| Step | Hyp | Ref
| Expression |
| 1 | | unieq 4918 |
. . . . . . . . 9
⊢ (𝑎 = ∅ → ∪ 𝑎 =
∪ ∅) |
| 2 | | uni0 4935 |
. . . . . . . . 9
⊢ ∪ ∅ = ∅ |
| 3 | 1, 2 | eqtrdi 2793 |
. . . . . . . 8
⊢ (𝑎 = ∅ → ∪ 𝑎 =
∅) |
| 4 | 3 | eleq1d 2826 |
. . . . . . 7
⊢ (𝑎 = ∅ → (∪ 𝑎
∈ 𝐽 ↔ ∅
∈ 𝐽)) |
| 5 | | mretopd.j |
. . . . . . . . . . . . . 14
⊢ 𝐽 = {𝑧 ∈ 𝒫 𝐵 ∣ (𝐵 ∖ 𝑧) ∈ 𝑀} |
| 6 | 5 | ssrab3 4082 |
. . . . . . . . . . . . 13
⊢ 𝐽 ⊆ 𝒫 𝐵 |
| 7 | | sstr 3992 |
. . . . . . . . . . . . 13
⊢ ((𝑎 ⊆ 𝐽 ∧ 𝐽 ⊆ 𝒫 𝐵) → 𝑎 ⊆ 𝒫 𝐵) |
| 8 | 6, 7 | mpan2 691 |
. . . . . . . . . . . 12
⊢ (𝑎 ⊆ 𝐽 → 𝑎 ⊆ 𝒫 𝐵) |
| 9 | 8 | adantl 481 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑎 ⊆ 𝐽) → 𝑎 ⊆ 𝒫 𝐵) |
| 10 | | sspwuni 5100 |
. . . . . . . . . . 11
⊢ (𝑎 ⊆ 𝒫 𝐵 ↔ ∪ 𝑎
⊆ 𝐵) |
| 11 | 9, 10 | sylib 218 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑎 ⊆ 𝐽) → ∪ 𝑎 ⊆ 𝐵) |
| 12 | | vuniex 7759 |
. . . . . . . . . . 11
⊢ ∪ 𝑎
∈ V |
| 13 | 12 | elpw 4604 |
. . . . . . . . . 10
⊢ (∪ 𝑎
∈ 𝒫 𝐵 ↔
∪ 𝑎 ⊆ 𝐵) |
| 14 | 11, 13 | sylibr 234 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑎 ⊆ 𝐽) → ∪ 𝑎 ∈ 𝒫 𝐵) |
| 15 | 14 | adantr 480 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑎 ⊆ 𝐽) ∧ 𝑎 ≠ ∅) → ∪ 𝑎
∈ 𝒫 𝐵) |
| 16 | | uniiun 5058 |
. . . . . . . . . 10
⊢ ∪ 𝑎 =
∪ 𝑏 ∈ 𝑎 𝑏 |
| 17 | 16 | difeq2i 4123 |
. . . . . . . . 9
⊢ (𝐵 ∖ ∪ 𝑎) =
(𝐵 ∖ ∪ 𝑏 ∈ 𝑎 𝑏) |
| 18 | | iindif2 5077 |
. . . . . . . . . . 11
⊢ (𝑎 ≠ ∅ → ∩ 𝑏 ∈ 𝑎 (𝐵 ∖ 𝑏) = (𝐵 ∖ ∪
𝑏 ∈ 𝑎 𝑏)) |
| 19 | 18 | adantl 481 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑎 ⊆ 𝐽) ∧ 𝑎 ≠ ∅) → ∩ 𝑏 ∈ 𝑎 (𝐵 ∖ 𝑏) = (𝐵 ∖ ∪
𝑏 ∈ 𝑎 𝑏)) |
