Step | Hyp | Ref
| Expression |
1 | | cvmlift2.b |
. 2
⊢ 𝐵 = ∪
𝐶 |
2 | | cvmlift2.f |
. 2
⊢ (𝜑 → 𝐹 ∈ (𝐶 CovMap 𝐽)) |
3 | | cvmlift2.g |
. 2
⊢ (𝜑 → 𝐺 ∈ ((II ×t II) Cn
𝐽)) |
4 | | cvmlift2.p |
. 2
⊢ (𝜑 → 𝑃 ∈ 𝐵) |
5 | | cvmlift2.i |
. 2
⊢ (𝜑 → (𝐹‘𝑃) = (0𝐺0)) |
6 | | coeq2 5711 |
. . . . 5
⊢ (ℎ = 𝑔 → (𝐹 ∘ ℎ) = (𝐹 ∘ 𝑔)) |
7 | | oveq1 7190 |
. . . . . . 7
⊢ (𝑤 = 𝑧 → (𝑤𝐺0) = (𝑧𝐺0)) |
8 | 7 | cbvmptv 5143 |
. . . . . 6
⊢ (𝑤 ∈ (0[,]1) ↦ (𝑤𝐺0)) = (𝑧 ∈ (0[,]1) ↦ (𝑧𝐺0)) |
9 | 8 | a1i 11 |
. . . . 5
⊢ (ℎ = 𝑔 → (𝑤 ∈ (0[,]1) ↦ (𝑤𝐺0)) = (𝑧 ∈ (0[,]1) ↦ (𝑧𝐺0))) |
10 | 6, 9 | eqeq12d 2755 |
. . . 4
⊢ (ℎ = 𝑔 → ((𝐹 ∘ ℎ) = (𝑤 ∈ (0[,]1) ↦ (𝑤𝐺0)) ↔ (𝐹 ∘ 𝑔) = (𝑧 ∈ (0[,]1) ↦ (𝑧𝐺0)))) |
11 | | fveq1 6686 |
. . . . 5
⊢ (ℎ = 𝑔 → (ℎ‘0) = (𝑔‘0)) |
12 | 11 | eqeq1d 2741 |
. . . 4
⊢ (ℎ = 𝑔 → ((ℎ‘0) = 𝑃 ↔ (𝑔‘0) = 𝑃)) |
13 | 10, 12 | anbi12d 634 |
. . 3
⊢ (ℎ = 𝑔 → (((𝐹 ∘ ℎ) = (𝑤 ∈ (0[,]1) ↦ (𝑤𝐺0)) ∧ (ℎ‘0) = 𝑃) ↔ ((𝐹 ∘ 𝑔) = (𝑧 ∈ (0[,]1) ↦ (𝑧𝐺0)) ∧ (𝑔‘0) = 𝑃))) |
14 | 13 | cbvriotavw 7150 |
. 2
⊢
(℩ℎ
∈ (II Cn 𝐶)((𝐹 ∘ ℎ) = (𝑤 ∈ (0[,]1) ↦ (𝑤𝐺0)) ∧ (ℎ‘0) = 𝑃)) = (℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝑧 ∈ (0[,]1) ↦ (𝑧𝐺0)) ∧ (𝑔‘0) = 𝑃)) |
15 | | coeq2 5711 |
. . . . . . . 8
⊢ (𝑘 = 𝑔 → (𝐹 ∘ 𝑘) = (𝐹 ∘ 𝑔)) |
16 | | oveq2 7191 |
. . . . . . . . . 10
⊢ (𝑤 = 𝑧 → (𝑢𝐺𝑤) = (𝑢𝐺𝑧)) |
17 | 16 | cbvmptv 5143 |
. . . . . . . . 9
⊢ (𝑤 ∈ (0[,]1) ↦ (𝑢𝐺𝑤)) = (𝑧 ∈ (0[,]1) ↦ (𝑢𝐺𝑧)) |
