Step | Hyp | Ref
| Expression |
1 | | cvxpconn.4 |
. . 3
⊢ 𝐾 = (𝐽 ↾t 𝑆) |
2 | | cvxpconn.3 |
. . . . 5
⊢ 𝐽 =
(TopOpen‘ℂfld) |
3 | 2 | cnfldtop 22587 |
. . . 4
⊢ 𝐽 ∈ Top |
4 | | cvxpconn.1 |
. . . . 5
⊢ (𝜑 → 𝑆 ⊆ ℂ) |
5 | | cnex 10017 |
. . . . 5
⊢ ℂ
∈ V |
6 | | ssexg 4804 |
. . . . 5
⊢ ((𝑆 ⊆ ℂ ∧ ℂ
∈ V) → 𝑆 ∈
V) |
7 | 4, 5, 6 | sylancl 694 |
. . . 4
⊢ (𝜑 → 𝑆 ∈ V) |
8 | | resttop 20964 |
. . . 4
⊢ ((𝐽 ∈ Top ∧ 𝑆 ∈ V) → (𝐽 ↾t 𝑆) ∈ Top) |
9 | 3, 7, 8 | sylancr 695 |
. . 3
⊢ (𝜑 → (𝐽 ↾t 𝑆) ∈ Top) |
10 | 1, 9 | syl5eqel 2705 |
. 2
⊢ (𝜑 → 𝐾 ∈ Top) |
11 | 2 | dfii3 22686 |
. . . . . . . 8
⊢ II =
(𝐽 ↾t
(0[,]1)) |
12 | 2 | cnfldtopon 22586 |
. . . . . . . . 9
⊢ 𝐽 ∈
(TopOn‘ℂ) |
13 | 12 | a1i 11 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑥 ∈ 𝑆)) → 𝐽 ∈
(TopOn‘ℂ)) |
14 | | unitssre 12319 |
. . . . . . . . . 10
⊢ (0[,]1)
⊆ ℝ |
15 | | ax-resscn 9993 |
. . . . . . . . . 10
⊢ ℝ
⊆ ℂ |
16 | 14, 15 | sstri 3612 |
. . . . . . . . 9
⊢ (0[,]1)
⊆ ℂ |
17 | 16 | a1i 11 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑥 ∈ 𝑆)) → (0[,]1) ⊆
ℂ) |
18 | 13 | cnmptid 21464 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑥 ∈ 𝑆)) → (𝑡 ∈ ℂ ↦ 𝑡) ∈ (𝐽 Cn 𝐽)) |
19 | 4 | adantr 481 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑥 ∈ 𝑆)) → 𝑆 ⊆ ℂ) |
20 | | simprr 796 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑥 ∈ 𝑆)) → 𝑥 ∈ 𝑆) |
21 | 19, 20 | sseldd 3604 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑥 ∈ 𝑆)) → 𝑥 ∈ ℂ) |
22 | 13, 13, 21 | cnmptc 21465 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑥 ∈ 𝑆)) → (𝑡 ∈ ℂ ↦ 𝑥) ∈ (𝐽 Cn 𝐽)) |
23 | 2 | mulcn 22670 |
. . . . . . . . . . 11
⊢ ·
∈ ((𝐽
×t 𝐽) Cn
𝐽) |
24 | 23 | a1i 11 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑥 ∈ 𝑆)) → · ∈ ((𝐽 ×t 𝐽) Cn 𝐽)) |
25 | 13, 18, 22, 24 | cnmpt12f 21469 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑥 ∈ 𝑆)) → (𝑡 ∈ ℂ ↦ (𝑡 · 𝑥)) ∈ (𝐽 Cn 𝐽)) |
26 | | 1cnd 10056 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑥 ∈ 𝑆)) → 1 ∈ ℂ) |
27 | 13, 13, 26 | cnmptc 21465 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑥 ∈ 𝑆)) → (𝑡 ∈ ℂ ↦ 1) ∈ (𝐽 Cn 𝐽)) |
28 | 2 | subcn 22669 |
. . . . . . . . . . . 12
⊢ −
∈ ((𝐽
×t 𝐽) Cn
𝐽) |
