| Step | Hyp | Ref
| Expression |
| 1 | | fmptcof.1 |
. . . . 5
⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝑅 ∈ 𝐵) |
| 2 | | nfcsb1v 3117 |
. . . . . . 7
⊢
Ⅎ𝑥⦋𝑧 / 𝑥⦌𝑅 |
| 3 | 2 | nfel1 2350 |
. . . . . 6
⊢
Ⅎ𝑥⦋𝑧 / 𝑥⦌𝑅 ∈ 𝐵 |
| 4 | | csbeq1a 3093 |
. . . . . . 7
⊢ (𝑥 = 𝑧 → 𝑅 = ⦋𝑧 / 𝑥⦌𝑅) |
| 5 | 4 | eleq1d 2265 |
. . . . . 6
⊢ (𝑥 = 𝑧 → (𝑅 ∈ 𝐵 ↔ ⦋𝑧 / 𝑥⦌𝑅 ∈ 𝐵)) |
| 6 | 3, 5 | rspc 2862 |
. . . . 5
⊢ (𝑧 ∈ 𝐴 → (∀𝑥 ∈ 𝐴 𝑅 ∈ 𝐵 → ⦋𝑧 / 𝑥⦌𝑅 ∈ 𝐵)) |
| 7 | 1, 6 | mpan9 281 |
. . . 4
⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → ⦋𝑧 / 𝑥⦌𝑅 ∈ 𝐵) |
| 8 | | fmptcof.2 |
. . . . 5
⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝑅)) |
| 9 | | nfcv 2339 |
. . . . . 6
⊢
Ⅎ𝑧𝑅 |
| 10 | 9, 2, 4 | cbvmpt 4128 |
. . . . 5
⊢ (𝑥 ∈ 𝐴 ↦ 𝑅) = (𝑧 ∈ 𝐴 ↦ ⦋𝑧 / 𝑥⦌𝑅) |
| 11 | 8, 10 | eqtrdi 2245 |
. . . 4
⊢ (𝜑 → 𝐹 = (𝑧 ∈ 𝐴 ↦ ⦋𝑧 / 𝑥⦌𝑅)) |
| 12 | | fmptcof.3 |
. . . . 5
⊢ (𝜑 → 𝐺 = (𝑦 ∈ 𝐵 ↦ 𝑆)) |
| 13 | | nfcv 2339 |
. . . . . 6
⊢
Ⅎ𝑤𝑆 |
| 14 | | nfcsb1v 3117 |
. . . . . 6
⊢
Ⅎ𝑦⦋𝑤 / 𝑦⦌𝑆 |
| 15 | | csbeq1a 3093 |
. . . . . 6
⊢ (𝑦 = 𝑤 → 𝑆 = ⦋𝑤 / 𝑦⦌𝑆) |
| 16 | 13, 14, 15 | cbvmpt 4128 |
. . . . 5
⊢ (𝑦 ∈ 𝐵 ↦ 𝑆) = (𝑤 ∈ 𝐵 ↦ ⦋𝑤 / 𝑦⦌𝑆) |
| 17 | 12, 16 | eqtrdi 2245 |
. . . 4
⊢ (𝜑 → 𝐺 = (𝑤 ∈ 𝐵 ↦ ⦋𝑤 / 𝑦⦌𝑆)) |
| 18 | | csbeq1 3087 |
. . . 4
⊢ (𝑤 = ⦋𝑧 / 𝑥⦌𝑅 → ⦋𝑤 / 𝑦⦌𝑆 = ⦋⦋𝑧 / 𝑥⦌𝑅 / 𝑦⦌𝑆) |
| 19 | 7, 11, 17, 18 | fmptco 5728 |
. . 3
⊢ (𝜑 → (𝐺 ∘ 𝐹) = (𝑧 ∈ 𝐴 ↦ ⦋⦋𝑧 / 𝑥⦌𝑅 / 𝑦⦌𝑆)) |
| 20 | | nfcv 2339 |
. . . 4
⊢
Ⅎ𝑧⦋𝑅 / 𝑦⦌𝑆 |
| 21 | | nfcv 2339 |
. . . . 5
⊢
Ⅎ𝑥𝑆 |
| 22 | 2, 21 | nfcsb 3122 |
. . . 4
⊢
Ⅎ𝑥⦋⦋𝑧 / 𝑥⦌𝑅 / 𝑦⦌𝑆 |
| 23 | 4 | csbeq1d 3091 |
. . . 4
⊢ (𝑥 = 𝑧 → ⦋𝑅 / 𝑦⦌𝑆 = ⦋⦋𝑧 / 𝑥⦌𝑅 / 𝑦⦌𝑆) |
| 24 | 20, 22, 23 | cbvmpt 4128 |
. . 3
⊢ (𝑥 ∈ 𝐴 ↦ ⦋𝑅 / 𝑦⦌𝑆) = (𝑧 ∈ 𝐴 ↦ ⦋⦋𝑧 / 𝑥⦌𝑅 / 𝑦⦌𝑆) |
| 25 | 19, 24 | eqtr4di 2247 |
. 2
⊢ (𝜑 → (𝐺 ∘ 𝐹) = (𝑥 ∈ 𝐴 ↦ ⦋𝑅 / 𝑦⦌𝑆)) |
| 26 | | eqid 2196 |
. . . 4
⊢ 𝐴 = 𝐴 |
| 27 | | nfcvd 2340 |
. . . . . 6
⊢ (𝑅 ∈ 𝐵 → Ⅎ𝑦𝑇) |
| 28 | | fmptcof.4 |
. . . . . 6
⊢ (𝑦 = 𝑅 → 𝑆 = 𝑇) |
| 29 | 27, 28 | csbiegf 3128 |
. . . . 5
⊢ (𝑅 ∈ 𝐵 → ⦋𝑅 / 𝑦⦌𝑆 = 𝑇) |
| 30 | 29 | ralimi 2560 |
. . . 4
⊢
(∀𝑥 ∈
𝐴 𝑅 ∈ 𝐵 → ∀𝑥 ∈ 𝐴 ⦋𝑅 / 𝑦⦌𝑆 = 𝑇) |
| 31 | | mpteq12 4116 |
. . . 4
⊢ ((𝐴 = 𝐴 ∧ ∀𝑥 ∈ 𝐴 ⦋𝑅 / 𝑦⦌𝑆 = 𝑇) → (𝑥 ∈ 𝐴 ↦ ⦋𝑅 / 𝑦⦌𝑆) = (𝑥 ∈ 𝐴 ↦ 𝑇)) |
| 32 | 26, 30, 31 | sylancr 414 |
. . 3
⊢
(∀𝑥 ∈
𝐴 𝑅 ∈ 𝐵 → (𝑥 ∈ 𝐴 ↦ ⦋𝑅 / 𝑦⦌𝑆) = (𝑥 ∈ 𝐴 ↦ 𝑇)) |
| 33 | 1, 32 | syl 14 |
. 2
⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ ⦋𝑅 / 𝑦⦌𝑆) = (𝑥 ∈ 𝐴 ↦ 𝑇)) |
| 34 | 25, 33 | eqtrd 2229 |
1
⊢ (𝜑 → (𝐺 ∘ 𝐹) = (𝑥 ∈ 𝐴 ↦ 𝑇)) |