| Step | Hyp | Ref
| Expression |
| 1 | | nllytop 23481 |
. . 3
⊢ (𝑅 ∈ 𝑛-Locally Comp
→ 𝑅 ∈
Top) |
| 2 | | elinel1 4201 |
. . . 4
⊢ (𝑆 ∈ (ran 𝑘Gen ∩
Haus) → 𝑆 ∈ ran
𝑘Gen) |
| 3 | | kgentop 23550 |
. . . 4
⊢ (𝑆 ∈ ran 𝑘Gen →
𝑆 ∈
Top) |
| 4 | 2, 3 | syl 17 |
. . 3
⊢ (𝑆 ∈ (ran 𝑘Gen ∩
Haus) → 𝑆 ∈
Top) |
| 5 | | txtop 23577 |
. . 3
⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑅 ×t 𝑆) ∈ Top) |
| 6 | 1, 4, 5 | syl2an 596 |
. 2
⊢ ((𝑅 ∈ 𝑛-Locally Comp
∧ 𝑆 ∈ (ran
𝑘Gen ∩ Haus)) → (𝑅 ×t 𝑆) ∈ Top) |
| 7 | | simplll 775 |
. . . . . . . 8
⊢ ((((𝑅 ∈ 𝑛-Locally Comp
∧ 𝑆 ∈ (ran
𝑘Gen ∩ Haus)) ∧ 𝑥 ∈ (𝑘Gen‘(𝑅 ×t 𝑆))) ∧ 𝑦 ∈ 𝑥) → 𝑅 ∈ 𝑛-Locally
Comp) |
| 8 | | eqid 2737 |
. . . . . . . . . 10
⊢ (𝑡 ∈ ∪ 𝑅
↦ 〈𝑡,
(2nd ‘𝑦)〉) = (𝑡 ∈ ∪ 𝑅 ↦ 〈𝑡, (2nd ‘𝑦)〉) |
| 9 | 8 | mptpreima 6258 |
. . . . . . . . 9
⊢ (◡(𝑡 ∈ ∪ 𝑅 ↦ 〈𝑡, (2nd ‘𝑦)〉) “ 𝑥) = {𝑡 ∈ ∪ 𝑅 ∣ 〈𝑡, (2nd ‘𝑦)〉 ∈ 𝑥} |
| 10 | 1 | ad3antrrr 730 |
. . . . . . . . . . . . . 14
⊢ ((((𝑅 ∈ 𝑛-Locally Comp
∧ 𝑆 ∈ (ran
𝑘Gen ∩ Haus)) ∧ 𝑥 ∈ (𝑘Gen‘(𝑅 ×t 𝑆))) ∧ 𝑦 ∈ 𝑥) → 𝑅 ∈ Top) |
| 11 | | toptopon2 22924 |
. . . . . . . . . . . . . 14
⊢ (𝑅 ∈ Top ↔ 𝑅 ∈ (TopOn‘∪ 𝑅)) |
| 12 | 10, 11 | sylib 218 |
. . . . . . . . . . . . 13
⊢ ((((𝑅 ∈ 𝑛-Locally Comp
∧ 𝑆 ∈ (ran
𝑘Gen ∩ Haus)) ∧ 𝑥 ∈ (𝑘Gen‘(𝑅 ×t 𝑆))) ∧ 𝑦 ∈ 𝑥) → 𝑅 ∈ (TopOn‘∪ 𝑅)) |
| 13 | | idcn 23265 |
. . . . . . . . . . . . 13
⊢ (𝑅 ∈ (TopOn‘∪ 𝑅)