18 | 17 | a1i 11 |
. . . . . . . 8
⊢ (𝑘 = 𝑔 → (𝑤 ∈ (0[,]1) ↦ (𝑢𝐺𝑤)) = (𝑧 ∈ (0[,]1) ↦ (𝑢𝐺𝑧))) |
19 | 15, 18 | eqeq12d 2755 |
. . . . . . 7
⊢ (𝑘 = 𝑔 → ((𝐹 ∘ 𝑘) = (𝑤 ∈ (0[,]1) ↦ (𝑢𝐺𝑤)) ↔ (𝐹 ∘ 𝑔) = (𝑧 ∈ (0[,]1) ↦ (𝑢𝐺𝑧)))) |
20 | | fveq1 6686 |
. . . . . . . 8
⊢ (𝑘 = 𝑔 → (𝑘‘0) = (𝑔‘0)) |
21 | 20 | eqeq1d 2741 |
. . . . . . 7
⊢ (𝑘 = 𝑔 → ((𝑘‘0) = ((℩ℎ ∈ (II Cn 𝐶)((𝐹 ∘ ℎ) = (𝑤 ∈ (0[,]1) ↦ (𝑤𝐺0)) ∧ (ℎ‘0) = 𝑃))‘𝑢) ↔ (𝑔‘0) = ((℩ℎ ∈ (II Cn 𝐶)((𝐹 ∘ ℎ) = (𝑤 ∈ (0[,]1) ↦ (𝑤𝐺0)) ∧ (ℎ‘0) = 𝑃))‘𝑢))) |
22 | 19, 21 | anbi12d 634 |
. . . . . 6
⊢ (𝑘 = 𝑔 → (((𝐹 ∘ 𝑘) = (𝑤 ∈ (0[,]1) ↦ (𝑢𝐺𝑤)) ∧ (𝑘‘0) = ((℩ℎ ∈ (II Cn 𝐶)((𝐹 ∘ ℎ) = (𝑤 ∈ (0[,]1) ↦ (𝑤𝐺0)) ∧ (ℎ‘0) = 𝑃))‘𝑢)) ↔ ((𝐹 ∘ 𝑔) = (𝑧 ∈ (0[,]1) ↦ (𝑢𝐺𝑧)) ∧ (𝑔‘0) = ((℩ℎ ∈ (II Cn 𝐶)((𝐹 ∘ ℎ) = (𝑤 ∈ (0[,]1) ↦ (𝑤𝐺0)) ∧ (ℎ‘0) = 𝑃))‘𝑢)))) |
23 | 22 | cbvriotavw 7150 |
. . . . 5
⊢
(℩𝑘
∈ (II Cn 𝐶)((𝐹 ∘ 𝑘) = (𝑤 ∈ (0[,]1) ↦ (𝑢𝐺𝑤)) ∧ (𝑘‘0) = ((℩ℎ ∈ (II Cn 𝐶)((𝐹 ∘ ℎ) = (𝑤 ∈ (0[,]1) ↦ (𝑤𝐺0)) ∧ (ℎ‘0) = 𝑃))‘𝑢))) = (℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝑧 ∈ (0[,]1) ↦ (𝑢𝐺𝑧)) ∧ (𝑔‘0) = ((℩ℎ ∈ (II Cn 𝐶)((𝐹 ∘ ℎ) = (𝑤 ∈ (0[,]1) ↦ (𝑤𝐺0)) ∧ (ℎ‘0) = 𝑃))‘𝑢))) |
24 | | oveq1 7190 |
. . . . . . . . 9
⊢ (𝑢 = 𝑥 → (𝑢𝐺𝑧) = (𝑥𝐺𝑧)) |
25 | 24 | mpteq2dv 5136 |
. . . . . . . 8
⊢ (𝑢 = 𝑥 → (𝑧 ∈ (0[,]1) ↦ (𝑢𝐺𝑧)) = (𝑧 ∈ (0[,]1) ↦ (𝑥𝐺𝑧))) |
26 | 25 | eqeq2d 2750 |
. . . . . . 7
⊢ (𝑢 = 𝑥 → ((𝐹 ∘ 𝑔) = (𝑧 ∈ (0[,]1) ↦ (𝑢𝐺𝑧)) ↔ (𝐹 ∘ 𝑔) = (𝑧 ∈ (0[,]1) ↦ (𝑥𝐺𝑧)))) |
