1v 3923 | . . . . . 6
⊢
Ⅎ𝑦⦋𝑤 / 𝑦⦌𝑆 | 
| 15 |  | csbeq1a 3913 | . . . . . 6
⊢ (𝑦 = 𝑤 → 𝑆 = ⦋𝑤 / 𝑦⦌𝑆) | 
| 16 | 13, 14, 15 | cbvmpt 5253 | . . . . 5
⊢ (𝑦 ∈ 𝐵 ↦ 𝑆) = (𝑤 ∈ 𝐵 ↦ ⦋𝑤 / 𝑦⦌𝑆) | 
| 17 | 12, 16 | eqtrdi 2793 | . . . 4
⊢ (𝜑 → 𝐺 = (𝑤 ∈ 𝐵 ↦ ⦋𝑤 / 𝑦⦌𝑆)) | 
| 18 |  | csbeq1 3902 | . . . 4
⊢ (𝑤 = ⦋𝑧 / 𝑥⦌𝑅 → ⦋𝑤 / 𝑦⦌𝑆 = ⦋⦋𝑧 / 𝑥⦌𝑅 / 𝑦⦌𝑆) | 
| 19 | 7, 11, 17, 18 | fmptco 7149 | . . 3
⊢ (𝜑 → (𝐺 ∘ 𝐹) = (𝑧 ∈ 𝐴 ↦ ⦋⦋𝑧 / 𝑥⦌𝑅 / 𝑦⦌𝑆)) | 
| 20 |  | nfcv 2905 | . . . 4
⊢
Ⅎ𝑧⦋𝑅 / 𝑦⦌𝑆 | 
| 21 |  | nfcv 2905 | . . . . 5
⊢
Ⅎ𝑥𝑆 | 
| 22 | 2, 21 | nfcsbw 3925 | . . . 4
⊢
Ⅎ𝑥⦋⦋𝑧 / 𝑥⦌𝑅 / 𝑦⦌𝑆 | 
| 23 | 4 | csbeq1d 3903 | . . . 4
⊢ (𝑥 = 𝑧 → ⦋𝑅 / 𝑦⦌𝑆 = ⦋⦋𝑧 / 𝑥⦌𝑅 / 𝑦⦌𝑆) | 
| 24 | 20, 22, 23 | cbvmpt 5253 | . . 3
⊢ (𝑥 ∈ 𝐴 ↦ ⦋𝑅 / 𝑦⦌𝑆) = (𝑧 ∈ 𝐴 ↦ ⦋⦋𝑧 / 𝑥⦌𝑅 / 𝑦⦌𝑆) | 
| 25 | 19, 24 | eqtr4di 2795 | . 2
⊢ (𝜑 → (𝐺 ∘ 𝐹) = (𝑥 ∈ 𝐴 ↦ ⦋𝑅 / 𝑦⦌𝑆)) | 
| 26 |  | eqid 2737 | . . . 4
⊢ 𝐴 = 𝐴 | 
| 27 |  | nfcvd 2906 | . . . . . 6
⊢ (𝑅 ∈ 𝐵 → Ⅎ𝑦𝑇) | 
| 28 |  | fmptcof.4 | . . . . . 6
⊢ (𝑦 = 𝑅 → 𝑆 = 𝑇) | 
| 29 | 27, 28 | csbiegf 3932 | . . . . 5
⊢ (𝑅 ∈ 𝐵 → ⦋𝑅 / 𝑦⦌𝑆 = 𝑇) | 
| 30 | 29 | ralimi 3083 | . . . 4
⊢
(∀𝑥 ∈
𝐴 𝑅 ∈ 𝐵 → ∀𝑥 ∈ 𝐴 ⦋𝑅 / 𝑦⦌𝑆 = 𝑇) | 
| 31 |  | mpteq12 5234 | . . . 4
⊢ ((𝐴 = 𝐴 ∧ ∀𝑥 ∈ 𝐴 ⦋𝑅 / 𝑦⦌𝑆 = 𝑇) → (𝑥 ∈ 𝐴 ↦ ⦋𝑅 / 𝑦⦌𝑆) = (𝑥 ∈ 𝐴 ↦ 𝑇)) | 
| 32 | 26, 30, 31 | sylancr 587 | . . 3
⊢
(∀𝑥 ∈
𝐴 𝑅 ∈ 𝐵 → (𝑥 ∈ 𝐴 ↦ ⦋𝑅 / 𝑦⦌𝑆) = (𝑥 ∈ 𝐴 ↦ 𝑇)) | 
| 33 | 1, 32 | syl 17 | . 2
⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ ⦋𝑅 / 𝑦⦌𝑆) = (𝑥 ∈ 𝐴 ↦ 𝑇)) | 
| 34 | 25, 33 | eqtrd 2777 | 1
⊢ (𝜑 → (𝐺 ∘ 𝐹) = (𝑥 ∈ 𝐴 ↦ 𝑇)) |