→ ( I ↾ ∪ 𝑅) ∈ (𝑅 Cn 𝑅)) |
| 14 | 12, 13 | syl 17 |
. . . . . . . . . . . 12
⊢ ((((𝑅 ∈ 𝑛-Locally Comp
∧ 𝑆 ∈ (ran
𝑘Gen ∩ Haus)) ∧ 𝑥 ∈ (𝑘Gen‘(𝑅 ×t 𝑆))) ∧ 𝑦 ∈ 𝑥) → ( I ↾ ∪ 𝑅)
∈ (𝑅 Cn 𝑅)) |
| 15 | | simpllr 776 |
. . . . . . . . . . . . . . 15
⊢ ((((𝑅 ∈ 𝑛-Locally Comp
∧ 𝑆 ∈ (ran
𝑘Gen ∩ Haus)) ∧ 𝑥 ∈ (𝑘Gen‘(𝑅 ×t 𝑆))) ∧ 𝑦 ∈ 𝑥) → 𝑆 ∈ (ran 𝑘Gen ∩
Haus)) |
| 16 | 15, 4 | syl 17 |
. . . . . . . . . . . . . 14
⊢ ((((𝑅 ∈ 𝑛-Locally Comp
∧ 𝑆 ∈ (ran
𝑘Gen ∩ Haus)) ∧ 𝑥 ∈ (𝑘Gen‘(𝑅 ×t 𝑆))) ∧ 𝑦 ∈ 𝑥) → 𝑆 ∈ Top) |
| 17 | | toptopon2 22924 |
. . . . . . . . . . . . . 14
⊢ (𝑆 ∈ Top ↔ 𝑆 ∈ (TopOn‘∪ 𝑆)) |
| 18 | 16, 17 | sylib 218 |
. . . . . . . . . . . . 13
⊢ ((((𝑅 ∈ 𝑛-Locally Comp
∧ 𝑆 ∈ (ran
𝑘Gen ∩ Haus)) ∧ 𝑥 ∈ (𝑘Gen‘(𝑅 ×t 𝑆))) ∧ 𝑦 ∈ 𝑥) → 𝑆 ∈ (TopOn‘∪ 𝑆)) |
| 19 | | simpr 484 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝑅 ∈ 𝑛-Locally Comp
∧ 𝑆 ∈ (ran
𝑘Gen ∩ Haus)) ∧ 𝑥 ∈ (𝑘Gen‘(𝑅 ×t 𝑆))) ∧ 𝑦 ∈ 𝑥) → 𝑦 ∈ 𝑥) |
| 20 | | simplr 769 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝑅 ∈ 𝑛-Locally Comp
∧ 𝑆 ∈ (ran
𝑘Gen ∩ Haus)) ∧ 𝑥 ∈ (𝑘Gen‘(𝑅 ×t 𝑆))) ∧ 𝑦 ∈ 𝑥) → 𝑥 ∈ (𝑘Gen‘(𝑅 ×t 𝑆))) |
| 21 | | elunii 4912 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑦 ∈ 𝑥 ∧ 𝑥 ∈ (𝑘Gen‘(𝑅 ×t 𝑆))) → 𝑦 ∈ ∪
(𝑘Gen‘(𝑅
×t 𝑆))) |
| 22 | 19, 20, 21 | syl2anc 584 |
. . . . . . . . . . . . . . 15
⊢ ((((𝑅 ∈ 𝑛-Locally Comp
∧ 𝑆 ∈ (ran