27 | | fveq2 6687 |
. . . . . . . 8
⊢ (𝑢 = 𝑥 → ((℩ℎ ∈ (II Cn 𝐶)((𝐹 ∘ ℎ) = (𝑤 ∈ (0[,]1) ↦ (𝑤𝐺0)) ∧ (ℎ‘0) = 𝑃))‘𝑢) = ((℩ℎ ∈ (II Cn 𝐶)((𝐹 ∘ ℎ) = (𝑤 ∈ (0[,]1) ↦ (𝑤𝐺0)) ∧ (ℎ‘0) = 𝑃))‘𝑥)) |
28 | 27 | eqeq2d 2750 |
. . . . . . 7
⊢ (𝑢 = 𝑥 → ((𝑔‘0) = ((℩ℎ ∈ (II Cn 𝐶)((𝐹 ∘ ℎ) = (𝑤 ∈ (0[,]1) ↦ (𝑤𝐺0)) ∧ (ℎ‘0) = 𝑃))‘𝑢) ↔ (𝑔‘0) = ((℩ℎ ∈ (II Cn 𝐶)((𝐹 ∘ ℎ) = (𝑤 ∈ (0[,]1) ↦ (𝑤𝐺0)) ∧ (ℎ‘0) = 𝑃))‘𝑥))) |
29 | 26, 28 | anbi12d 634 |
. . . . . 6
⊢ (𝑢 = 𝑥 → (((𝐹 ∘ 𝑔) = (𝑧 ∈ (0[,]1) ↦ (𝑢𝐺𝑧)) ∧ (𝑔‘0) = ((℩ℎ ∈ (II Cn 𝐶)((𝐹 ∘ ℎ) = (𝑤 ∈ (0[,]1) ↦ (𝑤𝐺0)) ∧ (ℎ‘0) = 𝑃))‘𝑢)) ↔ ((𝐹 ∘ 𝑔) = (𝑧 ∈ (0[,]1) ↦ (𝑥𝐺𝑧)) ∧ (𝑔‘0) = ((℩ℎ ∈ (II Cn 𝐶)((𝐹 ∘ ℎ) = (𝑤 ∈ (0[,]1) ↦ (𝑤𝐺0)) ∧ (ℎ‘0) = 𝑃))‘𝑥)))) |
30 | 29 | riotabidv 7142 |
. . . . 5
⊢ (𝑢 = 𝑥 → (℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝑧 ∈ (0[,]1) ↦ (𝑢𝐺𝑧)) ∧ (𝑔‘0) = ((℩ℎ ∈ (II Cn 𝐶)((𝐹 ∘ ℎ) = (𝑤 ∈ (0[,]1) ↦ (𝑤𝐺0)) ∧ (ℎ‘0) = 𝑃))‘𝑢))) = (℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝑧 ∈ (0[,]1) ↦ (𝑥𝐺𝑧)) ∧ (𝑔‘0) = ((℩ℎ ∈ (II Cn 𝐶)((𝐹 ∘ ℎ) = (𝑤 ∈ (0[,]1) ↦ (𝑤𝐺0)) ∧ (ℎ‘0) = 𝑃))‘𝑥)))) |
31 | 23, 30 | syl5eq 2786 |
. . . 4
⊢ (𝑢 = 𝑥 → (℩𝑘 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑘) = (𝑤 ∈ (0[,]1) ↦ (𝑢𝐺𝑤)) ∧ (𝑘‘0) = ((℩ℎ ∈ (II Cn 𝐶)((𝐹 ∘ ℎ) = (𝑤 ∈ (0[,]1) ↦ (𝑤𝐺0)) ∧ (ℎ‘0) = 𝑃))‘𝑢))) = (℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝑧 ∈ (0[,]1) ↦ (𝑥𝐺𝑧)) ∧ (𝑔‘0) = ((℩ℎ ∈ (II Cn 𝐶)((𝐹 ∘ ℎ) = (𝑤 ∈ (0[,]1) ↦ (𝑤𝐺0)) ∧ (ℎ‘0) = 𝑃))‘𝑥)))) |