| 20 | | mretopd.m |
. . . . . . . . . . . 12
⊢ (𝜑 → 𝑀 ∈ (Moore‘𝐵)) |
| 21 | 20 | ad2antrr 726 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑎 ⊆ 𝐽) ∧ 𝑎 ≠ ∅) → 𝑀 ∈ (Moore‘𝐵)) |
| 22 | | simpr 484 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑎 ⊆ 𝐽) ∧ 𝑎 ≠ ∅) → 𝑎 ≠ ∅) |
| 23 | | difeq2 4120 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑧 = 𝑏 → (𝐵 ∖ 𝑧) = (𝐵 ∖ 𝑏)) |
| 24 | 23 | eleq1d 2826 |
. . . . . . . . . . . . . . . 16
⊢ (𝑧 = 𝑏 → ((𝐵 ∖ 𝑧) ∈ 𝑀 ↔ (𝐵 ∖ 𝑏) ∈ 𝑀)) |
| 25 | 24, 5 | elrab2 3695 |
. . . . . . . . . . . . . . 15
⊢ (𝑏 ∈ 𝐽 ↔ (𝑏 ∈ 𝒫 𝐵 ∧ (𝐵 ∖ 𝑏) ∈ 𝑀)) |
| 26 | 25 | simprbi 496 |
. . . . . . . . . . . . . 14
⊢ (𝑏 ∈ 𝐽 → (𝐵 ∖ 𝑏) ∈ 𝑀) |
| 27 | 26 | rgen 3063 |
. . . . . . . . . . . . 13
⊢
∀𝑏 ∈
𝐽 (𝐵 ∖ 𝑏) ∈ 𝑀 |
| 28 | | ssralv 4052 |
. . . . . . . . . . . . . 14
⊢ (𝑎 ⊆ 𝐽 → (∀𝑏 ∈ 𝐽 (𝐵 ∖ 𝑏) ∈ 𝑀 → ∀𝑏 ∈ 𝑎 (𝐵 ∖ 𝑏) ∈ 𝑀)) |
| 29 | 28 | adantl 481 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑎 ⊆ 𝐽) → (∀𝑏 ∈ 𝐽 (𝐵 ∖ 𝑏) ∈ 𝑀 → ∀𝑏 ∈ 𝑎 (𝐵 ∖ 𝑏) ∈ 𝑀)) |
| 30 | 27, 29 | mpi 20 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑎 ⊆ 𝐽) → ∀𝑏 ∈ 𝑎 (𝐵 ∖ 𝑏) ∈ 𝑀) |
| 31 | 30 | adantr 480 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑎 ⊆ 𝐽) ∧ 𝑎 ≠ ∅) → ∀𝑏 ∈ 𝑎 (𝐵 ∖ 𝑏) ∈ 𝑀) |
| 32 | | mreiincl 17639 |
. . . . . . . . . . 11
⊢ ((𝑀 ∈ (Moore‘𝐵) ∧ 𝑎 ≠ ∅ ∧ ∀𝑏 ∈ 𝑎 (𝐵 ∖ 𝑏) ∈ 𝑀) → ∩
𝑏 ∈ 𝑎 (𝐵 ∖ 𝑏) ∈ 𝑀) |
| 33 | 21, 22, 31, 32 | syl3anc 1373 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑎 ⊆ 𝐽) ∧ 𝑎 ≠ ∅) → ∩ 𝑏 ∈ 𝑎 (𝐵 ∖ 𝑏) ∈ 𝑀) |
| 34 | 19, 33 | eqeltrrd 2842 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑎 ⊆ 𝐽) ∧ 𝑎 ≠ ∅) → (𝐵 ∖ ∪