𝑘Gen ∩ Haus)) ∧ 𝑥 ∈ (𝑘Gen‘(𝑅 ×t 𝑆))) ∧ 𝑦 ∈ 𝑥) → 𝑦 ∈ ∪
(𝑘Gen‘(𝑅
×t 𝑆))) |
| 23 | | eqid 2737 |
. . . . . . . . . . . . . . . . . 18
⊢ ∪ 𝑅 =
∪ 𝑅 |
| 24 | | eqid 2737 |
. . . . . . . . . . . . . . . . . 18
⊢ ∪ 𝑆 =
∪ 𝑆 |
| 25 | 23, 24 | txuni 23600 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (∪ 𝑅
× ∪ 𝑆) = ∪ (𝑅 ×t 𝑆)) |
| 26 | 10, 16, 25 | syl2anc 584 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝑅 ∈ 𝑛-Locally Comp
∧ 𝑆 ∈ (ran
𝑘Gen ∩ Haus)) ∧ 𝑥 ∈ (𝑘Gen‘(𝑅 ×t 𝑆))) ∧ 𝑦 ∈ 𝑥) → (∪ 𝑅 × ∪ 𝑆) =
∪ (𝑅 ×t 𝑆)) |
| 27 | 10, 16, 5 | syl2anc 584 |
. . . . . . . . . . . . . . . . 17
⊢ ((((𝑅 ∈ 𝑛-Locally Comp
∧ 𝑆 ∈ (ran
𝑘Gen ∩ Haus)) ∧ 𝑥 ∈ (𝑘Gen‘(𝑅 ×t 𝑆))) ∧ 𝑦 ∈ 𝑥) → (𝑅 ×t 𝑆) ∈ Top) |
| 28 | | eqid 2737 |
. . . . . . . . . . . . . . . . . 18
⊢ ∪ (𝑅
×t 𝑆) =
∪ (𝑅 ×t 𝑆) |
| 29 | 28 | kgenuni 23547 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑅 ×t 𝑆) ∈ Top → ∪ (𝑅
×t 𝑆) =
∪ (𝑘Gen‘(𝑅 ×t 𝑆))) |
| 30 | 27, 29 | syl 17 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝑅 ∈ 𝑛-Locally Comp
∧ 𝑆 ∈ (ran
𝑘Gen ∩ Haus)) ∧ 𝑥 ∈ (𝑘Gen‘(𝑅 ×t 𝑆))) ∧ 𝑦 ∈ 𝑥) → ∪ (𝑅 ×t 𝑆) = ∪
(𝑘Gen‘(𝑅
×t 𝑆))) |
| 31 | 26, 30 | eqtrd 2777 |
. . . . . . . . . . . . . . 15
⊢ ((((𝑅 ∈ 𝑛-Locally Comp
∧ 𝑆 ∈ (ran
𝑘Gen ∩ Haus)) ∧ 𝑥 ∈ (𝑘Gen‘(𝑅 ×t 𝑆))) ∧ 𝑦 ∈ 𝑥) → (∪ 𝑅 × ∪ 𝑆) =
∪ (𝑘Gen‘(𝑅 ×t 𝑆))) |
| 32 | 22, 31 | eleqtrrd 2844 |
. . . . . . . . . . . . . 14