32 | 31 | fveq1d 6689 |
. . 3
⊢ (𝑢 = 𝑥 → ((℩𝑘 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑘) = (𝑤 ∈ (0[,]1) ↦ (𝑢𝐺𝑤)) ∧ (𝑘‘0) = ((℩ℎ ∈ (II Cn 𝐶)((𝐹 ∘ ℎ) = (𝑤 ∈ (0[,]1) ↦ (𝑤𝐺0)) ∧ (ℎ‘0) = 𝑃))‘𝑢)))‘𝑣) = ((℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝑧 ∈ (0[,]1) ↦ (𝑥𝐺𝑧)) ∧ (𝑔‘0) = ((℩ℎ ∈ (II Cn 𝐶)((𝐹 ∘ ℎ) = (𝑤 ∈ (0[,]1) ↦ (𝑤𝐺0)) ∧ (ℎ‘0) = 𝑃))‘𝑥)))‘𝑣)) |
33 | | fveq2 6687 |
. . 3
⊢ (𝑣 = 𝑦 → ((℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝑧 ∈ (0[,]1) ↦ (𝑥𝐺𝑧)) ∧ (𝑔‘0) = ((℩ℎ ∈ (II Cn 𝐶)((𝐹 ∘ ℎ) = (𝑤 ∈ (0[,]1) ↦ (𝑤𝐺0)) ∧ (ℎ‘0) = 𝑃))‘𝑥)))‘𝑣) = ((℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝑧 ∈ (0[,]1) ↦ (𝑥𝐺𝑧)) ∧ (𝑔‘0) = ((℩ℎ ∈ (II Cn 𝐶)((𝐹 ∘ ℎ) = (𝑤 ∈ (0[,]1) ↦ (𝑤𝐺0)) ∧ (ℎ‘0) = 𝑃))‘𝑥)))‘𝑦)) |
34 | 32, 33 | cbvmpov 7276 |
. 2
⊢ (𝑢 ∈ (0[,]1), 𝑣 ∈ (0[,]1) ↦
((℩𝑘 ∈
(II Cn 𝐶)((𝐹 ∘ 𝑘) = (𝑤 ∈ (0[,]1) ↦ (𝑢𝐺𝑤)) ∧ (𝑘‘0) = ((℩ℎ ∈ (II Cn 𝐶)((𝐹 ∘ ℎ) = (𝑤 ∈ (0[,]1) ↦ (𝑤𝐺0)) ∧ (ℎ‘0) = 𝑃))‘𝑢)))‘𝑣)) = (𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ ((℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝑧 ∈ (0[,]1) ↦ (𝑥𝐺𝑧)) ∧ (𝑔‘0) = ((℩ℎ ∈ (II Cn 𝐶)((𝐹 ∘ ℎ) = (𝑤 ∈ (0[,]1) ↦ (𝑤𝐺0)) ∧ (ℎ‘0) = 𝑃))‘𝑥)))‘𝑦)) |
35 | 1, 2, 3, 4, 5, 14,
34 | cvmlift2lem13 32861 |
1
⊢ (𝜑 → ∃!𝑓 ∈ ((II ×t II) Cn 𝐶)((𝐹 ∘ 𝑓) = 𝐺 ∧ (0𝑓0) = 𝑃)) |