29 | 28 | a1i 11 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑥 ∈ 𝑆)) → − ∈ ((𝐽 ×t 𝐽) Cn 𝐽)) |
30 | 13, 27, 18, 29 | cnmpt12f 21469 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑥 ∈ 𝑆)) → (𝑡 ∈ ℂ ↦ (1 − 𝑡)) ∈ (𝐽 Cn 𝐽)) |
31 | | simprl 794 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑥 ∈ 𝑆)) → 𝑦 ∈ 𝑆) |
32 | 19, 31 | sseldd 3604 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑥 ∈ 𝑆)) → 𝑦 ∈ ℂ) |
33 | 13, 13, 32 | cnmptc 21465 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑥 ∈ 𝑆)) → (𝑡 ∈ ℂ ↦ 𝑦) ∈ (𝐽 Cn 𝐽)) |
34 | 13, 30, 33, 24 | cnmpt12f 21469 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑥 ∈ 𝑆)) → (𝑡 ∈ ℂ ↦ ((1 − 𝑡) · 𝑦)) ∈ (𝐽 Cn 𝐽)) |
35 | 2 | addcn 22668 |
. . . . . . . . . 10
⊢ + ∈
((𝐽 ×t
𝐽) Cn 𝐽) |
36 | 35 | a1i 11 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑥 ∈ 𝑆)) → + ∈ ((𝐽 ×t 𝐽) Cn 𝐽)) |
37 | 13, 25, 34, 36 | cnmpt12f 21469 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑥 ∈ 𝑆)) → (𝑡 ∈ ℂ ↦ ((𝑡 · 𝑥) + ((1 − 𝑡) · 𝑦))) ∈ (𝐽 Cn 𝐽)) |
38 | 11, 13, 17, 37 | cnmpt1res 21479 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑥 ∈ 𝑆)) → (𝑡 ∈ (0[,]1) ↦ ((𝑡 · 𝑥) + ((1 − 𝑡) · 𝑦))) ∈ (II Cn 𝐽)) |
39 | | cvxpconn.2 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑡 ∈ (0[,]1))) → ((𝑡 · 𝑥) + ((1 − 𝑡) · 𝑦)) ∈ 𝑆) |
40 | 39 | 3exp2 1285 |
. . . . . . . . . . . 12
⊢ (𝜑 → (𝑥 ∈ 𝑆 → (𝑦 ∈ 𝑆 → (𝑡 ∈ (0[,]1) → ((𝑡 · 𝑥) + ((1 − 𝑡) · 𝑦)) ∈ 𝑆)))) |
41 | 40 | com23 86 |
. . . . . . . . . . 11
⊢ (𝜑 → (𝑦 ∈ 𝑆 → (𝑥 ∈ 𝑆 → (𝑡 ∈ (0[,]1) → ((𝑡 · 𝑥) + ((1 − 𝑡) · 𝑦)) ∈ 𝑆)))) |
42 | 41 | imp42 620 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑥 ∈ 𝑆)) ∧ 𝑡 ∈ (0[,]1)) → ((𝑡 · 𝑥) + ((1 − 𝑡) · 𝑦)) ∈ 𝑆) |
43 | | eqid 2622 |
. . . . . . . . . 10
⊢ (𝑡 ∈ (0[,]1) ↦ ((𝑡 · 𝑥) + ((1 − 𝑡) · 𝑦))) = (𝑡 ∈ (0[,]1) ↦ ((𝑡 · 𝑥) + ((1 − 𝑡) · 𝑦))) |
44 | 42, 43 | fmptd 6385 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑥 ∈ 𝑆)) → (𝑡 ∈ (0[,]1) ↦ ((𝑡 · 𝑥) + ((1 − 𝑡) · 𝑦))):(0[,]1)⟶𝑆) |
45 | | frn 6053 |
. . . . . . . . 9
⊢ ((𝑡 ∈ (0[,]1) ↦ ((𝑡 · 𝑥) + ((1 − 𝑡) · 𝑦))):(0[,]1)⟶𝑆 → ran (𝑡 ∈ (0[,]1) ↦ ((𝑡 · 𝑥) + ((1 − 𝑡) · 𝑦))) ⊆ 𝑆) |
46 | 44, 45 | syl 17 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑥 ∈ 𝑆)) → ran (𝑡 ∈ (0[,]1) ↦ ((𝑡 · 𝑥) + ((1 − 𝑡) · 𝑦))) ⊆ 𝑆) |