⊢ ((((𝑅 ∈ 𝑛-Locally Comp
∧ 𝑆 ∈ (ran
𝑘Gen ∩ Haus)) ∧ 𝑥 ∈ (𝑘Gen‘(𝑅 ×t 𝑆))) ∧ 𝑦 ∈ 𝑥) → 𝑦 ∈ (∪ 𝑅 × ∪ 𝑆)) |
| 33 | | xp2nd 8047 |
. . . . . . . . . . . . . 14
⊢ (𝑦 ∈ (∪ 𝑅
× ∪ 𝑆) → (2nd ‘𝑦) ∈ ∪ 𝑆) |
| 34 | 32, 33 | syl 17 |
. . . . . . . . . . . . 13
⊢ ((((𝑅 ∈ 𝑛-Locally Comp
∧ 𝑆 ∈ (ran
𝑘Gen ∩ Haus)) ∧ 𝑥 ∈ (𝑘Gen‘(𝑅 ×t 𝑆))) ∧ 𝑦 ∈ 𝑥) → (2nd ‘𝑦) ∈ ∪ 𝑆) |
| 35 | | cnconst2 23291 |
. . . . . . . . . . . . 13
⊢ ((𝑅 ∈ (TopOn‘∪ 𝑅)
∧ 𝑆 ∈
(TopOn‘∪ 𝑆) ∧ (2nd ‘𝑦) ∈ ∪ 𝑆)
→ (∪ 𝑅 × {(2nd ‘𝑦)}) ∈ (𝑅 Cn 𝑆)) |
| 36 | 12, 18, 34, 35 | syl3anc 1373 |
. . . . . . . . . . . 12
⊢ ((((𝑅 ∈ 𝑛-Locally Comp
∧ 𝑆 ∈ (ran
𝑘Gen ∩ Haus)) ∧ 𝑥 ∈ (𝑘Gen‘(𝑅 ×t 𝑆))) ∧ 𝑦 ∈ 𝑥) → (∪ 𝑅 × {(2nd
‘𝑦)}) ∈ (𝑅 Cn 𝑆)) |
| 37 | | fvresi 7193 |
. . . . . . . . . . . . . . . 16
⊢ (𝑡 ∈ ∪ 𝑅
→ (( I ↾ ∪ 𝑅)‘𝑡) = 𝑡) |
| 38 | | fvex 6919 |
. . . . . . . . . . . . . . . . 17
⊢
(2nd ‘𝑦) ∈ V |
| 39 | 38 | fvconst2 7224 |
. . . . . . . . . . . . . . . 16
⊢ (𝑡 ∈ ∪ 𝑅
→ ((∪ 𝑅 × {(2nd ‘𝑦)})‘𝑡) = (2nd ‘𝑦)) |
| 40 | 37, 39 | opeq12d 4881 |
. . . . . . . . . . . . . . 15
⊢ (𝑡 ∈ ∪ 𝑅
→ 〈(( I ↾ ∪ 𝑅)‘𝑡), ((∪ 𝑅 × {(2nd
‘𝑦)})‘𝑡)〉 = 〈𝑡, (2nd ‘𝑦)〉) |
| 41 | 40 | mpteq2ia 5245 |
. . . . . . . . . . . . . 14
⊢ (𝑡 ∈ ∪ 𝑅
↦ 〈(( I ↾ ∪ 𝑅)‘𝑡), ((∪ 𝑅 × {(2nd
‘𝑦)})‘𝑡)〉) = (𝑡 ∈ ∪ 𝑅 ↦ 〈𝑡, (2nd ‘𝑦)〉) |
| 42 | 41 | eqcomi 2746 |
. . . . . . . . . . . . 13
⊢ (𝑡 ∈ ∪ 𝑅
↦ 〈𝑡,
(2nd ‘𝑦)〉) = (𝑡 ∈ ∪ 𝑅 ↦ 〈(( I ↾