𝑏 ∈ 𝑎 𝑏) ∈ 𝑀) |
| 35 | 17, 34 | eqeltrid 2845 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑎 ⊆ 𝐽) ∧ 𝑎 ≠ ∅) → (𝐵 ∖ ∪ 𝑎) ∈ 𝑀) |
| 36 | | difeq2 4120 |
. . . . . . . . . 10
⊢ (𝑧 = ∪
𝑎 → (𝐵 ∖ 𝑧) = (𝐵 ∖ ∪ 𝑎)) |
| 37 | 36 | eleq1d 2826 |
. . . . . . . . 9
⊢ (𝑧 = ∪
𝑎 → ((𝐵 ∖ 𝑧) ∈ 𝑀 ↔ (𝐵 ∖ ∪ 𝑎) ∈ 𝑀)) |
| 38 | 37, 5 | elrab2 3695 |
. . . . . . . 8
⊢ (∪ 𝑎
∈ 𝐽 ↔ (∪ 𝑎
∈ 𝒫 𝐵 ∧
(𝐵 ∖ ∪ 𝑎)
∈ 𝑀)) |
| 39 | 15, 35, 38 | sylanbrc 583 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑎 ⊆ 𝐽) ∧ 𝑎 ≠ ∅) → ∪ 𝑎
∈ 𝐽) |
| 40 | | 0elpw 5356 |
. . . . . . . . . 10
⊢ ∅
∈ 𝒫 𝐵 |
| 41 | 40 | a1i 11 |
. . . . . . . . 9
⊢ (𝜑 → ∅ ∈ 𝒫
𝐵) |
| 42 | | mre1cl 17637 |
. . . . . . . . . 10
⊢ (𝑀 ∈ (Moore‘𝐵) → 𝐵 ∈ 𝑀) |
| 43 | 20, 42 | syl 17 |
. . . . . . . . 9
⊢ (𝜑 → 𝐵 ∈ 𝑀) |
| 44 | | difeq2 4120 |
. . . . . . . . . . . 12
⊢ (𝑧 = ∅ → (𝐵 ∖ 𝑧) = (𝐵 ∖ ∅)) |
| 45 | | dif0 4378 |
. . . . . . . . . . . 12
⊢ (𝐵 ∖ ∅) = 𝐵 |
| 46 | 44, 45 | eqtrdi 2793 |
. . . . . . . . . . 11
⊢ (𝑧 = ∅ → (𝐵 ∖ 𝑧) = 𝐵) |
| 47 | 46 | eleq1d 2826 |
. . . . . . . . . 10
⊢ (𝑧 = ∅ → ((𝐵 ∖ 𝑧) ∈ 𝑀 ↔ 𝐵 ∈ 𝑀)) |
| 48 | 47, 5 | elrab2 3695 |
. . . . . . . . 9
⊢ (∅
∈ 𝐽 ↔ (∅
∈ 𝒫 𝐵 ∧
𝐵 ∈ 𝑀)) |
| 49 | 41, 43, 48 | sylanbrc 583 |
. . . . . . . 8
⊢ (𝜑 → ∅ ∈ 𝐽) |
| 50 | 49 | adantr 480 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑎 ⊆ 𝐽) → ∅ ∈ 𝐽) |
| 51 | 4, 39, 50 | pm2.61ne 3027 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑎 ⊆ 𝐽) → ∪ 𝑎 ∈ 𝐽) |
| 52 | 51 | ex 412 |
. . . . 5
⊢ (𝜑 → (𝑎 ⊆ 𝐽 → ∪ 𝑎 ∈ 𝐽)) |
| 53 | 52 | alrimiv 1927 |
. . . 4
⊢ (𝜑 → ∀𝑎(𝑎 ⊆ 𝐽 → ∪ 𝑎 ∈ 𝐽)) |
| 54 | | inss1 4237 |