47 | | cnrest2 21090 |
. . . . . . . 8
⊢ ((𝐽 ∈ (TopOn‘ℂ)
∧ ran (𝑡 ∈ (0[,]1)
↦ ((𝑡 · 𝑥) + ((1 − 𝑡) · 𝑦))) ⊆ 𝑆 ∧ 𝑆 ⊆ ℂ) → ((𝑡 ∈ (0[,]1) ↦ ((𝑡 · 𝑥) + ((1 − 𝑡) · 𝑦))) ∈ (II Cn 𝐽) ↔ (𝑡 ∈ (0[,]1) ↦ ((𝑡 · 𝑥) + ((1 − 𝑡) · 𝑦))) ∈ (II Cn (𝐽 ↾t 𝑆)))) |
48 | 13, 46, 19, 47 | syl3anc 1326 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑥 ∈ 𝑆)) → ((𝑡 ∈ (0[,]1) ↦ ((𝑡 · 𝑥) + ((1 − 𝑡) · 𝑦))) ∈ (II Cn 𝐽) ↔ (𝑡 ∈ (0[,]1) ↦ ((𝑡 · 𝑥) + ((1 − 𝑡) · 𝑦))) ∈ (II Cn (𝐽 ↾t 𝑆)))) |
49 | 38, 48 | mpbid 222 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑥 ∈ 𝑆)) → (𝑡 ∈ (0[,]1) ↦ ((𝑡 · 𝑥) + ((1 − 𝑡) · 𝑦))) ∈ (II Cn (𝐽 ↾t 𝑆))) |
50 | 1 | oveq2i 6661 |
. . . . . 6
⊢ (II Cn
𝐾) = (II Cn (𝐽 ↾t 𝑆)) |
51 | 49, 50 | syl6eleqr 2712 |
. . . . 5
⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑥 ∈ 𝑆)) → (𝑡 ∈ (0[,]1) ↦ ((𝑡 · 𝑥) + ((1 − 𝑡) · 𝑦))) ∈ (II Cn 𝐾)) |
52 | | 0elunit 12290 |
. . . . . . 7
⊢ 0 ∈
(0[,]1) |
53 | | oveq1 6657 |
. . . . . . . . 9
⊢ (𝑡 = 0 → (𝑡 · 𝑥) = (0 · 𝑥)) |
54 | | oveq2 6658 |
. . . . . . . . . . 11
⊢ (𝑡 = 0 → (1 − 𝑡) = (1 −
0)) |
55 | | 1m0e1 11131 |
. . . . . . . . . . 11
⊢ (1
− 0) = 1 |
56 | 54, 55 | syl6eq 2672 |
. . . . . . . . . 10
⊢ (𝑡 = 0 → (1 − 𝑡) = 1) |
57 | 56 | oveq1d 6665 |
. . . . . . . . 9
⊢ (𝑡 = 0 → ((1 − 𝑡) · 𝑦) = (1 · 𝑦)) |
58 | 53, 57 | oveq12d 6668 |
. . . . . . . 8
⊢ (𝑡 = 0 → ((𝑡 · 𝑥) + ((1 − 𝑡) · 𝑦)) = ((0 · 𝑥) + (1 · 𝑦))) |
59 | | ovex 6678 |
. . . . . . . 8
⊢ ((0
· 𝑥) + (1 ·
𝑦)) ∈
V |
60 | 58, 43, 59 | fvmpt 6282 |
. . . . . . 7
⊢ (0 ∈
(0[,]1) → ((𝑡 ∈
(0[,]1) ↦ ((𝑡
· 𝑥) + ((1 −
𝑡) · 𝑦)))‘0) = ((0 ·
𝑥) + (1 · 𝑦))) |
61 | 52, 60 | ax-mp 5 |
. . . . . 6
⊢ ((𝑡 ∈ (0[,]1) ↦ ((𝑡 · 𝑥) + ((1 − 𝑡) · 𝑦)))‘0) = ((0 · 𝑥) + (1 · 𝑦)) |
62 | 21 | mul02d 10234 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑥 ∈ 𝑆)) → (0 · 𝑥) = 0) |
63 | 32 | mulid2d 10058 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑥 ∈ 𝑆)) → (1 · 𝑦) = 𝑦) |
64 | 62, 63 | oveq12d 6668 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑥 ∈ 𝑆)) → ((0 · 𝑥) + (1 · 𝑦)) = (0 + 𝑦)) |
65 | 32 | addid2d 10237 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑥 ∈ 𝑆)) → (0 + 𝑦) = 𝑦) |