∪ 𝑅)‘𝑡), ((∪ 𝑅 × {(2nd
‘𝑦)})‘𝑡)〉) |
| 43 | 23, 42 | txcnmpt 23632 |
. . . . . . . . . . . 12
⊢ ((( I
↾ ∪ 𝑅) ∈ (𝑅 Cn 𝑅) ∧ (∪ 𝑅 × {(2nd
‘𝑦)}) ∈ (𝑅 Cn 𝑆)) → (𝑡 ∈ ∪ 𝑅 ↦ 〈𝑡, (2nd ‘𝑦)〉) ∈ (𝑅 Cn (𝑅 ×t 𝑆))) |
| 44 | 14, 36, 43 | syl2anc 584 |
. . . . . . . . . . 11
⊢ ((((𝑅 ∈ 𝑛-Locally Comp
∧ 𝑆 ∈ (ran
𝑘Gen ∩ Haus)) ∧ 𝑥 ∈ (𝑘Gen‘(𝑅 ×t 𝑆))) ∧ 𝑦 ∈ 𝑥) → (𝑡 ∈ ∪ 𝑅 ↦ 〈𝑡, (2nd ‘𝑦)〉) ∈ (𝑅 Cn (𝑅 ×t 𝑆))) |
| 45 | | llycmpkgen 23560 |
. . . . . . . . . . . . 13
⊢ (𝑅 ∈ 𝑛-Locally Comp
→ 𝑅 ∈ ran
𝑘Gen) |
| 46 | 45 | ad3antrrr 730 |
. . . . . . . . . . . 12
⊢ ((((𝑅 ∈ 𝑛-Locally Comp
∧ 𝑆 ∈ (ran
𝑘Gen ∩ Haus)) ∧ 𝑥 ∈ (𝑘Gen‘(𝑅 ×t 𝑆))) ∧ 𝑦 ∈ 𝑥) → 𝑅 ∈ ran 𝑘Gen) |
| 47 | 6 | ad2antrr 726 |
. . . . . . . . . . . 12
⊢ ((((𝑅 ∈ 𝑛-Locally Comp
∧ 𝑆 ∈ (ran
𝑘Gen ∩ Haus)) ∧ 𝑥 ∈ (𝑘Gen‘(𝑅 ×t 𝑆))) ∧ 𝑦 ∈ 𝑥) → (𝑅 ×t 𝑆) ∈ Top) |
| 48 | | kgencn3 23566 |
. . . . . . . . . . . 12
⊢ ((𝑅 ∈ ran 𝑘Gen ∧
(𝑅 ×t
𝑆) ∈ Top) →
(𝑅 Cn (𝑅 ×t 𝑆)) = (𝑅 Cn (𝑘Gen‘(𝑅 ×t 𝑆)))) |
| 49 | 46, 47, 48 | syl2anc 584 |
. . . . . . . . . . 11
⊢ ((((𝑅 ∈ 𝑛-Locally Comp
∧ 𝑆 ∈ (ran
𝑘Gen ∩ Haus)) ∧ 𝑥 ∈ (𝑘Gen‘(𝑅 ×t 𝑆))) ∧ 𝑦 ∈ 𝑥) → (𝑅 Cn (𝑅 ×t 𝑆)) = (𝑅 Cn (𝑘Gen‘(𝑅 ×t 𝑆)))) |
| 50 | 44, 49 | eleqtrd 2843 |
. . . . . . . . . 10
⊢ ((((𝑅 ∈ 𝑛-Locally Comp
∧ 𝑆 ∈ (ran
𝑘Gen ∩ Haus)) ∧ 𝑥 ∈ (𝑘Gen‘(𝑅 ×t 𝑆))) ∧ 𝑦 ∈ 𝑥) → (𝑡 ∈ ∪ 𝑅 ↦ 〈𝑡, (2nd ‘𝑦)〉) ∈ (𝑅 Cn (𝑘Gen‘(𝑅 ×t 𝑆)))) |
| 51 | | cnima 23273 |