. . . . . . . 8
⊢ (𝑎 ∩ 𝑏) ⊆ 𝑎 |
| 55 | | difeq2 4120 |
. . . . . . . . . . . . 13
⊢ (𝑧 = 𝑎 → (𝐵 ∖ 𝑧) = (𝐵 ∖ 𝑎)) |
| 56 | 55 | eleq1d 2826 |
. . . . . . . . . . . 12
⊢ (𝑧 = 𝑎 → ((𝐵 ∖ 𝑧) ∈ 𝑀 ↔ (𝐵 ∖ 𝑎) ∈ 𝑀)) |
| 57 | 56, 5 | elrab2 3695 |
. . . . . . . . . . 11
⊢ (𝑎 ∈ 𝐽 ↔ (𝑎 ∈ 𝒫 𝐵 ∧ (𝐵 ∖ 𝑎) ∈ 𝑀)) |
| 58 | 57 | simplbi 497 |
. . . . . . . . . 10
⊢ (𝑎 ∈ 𝐽 → 𝑎 ∈ 𝒫 𝐵) |
| 59 | 58 | elpwid 4609 |
. . . . . . . . 9
⊢ (𝑎 ∈ 𝐽 → 𝑎 ⊆ 𝐵) |
| 60 | 59 | ad2antrl 728 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑎 ∈ 𝐽 ∧ 𝑏 ∈ 𝐽)) → 𝑎 ⊆ 𝐵) |
| 61 | 54, 60 | sstrid 3995 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑎 ∈ 𝐽 ∧ 𝑏 ∈ 𝐽)) → (𝑎 ∩ 𝑏) ⊆ 𝐵) |
| 62 | | vex 3484 |
. . . . . . . . 9
⊢ 𝑎 ∈ V |
| 63 | 62 | inex1 5317 |
. . . . . . . 8
⊢ (𝑎 ∩ 𝑏) ∈ V |
| 64 | 63 | elpw 4604 |
. . . . . . 7
⊢ ((𝑎 ∩ 𝑏) ∈ 𝒫 𝐵 ↔ (𝑎 ∩ 𝑏) ⊆ 𝐵) |
| 65 | 61, 64 | sylibr 234 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑎 ∈ 𝐽 ∧ 𝑏 ∈ 𝐽)) → (𝑎 ∩ 𝑏) ∈ 𝒫 𝐵) |
| 66 | | difindi 4292 |
. . . . . . 7
⊢ (𝐵 ∖ (𝑎 ∩ 𝑏)) = ((𝐵 ∖ 𝑎) ∪ (𝐵 ∖ 𝑏)) |
| 67 | 57 | simprbi 496 |
. . . . . . . . 9
⊢ (𝑎 ∈ 𝐽 → (𝐵 ∖ 𝑎) ∈ 𝑀) |
| 68 | 67 | ad2antrl 728 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑎 ∈ 𝐽 ∧ 𝑏 ∈ 𝐽)) → (𝐵 ∖ 𝑎) ∈ 𝑀) |
| 69 | 26 | ad2antll 729 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑎 ∈ 𝐽 ∧ 𝑏 ∈ 𝐽)) → (𝐵 ∖ 𝑏) ∈ 𝑀) |
| 70 | | simpl 482 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑎 ∈ 𝐽 ∧ 𝑏 ∈ 𝐽)) → 𝜑) |
| 71 | | uneq1 4161 |
. . . . . . . . . . . 12
⊢ (𝑥 = (𝐵 ∖ 𝑎) → (𝑥 ∪ 𝑦) = ((𝐵 ∖ 𝑎) ∪ 𝑦)) |
| 72 | 71 | eleq1d 2826 |
. . . . . . . . . . 11