66 | 64, 65 | eqtrd 2656 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑥 ∈ 𝑆)) → ((0 · 𝑥) + (1 · 𝑦)) = 𝑦) |
67 | 61, 66 | syl5eq 2668 |
. . . . 5
⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑥 ∈ 𝑆)) → ((𝑡 ∈ (0[,]1) ↦ ((𝑡 · 𝑥) + ((1 − 𝑡) · 𝑦)))‘0) = 𝑦) |
68 | | 1elunit 12291 |
. . . . . . 7
⊢ 1 ∈
(0[,]1) |
69 | | oveq1 6657 |
. . . . . . . . 9
⊢ (𝑡 = 1 → (𝑡 · 𝑥) = (1 · 𝑥)) |
70 | | oveq2 6658 |
. . . . . . . . . . 11
⊢ (𝑡 = 1 → (1 − 𝑡) = (1 −
1)) |
71 | | 1m1e0 11089 |
. . . . . . . . . . 11
⊢ (1
− 1) = 0 |
72 | 70, 71 | syl6eq 2672 |
. . . . . . . . . 10
⊢ (𝑡 = 1 → (1 − 𝑡) = 0) |
73 | 72 | oveq1d 6665 |
. . . . . . . . 9
⊢ (𝑡 = 1 → ((1 − 𝑡) · 𝑦) = (0 · 𝑦)) |
74 | 69, 73 | oveq12d 6668 |
. . . . . . . 8
⊢ (𝑡 = 1 → ((𝑡 · 𝑥) + ((1 − 𝑡) · 𝑦)) = ((1 · 𝑥) + (0 · 𝑦))) |
75 | | ovex 6678 |
. . . . . . . 8
⊢ ((1
· 𝑥) + (0 ·
𝑦)) ∈
V |
76 | 74, 43, 75 | fvmpt 6282 |
. . . . . . 7
⊢ (1 ∈
(0[,]1) → ((𝑡 ∈
(0[,]1) ↦ ((𝑡
· 𝑥) + ((1 −
𝑡) · 𝑦)))‘1) = ((1 ·
𝑥) + (0 · 𝑦))) |
77 | 68, 76 | ax-mp 5 |
. . . . . 6
⊢ ((𝑡 ∈ (0[,]1) ↦ ((𝑡 · 𝑥) + ((1 − 𝑡) · 𝑦)))‘1) = ((1 · 𝑥) + (0 · 𝑦)) |
78 | 21 | mulid2d 10058 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑥 ∈ 𝑆)) → (1 · 𝑥) = 𝑥) |
79 | 32 | mul02d 10234 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑥 ∈ 𝑆)) → (0 · 𝑦) = 0) |
80 | 78, 79 | oveq12d 6668 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑥 ∈ 𝑆)) → ((1 · 𝑥) + (0 · 𝑦)) = (𝑥 + 0)) |
81 | 21 | addid1d 10236 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑥 ∈ 𝑆)) → (𝑥 + 0) = 𝑥) |
82 | 80, 81 | eqtrd 2656 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑥 ∈ 𝑆)) → ((1 · 𝑥) + (0 · 𝑦)) = 𝑥) |
83 | 77, 82 | syl5eq 2668 |
. . . . 5
⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑥 ∈ 𝑆)) → ((𝑡 ∈ (0[,]1) ↦ ((𝑡 · 𝑥) + ((1 − 𝑡) · 𝑦)))‘1) = 𝑥) |
84 | | fveq1 6190 |
. . . . . . . 8
⊢ (𝑓 = (𝑡 ∈ (0[,]1) ↦ ((𝑡 · 𝑥) + ((1 − 𝑡) · 𝑦))) → (𝑓‘0) = ((𝑡 ∈ (0[,]1) ↦ ((𝑡 · 𝑥) + ((1 − 𝑡) · 𝑦)))‘0)) |
85 | 84 | eqeq1d 2624 |
. . . . . . 7
⊢ (𝑓 = (𝑡 ∈ (0[,]1) ↦ ((𝑡 · 𝑥) + ((1 − 𝑡) · 𝑦))) → ((𝑓‘0) = 𝑦 ↔ ((𝑡 ∈ (0[,]1) ↦ ((𝑡 · 𝑥) + ((1 − 𝑡) · 𝑦)))‘0) = 𝑦)) |
86 | | fveq1 6190 |
. . . . . . . 8
⊢ (𝑓 = (𝑡 ∈ (0[,]1) ↦ ((𝑡 · 𝑥) + ((1 − 𝑡) · 𝑦))) → (𝑓‘1) = ((𝑡 ∈ (0[,]1) ↦ ((𝑡 · 𝑥) + ((1 − 𝑡) · 𝑦)))‘1)) |