. . . . . . . . . 10
⊢ (((𝑡 ∈ ∪ 𝑅
↦ 〈𝑡,
(2nd ‘𝑦)〉) ∈ (𝑅 Cn (𝑘Gen‘(𝑅 ×t 𝑆))) ∧ 𝑥 ∈ (𝑘Gen‘(𝑅 ×t 𝑆))) → (◡(𝑡 ∈ ∪ 𝑅 ↦ 〈𝑡, (2nd ‘𝑦)〉) “ 𝑥) ∈ 𝑅) |
| 52 | 50, 20, 51 | syl2anc 584 |
. . . . . . . . 9
⊢ ((((𝑅 ∈ 𝑛-Locally Comp
∧ 𝑆 ∈ (ran
𝑘Gen ∩ Haus)) ∧ 𝑥 ∈ (𝑘Gen‘(𝑅 ×t 𝑆))) ∧ 𝑦 ∈ 𝑥) → (◡(𝑡 ∈ ∪ 𝑅 ↦ 〈𝑡, (2nd ‘𝑦)〉) “ 𝑥) ∈ 𝑅) |
| 53 | 9, 52 | eqeltrrid 2846 |
. . . . . . . 8
⊢ ((((𝑅 ∈ 𝑛-Locally Comp
∧ 𝑆 ∈ (ran
𝑘Gen ∩ Haus)) ∧ 𝑥 ∈ (𝑘Gen‘(𝑅 ×t 𝑆))) ∧ 𝑦 ∈ 𝑥) → {𝑡 ∈ ∪ 𝑅 ∣ 〈𝑡, (2nd ‘𝑦)〉 ∈ 𝑥} ∈ 𝑅) |
| 54 | | opeq1 4873 |
. . . . . . . . . 10
⊢ (𝑡 = (1st ‘𝑦) → 〈𝑡, (2nd ‘𝑦)〉 = 〈(1st
‘𝑦), (2nd
‘𝑦)〉) |
| 55 | 54 | eleq1d 2826 |
. . . . . . . . 9
⊢ (𝑡 = (1st ‘𝑦) → (〈𝑡, (2nd ‘𝑦)〉 ∈ 𝑥 ↔ 〈(1st
‘𝑦), (2nd
‘𝑦)〉 ∈
𝑥)) |
| 56 | | xp1st 8046 |
. . . . . . . . . 10
⊢ (𝑦 ∈ (∪ 𝑅
× ∪ 𝑆) → (1st ‘𝑦) ∈ ∪ 𝑅) |
| 57 | 32, 56 | syl 17 |
. . . . . . . . 9
⊢ ((((𝑅 ∈ 𝑛-Locally Comp
∧ 𝑆 ∈ (ran
𝑘Gen ∩ Haus)) ∧ 𝑥 ∈ (𝑘Gen‘(𝑅 ×t 𝑆))) ∧ 𝑦 ∈ 𝑥) → (1st ‘𝑦) ∈ ∪ 𝑅) |
| 58 | | 1st2nd2 8053 |
. . . . . . . . . . 11
⊢ (𝑦 ∈ (∪ 𝑅
× ∪ 𝑆) → 𝑦 = 〈(1st ‘𝑦), (2nd ‘𝑦)〉) |
| 59 | 32, 58 | syl 17 |
. . . . . . . . . 10
⊢ ((((𝑅 ∈ 𝑛-Locally Comp
∧ 𝑆 ∈ (ran
𝑘Gen ∩ Haus)) ∧ 𝑥 ∈ (𝑘Gen‘(𝑅 ×t 𝑆))) ∧ 𝑦 ∈ 𝑥) → 𝑦 = 〈(1st ‘𝑦), (2nd ‘𝑦)〉) |
| 60 | 59, 19 | eqeltrrd 2842 |
. . . . . . . . 9
⊢ ((((𝑅 ∈ 𝑛-Locally Comp
∧ 𝑆 ∈ (ran
𝑘Gen ∩ Haus)) ∧ 𝑥 ∈ (𝑘Gen‘(𝑅 ×t 𝑆))) ∧ 𝑦 ∈ 𝑥) → 〈(1st ‘𝑦), (2nd ‘𝑦)〉 ∈ 𝑥) |