⊢ (𝑥 = (𝐵 ∖ 𝑎) → ((𝑥 ∪ 𝑦) ∈ 𝑀 ↔ ((𝐵 ∖ 𝑎) ∪ 𝑦) ∈ 𝑀)) |
| 73 | 72 | imbi2d 340 |
. . . . . . . . . 10
⊢ (𝑥 = (𝐵 ∖ 𝑎) → ((𝜑 → (𝑥 ∪ 𝑦) ∈ 𝑀) ↔ (𝜑 → ((𝐵 ∖ 𝑎) ∪ 𝑦) ∈ 𝑀))) |
| 74 | | uneq2 4162 |
. . . . . . . . . . . 12
⊢ (𝑦 = (𝐵 ∖ 𝑏) → ((𝐵 ∖ 𝑎) ∪ 𝑦) = ((𝐵 ∖ 𝑎) ∪ (𝐵 ∖ 𝑏))) |
| 75 | 74 | eleq1d 2826 |
. . . . . . . . . . 11
⊢ (𝑦 = (𝐵 ∖ 𝑏) → (((𝐵 ∖ 𝑎) ∪ 𝑦) ∈ 𝑀 ↔ ((𝐵 ∖ 𝑎) ∪ (𝐵 ∖ 𝑏)) ∈ 𝑀)) |
| 76 | 75 | imbi2d 340 |
. . . . . . . . . 10
⊢ (𝑦 = (𝐵 ∖ 𝑏) → ((𝜑 → ((𝐵 ∖ 𝑎) ∪ 𝑦) ∈ 𝑀) ↔ (𝜑 → ((𝐵 ∖ 𝑎) ∪ (𝐵 ∖ 𝑏)) ∈ 𝑀))) |
| 77 | | mretopd.u |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑀 ∧ 𝑦 ∈ 𝑀) → (𝑥 ∪ 𝑦) ∈ 𝑀) |
| 78 | 77 | 3expb 1121 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑀 ∧ 𝑦 ∈ 𝑀)) → (𝑥 ∪ 𝑦) ∈ 𝑀) |
| 79 | 78 | expcom 413 |
. . . . . . . . . 10
⊢ ((𝑥 ∈ 𝑀 ∧ 𝑦 ∈ 𝑀) → (𝜑 → (𝑥 ∪ 𝑦) ∈ 𝑀)) |
| 80 | 73, 76, 79 | vtocl2ga 3578 |
. . . . . . . . 9
⊢ (((𝐵 ∖ 𝑎) ∈ 𝑀 ∧ (𝐵 ∖ 𝑏) ∈ 𝑀) → (𝜑 → ((𝐵 ∖ 𝑎) ∪ (𝐵 ∖ 𝑏)) ∈ 𝑀)) |
| 81 | 80 | imp 406 |
. . . . . . . 8
⊢ ((((𝐵 ∖ 𝑎) ∈ 𝑀 ∧ (𝐵 ∖ 𝑏) ∈ 𝑀) ∧ 𝜑) → ((𝐵 ∖ 𝑎) ∪ (𝐵 ∖ 𝑏)) ∈ 𝑀) |
| 82 | 68, 69, 70, 81 | syl21anc 838 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑎 ∈ 𝐽 ∧ 𝑏 ∈ 𝐽)) → ((𝐵 ∖ 𝑎) ∪ (𝐵 ∖ 𝑏)) ∈ 𝑀) |
| 83 | 66, 82 | eqeltrid 2845 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑎 ∈ 𝐽 ∧ 𝑏 ∈ 𝐽)) → (𝐵 ∖ (𝑎 ∩ 𝑏)) ∈ 𝑀) |
| 84 | | difeq2 4120 |
. . . . . . . 8
⊢ (𝑧 = (𝑎 ∩ 𝑏) → (𝐵 ∖ 𝑧) = (𝐵 ∖ (𝑎 ∩ 𝑏))) |
| 85 | 84 | eleq1d 2826 |
. . . . . . 7
⊢ (𝑧 = (𝑎 ∩ 𝑏) → ((𝐵 ∖ 𝑧) ∈ 𝑀 ↔ (𝐵 ∖ (𝑎 ∩ 𝑏)) ∈ 𝑀)) |
| 86 | 85, 5 | elrab2 3695 |
. . . . . 6
⊢ ((𝑎 ∩ 𝑏) ∈ 𝐽 ↔ ((𝑎 ∩ 𝑏) ∈ 𝒫 𝐵 ∧ (𝐵 ∖ (𝑎 ∩ 𝑏)) ∈ 𝑀)) |