87 | 86 | eqeq1d 2624 |
. . . . . . 7
⊢ (𝑓 = (𝑡 ∈ (0[,]1) ↦ ((𝑡 · 𝑥) + ((1 − 𝑡) · 𝑦))) → ((𝑓‘1) = 𝑥 ↔ ((𝑡 ∈ (0[,]1) ↦ ((𝑡 · 𝑥) + ((1 − 𝑡) · 𝑦)))‘1) = 𝑥)) |
88 | 85, 87 | anbi12d 747 |
. . . . . 6
⊢ (𝑓 = (𝑡 ∈ (0[,]1) ↦ ((𝑡 · 𝑥) + ((1 − 𝑡) · 𝑦))) → (((𝑓‘0) = 𝑦 ∧ (𝑓‘1) = 𝑥) ↔ (((𝑡 ∈ (0[,]1) ↦ ((𝑡 · 𝑥) + ((1 − 𝑡) · 𝑦)))‘0) = 𝑦 ∧ ((𝑡 ∈ (0[,]1) ↦ ((𝑡 · 𝑥) + ((1 − 𝑡) · 𝑦)))‘1) = 𝑥))) |
89 | 88 | rspcev 3309 |
. . . . 5
⊢ (((𝑡 ∈ (0[,]1) ↦ ((𝑡 · 𝑥) + ((1 − 𝑡) · 𝑦))) ∈ (II Cn 𝐾) ∧ (((𝑡 ∈ (0[,]1) ↦ ((𝑡 · 𝑥) + ((1 − 𝑡) · 𝑦)))‘0) = 𝑦 ∧ ((𝑡 ∈ (0[,]1) ↦ ((𝑡 · 𝑥) + ((1 − 𝑡) · 𝑦)))‘1) = 𝑥)) → ∃𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = 𝑦 ∧ (𝑓‘1) = 𝑥)) |
90 | 51, 67, 83, 89 | syl12anc 1324 |
. . . 4
⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑥 ∈ 𝑆)) → ∃𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = 𝑦 ∧ (𝑓‘1) = 𝑥)) |
91 | 90 | ralrimivva 2971 |
. . 3
⊢ (𝜑 → ∀𝑦 ∈ 𝑆 ∀𝑥 ∈ 𝑆 ∃𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = 𝑦 ∧ (𝑓‘1) = 𝑥)) |
92 | | resttopon 20965 |
. . . . . . 7
⊢ ((𝐽 ∈ (TopOn‘ℂ)
∧ 𝑆 ⊆ ℂ)
→ (𝐽
↾t 𝑆)
∈ (TopOn‘𝑆)) |
93 | 12, 4, 92 | sylancr 695 |
. . . . . 6
⊢ (𝜑 → (𝐽 ↾t 𝑆) ∈ (TopOn‘𝑆)) |
94 | 1, 93 | syl5eqel 2705 |
. . . . 5
⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝑆)) |
95 | | toponuni 20719 |
. . . . 5
⊢ (𝐾 ∈ (TopOn‘𝑆) → 𝑆 = ∪ 𝐾) |
96 | 94, 95 | syl 17 |
. . . 4
⊢ (𝜑 → 𝑆 = ∪ 𝐾) |
97 | 96 | raleqdv 3144 |
. . . 4
⊢ (𝜑 → (∀𝑥 ∈ 𝑆 ∃𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = 𝑦 ∧ (𝑓‘1) = 𝑥) ↔ ∀𝑥 ∈ ∪ 𝐾∃𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = 𝑦 ∧ (𝑓‘1) = 𝑥))) |
98 | 96, 97 | raleqbidv 3152 |
. . 3
⊢ (𝜑 → (∀𝑦 ∈ 𝑆 ∀𝑥 ∈ 𝑆 ∃𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = 𝑦 ∧ (𝑓‘1) = 𝑥) ↔ ∀𝑦 ∈ ∪ 𝐾∀𝑥 ∈ ∪ 𝐾∃𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = 𝑦 ∧ (𝑓‘1) = 𝑥))) |
99 | 91, 98 | mpbid 222 |
. 2
⊢ (𝜑 → ∀𝑦 ∈ ∪ 𝐾∀𝑥 ∈ ∪ 𝐾∃𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = 𝑦 ∧ (𝑓‘1) = 𝑥)) |
100 | | eqid 2622 |
. . 3
⊢ ∪ 𝐾 =
∪ 𝐾 |
101 | 100 | ispconn 31205 |
. 2
⊢ (𝐾 ∈ PConn ↔ (𝐾 ∈ Top ∧ ∀𝑦 ∈ ∪ 𝐾∀𝑥 ∈ ∪ 𝐾∃𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = 𝑦 ∧ (𝑓‘1) = 𝑥))) |
102 | 10, 99, 101 | sylanbrc 698 |
1
⊢ (𝜑 → 𝐾 ∈ PConn) |