| 87 | 65, 83, 86 | sylanbrc 583 |
. . . . 5
⊢ ((𝜑 ∧ (𝑎 ∈ 𝐽 ∧ 𝑏 ∈ 𝐽)) → (𝑎 ∩ 𝑏) ∈ 𝐽) |
| 88 | 87 | ralrimivva 3202 |
. . . 4
⊢ (𝜑 → ∀𝑎 ∈ 𝐽 ∀𝑏 ∈ 𝐽 (𝑎 ∩ 𝑏) ∈ 𝐽) |
| 89 | 43 | pwexd 5379 |
. . . . . 6
⊢ (𝜑 → 𝒫 𝐵 ∈ V) |
| 90 | 5, 89 | rabexd 5340 |
. . . . 5
⊢ (𝜑 → 𝐽 ∈ V) |
| 91 | | istopg 22901 |
. . . . 5
⊢ (𝐽 ∈ V → (𝐽 ∈ Top ↔
(∀𝑎(𝑎 ⊆ 𝐽 → ∪ 𝑎 ∈ 𝐽) ∧ ∀𝑎 ∈ 𝐽 ∀𝑏 ∈ 𝐽 (𝑎 ∩ 𝑏) ∈ 𝐽))) |
| 92 | 90, 91 | syl 17 |
. . . 4
⊢ (𝜑 → (𝐽 ∈ Top ↔ (∀𝑎(𝑎 ⊆ 𝐽 → ∪ 𝑎 ∈ 𝐽) ∧ ∀𝑎 ∈ 𝐽 ∀𝑏 ∈ 𝐽 (𝑎 ∩ 𝑏) ∈ 𝐽))) |
| 93 | 53, 88, 92 | mpbir2and 713 |
. . 3
⊢ (𝜑 → 𝐽 ∈ Top) |
| 94 | 6 | unissi 4916 |
. . . . . 6
⊢ ∪ 𝐽
⊆ ∪ 𝒫 𝐵 |
| 95 | | unipw 5455 |
. . . . . 6
⊢ ∪ 𝒫 𝐵 = 𝐵 |
| 96 | 94, 95 | sseqtri 4032 |
. . . . 5
⊢ ∪ 𝐽
⊆ 𝐵 |
| 97 | | pwidg 4620 |
. . . . . . 7
⊢ (𝐵 ∈ 𝑀 → 𝐵 ∈ 𝒫 𝐵) |
| 98 | 43, 97 | syl 17 |
. . . . . 6
⊢ (𝜑 → 𝐵 ∈ 𝒫 𝐵) |
| 99 | | difid 4376 |
. . . . . . 7
⊢ (𝐵 ∖ 𝐵) = ∅ |
| 100 | | mretopd.z |
. . . . . . 7
⊢ (𝜑 → ∅ ∈ 𝑀) |
| 101 | 99, 100 | eqeltrid 2845 |
. . . . . 6
⊢ (𝜑 → (𝐵 ∖ 𝐵) ∈ 𝑀) |
| 102 | | difeq2 4120 |
. . . . . . . 8
⊢ (𝑧 = 𝐵 → (𝐵 ∖ 𝑧) = (𝐵 ∖ 𝐵)) |
| 103 | 102 | eleq1d 2826 |
. . . . . . 7
⊢ (𝑧 = 𝐵 → ((𝐵 ∖ 𝑧) ∈ 𝑀 ↔ (𝐵 ∖ 𝐵) ∈ 𝑀)) |
| 104 | 103, 5 | elrab2 3695 |
. . . . . 6
⊢ (𝐵 ∈ 𝐽 ↔ (𝐵 ∈ 𝒫 𝐵 ∧ (𝐵 ∖ 𝐵) ∈ 𝑀)) |
| 105 | 98, 101, 104 | sylanbrc 583 |
. . . . 5
⊢ (𝜑 → 𝐵 ∈ 𝐽) |
| 106 | | unissel 4938 |
. . . . 5
⊢ ((∪ 𝐽
⊆ 𝐵 ∧ 𝐵 ∈ 𝐽) → ∪ 𝐽 = 𝐵) |
| 107 | 96, 105, 106 | sylancr 587 |
. . . 4
⊢ (𝜑 → ∪ 𝐽 =
𝐵) |
| 108 | 107 | eqcomd 2743 |
. . 3
⊢ (𝜑 → 𝐵 = ∪ 𝐽) |
| 109 | | istopon 22918 |
. . 3
⊢ (𝐽 ∈ (TopOn‘𝐵) ↔ (𝐽 ∈ Top ∧ 𝐵 = ∪ 𝐽)) |
| 110 | 93, 108, 109 | sylanbrc 583 |
. 2
⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝐵)) |
| 111 | | eqid 2737 |
. . . . 5
⊢ ∪ 𝐽 =
∪ 𝐽 |
| 112 | 111 | cldval 23031 |
. . . 4
⊢ (𝐽 ∈ Top →
(Clsd‘𝐽) = {𝑥 ∈ 𝒫 ∪ 𝐽
∣ (∪ 𝐽 ∖ 𝑥) ∈ 𝐽}) |
| 113 | 93, 112 | syl 17 |
. . 3
⊢ (𝜑 → (Clsd‘𝐽) = {𝑥 ∈ 𝒫 ∪ 𝐽
∣ (∪ 𝐽 ∖ 𝑥) ∈ 𝐽}) |
| 114 | 107 | pweqd 4617 |
. . . 4
⊢ (𝜑 → 𝒫 ∪ 𝐽 =
𝒫 𝐵) |
| 115 | 107 | difeq1d 4125 |
. . . . 5
⊢ (𝜑 → (∪ 𝐽
∖ 𝑥) = (𝐵 ∖ 𝑥)) |
| 116 | 115 | eleq1d 2826 |
. . . 4
⊢ (𝜑 → ((∪ 𝐽
∖ 𝑥) ∈ 𝐽 ↔ (𝐵 ∖ 𝑥) ∈ 𝐽)) |
| 117 | 114, 116 | rabeqbidv 3455 |
. . 3
⊢ (𝜑 → {𝑥 ∈ 𝒫 ∪ 𝐽
∣ (∪ 𝐽 ∖ 𝑥) ∈ 𝐽} = {𝑥 ∈ 𝒫 𝐵 ∣ (𝐵 ∖ 𝑥) ∈ 𝐽}) |
| 118 | 5 | eleq2i 2833 |
. . . . . . 7
⊢ ((𝐵 ∖ 𝑥) ∈ 𝐽 ↔ (𝐵 ∖ 𝑥) ∈ {𝑧 ∈ 𝒫 𝐵 ∣ (𝐵 ∖ 𝑧) ∈ 𝑀}) |
| 119 | | difss 4136 |
. . . . . . . . . 10
⊢ (𝐵 ∖ 𝑥) ⊆ 𝐵 |
| 120 | | elpw2g 5333 |
. . . . . . . . . . 11
⊢ (𝐵 ∈ 𝑀 → ((𝐵 ∖ 𝑥) ∈ 𝒫 𝐵 ↔ (𝐵 ∖ 𝑥) ⊆ 𝐵)) |
| 121 | 43, 120 | syl 17 |
. . . . . . . . . 10
⊢ (𝜑 → ((𝐵 ∖ 𝑥) ∈ 𝒫 𝐵 ↔ (𝐵 ∖ 𝑥) ⊆ 𝐵)) |
| 122 | 119, 121 | mpbiri 258 |
. . . . . . . . 9
⊢ (𝜑 → (𝐵 ∖ 𝑥) ∈ 𝒫 𝐵) |
| 123 | | difeq2 4120 |
. . . . . . . . . . 11
⊢ (𝑧 = (𝐵 ∖ 𝑥) → (𝐵 ∖ 𝑧) = (𝐵 ∖ (𝐵 ∖ 𝑥))) |
| 124 | 123 | eleq1d 2826 |
. . . . . . . . . 10
⊢ (𝑧 = (𝐵 ∖ 𝑥) → ((𝐵 ∖ 𝑧) ∈ 𝑀 ↔ (𝐵 ∖ (𝐵 ∖ 𝑥)) ∈ 𝑀)) |
| 125 | 124 | elrab3 3693 |
. . . . . . . . 9
⊢ ((𝐵 ∖ 𝑥) ∈ 𝒫 𝐵 → ((𝐵 ∖ 𝑥) ∈ {𝑧 ∈ 𝒫 𝐵 ∣ (𝐵 ∖ 𝑧) ∈ 𝑀} ↔ (𝐵 ∖ (𝐵 ∖ 𝑥)) ∈ 𝑀)) |
| 126 | 122, 125 | syl 17 |
. . . . . . . 8
⊢ (𝜑 → ((𝐵 ∖ 𝑥) ∈ {𝑧 ∈ 𝒫 𝐵 ∣ (𝐵 ∖ 𝑧) ∈ 𝑀} ↔ (𝐵 ∖ (𝐵 ∖ 𝑥)) ∈ 𝑀)) |
| 127 | 126 | adantr 480 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ 𝒫 𝐵) → ((𝐵 ∖ 𝑥) ∈ {𝑧 ∈ 𝒫 𝐵 ∣ (𝐵 ∖ 𝑧) ∈ 𝑀} ↔ (𝐵 ∖ (𝐵 ∖ 𝑥)) ∈ 𝑀)) |
| 128 | 118, 127 | bitrid 283 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ 𝒫 𝐵) → ((𝐵 ∖ 𝑥) ∈ 𝐽 ↔ (𝐵 ∖ (𝐵 ∖ 𝑥)) ∈ 𝑀)) |
| 129 | | elpwi 4607 |
. . . . . . . . 9
⊢ (𝑥 ∈ 𝒫 𝐵 → 𝑥 ⊆ 𝐵) |
| 130 | | dfss4 4269 |
. . . . . . . . 9
⊢ (𝑥 ⊆ 𝐵 ↔ (𝐵 ∖ (𝐵 ∖ 𝑥)) = 𝑥) |
| 131 | 129, 130 | sylib 218 |
. . . . . . . 8
⊢ (𝑥 ∈ 𝒫 𝐵 → (𝐵 ∖ (𝐵 ∖ 𝑥)) = 𝑥) |
| 132 | 131 | adantl 481 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ 𝒫 𝐵) → (𝐵 ∖ (𝐵 ∖ 𝑥)) = 𝑥) |
| 133 | 132 | eleq1d 2826 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ 𝒫 𝐵) → ((𝐵 ∖ (𝐵 ∖ 𝑥)) ∈ 𝑀 ↔ 𝑥 ∈ 𝑀)) |
| 134 | 128, 133 | bitrd 279 |
. . . . 5
⊢ ((𝜑 ∧ 𝑥 ∈ 𝒫 𝐵) → ((𝐵 ∖ 𝑥) ∈ 𝐽 ↔ 𝑥 ∈ 𝑀)) |
| 135 | 134 | rabbidva 3443 |
. . . 4
⊢ (𝜑 → {𝑥 ∈ 𝒫 𝐵 ∣ (𝐵 ∖ 𝑥) ∈ 𝐽} = {𝑥 ∈ 𝒫 𝐵 ∣ 𝑥 ∈ 𝑀}) |
| 136 | | incom 4209 |
. . . . . 6
⊢ (𝑀 ∩ 𝒫 𝐵) = (𝒫 𝐵 ∩ 𝑀) |
| 137 | | dfin5 3959 |
. . . . . 6
⊢
(𝒫 𝐵 ∩
𝑀) = {𝑥 ∈ 𝒫 𝐵 ∣ 𝑥 ∈ 𝑀} |
| 138 | 136, 137 | eqtri 2765 |
. . . . 5
⊢ (𝑀 ∩ 𝒫 𝐵) = {𝑥 ∈ 𝒫 𝐵 ∣ 𝑥 ∈ 𝑀} |
| 139 | | mresspw 17635 |
. . . . . . 7
⊢ (𝑀 ∈ (Moore‘𝐵) → 𝑀 ⊆ 𝒫 𝐵) |
| 140 | 20, 139 | syl 17 |
. . . . . 6
⊢ (𝜑 → 𝑀 ⊆ 𝒫 𝐵) |
| 141 | | dfss2 3969 |
. . . . . 6
⊢ (𝑀 ⊆ 𝒫 𝐵 ↔ (𝑀 ∩ 𝒫 𝐵) = 𝑀) |
| 142 | 140, 141 | sylib 218 |
. . . . 5
⊢ (𝜑 → (𝑀 ∩ 𝒫 𝐵) = 𝑀) |
| 143 | 138, 142 | eqtr3id 2791 |
. . . 4
⊢ (𝜑 → {𝑥 ∈ 𝒫 𝐵 ∣ 𝑥 ∈ 𝑀} = 𝑀) |
| 144 | 135, 143 | eqtrd 2777 |
. . 3
⊢ (𝜑 → {𝑥 ∈ 𝒫 𝐵 ∣ (𝐵 ∖ 𝑥) ∈ 𝐽} = 𝑀) |
| 145 | 113, 117,
144 | 3eqtrrd 2782 |
. 2
⊢ (𝜑 → 𝑀 = (Clsd‘𝐽)) |
| 146 | 110, 145 | jca 511 |
1
⊢ (𝜑 → (𝐽 ∈ (TopOn‘𝐵) ∧ 𝑀 = (Clsd‘𝐽))) |