| 61 | 55, 57, 60 | elrabd 3694 |
. . . . . . . 8
⊢ ((((𝑅 ∈ 𝑛-Locally Comp
∧ 𝑆 ∈ (ran
𝑘Gen ∩ Haus)) ∧ 𝑥 ∈ (𝑘Gen‘(𝑅 ×t 𝑆))) ∧ 𝑦 ∈ 𝑥) → (1st ‘𝑦) ∈ {𝑡 ∈ ∪ 𝑅 ∣ 〈𝑡, (2nd ‘𝑦)〉 ∈ 𝑥}) |
| 62 | | nlly2i 23484 |
. . . . . . . 8
⊢ ((𝑅 ∈ 𝑛-Locally Comp
∧ {𝑡 ∈ ∪ 𝑅
∣ 〈𝑡,
(2nd ‘𝑦)〉 ∈ 𝑥} ∈ 𝑅 ∧ (1st ‘𝑦) ∈ {𝑡 ∈ ∪ 𝑅 ∣ 〈𝑡, (2nd ‘𝑦)〉 ∈ 𝑥}) → ∃𝑠 ∈ 𝒫 {𝑡 ∈ ∪ 𝑅
∣ 〈𝑡,
(2nd ‘𝑦)〉 ∈ 𝑥}∃𝑢 ∈ 𝑅 ((1st ‘𝑦) ∈ 𝑢 ∧ 𝑢 ⊆ 𝑠 ∧ (𝑅 ↾t 𝑠) ∈ Comp)) |
| 63 | 7, 53, 61, 62 | syl3anc 1373 |
. . . . . . 7
⊢ ((((𝑅 ∈ 𝑛-Locally Comp
∧ 𝑆 ∈ (ran
𝑘Gen ∩ Haus)) ∧ 𝑥 ∈ (𝑘Gen‘(𝑅 ×t 𝑆))) ∧ 𝑦 ∈ 𝑥) → ∃𝑠 ∈ 𝒫 {𝑡 ∈ ∪ 𝑅 ∣ 〈𝑡, (2nd ‘𝑦)〉 ∈ 𝑥}∃𝑢 ∈ 𝑅 ((1st ‘𝑦) ∈ 𝑢 ∧ 𝑢 ⊆ 𝑠 ∧ (𝑅 ↾t 𝑠) ∈ Comp)) |
| 64 | 10 | adantr 480 |
. . . . . . . . . . 11
⊢
(((((𝑅 ∈
𝑛-Locally Comp ∧ 𝑆 ∈ (ran 𝑘Gen ∩ Haus)) ∧
𝑥 ∈
(𝑘Gen‘(𝑅
×t 𝑆)))
∧ 𝑦 ∈ 𝑥) ∧ ((𝑠 ∈ 𝒫 {𝑡 ∈ ∪ 𝑅 ∣ 〈𝑡, (2nd ‘𝑦)〉 ∈ 𝑥} ∧ 𝑢 ∈ 𝑅) ∧ ((1st ‘𝑦) ∈ 𝑢 ∧ 𝑢 ⊆ 𝑠 ∧ (𝑅 ↾t 𝑠) ∈ Comp))) → 𝑅 ∈ Top) |
| 65 | 16 | adantr 480 |
. . . . . . . . . . 11
⊢
(((((𝑅 ∈
𝑛-Locally Comp ∧ 𝑆 ∈ (ran 𝑘Gen ∩ Haus)) ∧
𝑥 ∈
(𝑘Gen‘(𝑅
×t 𝑆)))
∧ 𝑦 ∈ 𝑥) ∧ ((𝑠 ∈ 𝒫 {𝑡 ∈ ∪ 𝑅 ∣ 〈𝑡, (2nd ‘𝑦)〉 ∈ 𝑥} ∧ 𝑢 ∈ 𝑅) ∧ ((1st ‘𝑦) ∈ 𝑢 ∧ 𝑢 ⊆ 𝑠 ∧ (𝑅 ↾t 𝑠) ∈ Comp))) → 𝑆 ∈ Top) |
| 66 | | simprlr 780 |
. . . . . . . . . . 11
⊢
(((((𝑅 ∈
𝑛-Locally Comp ∧ 𝑆 ∈ (ran 𝑘Gen ∩ Haus)) ∧
𝑥 ∈
(𝑘Gen‘(𝑅
×t 𝑆)))
∧ 𝑦 ∈ 𝑥) ∧ ((𝑠 ∈ 𝒫 {𝑡 ∈ ∪ 𝑅 ∣ 〈𝑡, (2nd ‘𝑦)〉 ∈ 𝑥} ∧ 𝑢 ∈ 𝑅) ∧ ((1st ‘𝑦) ∈ 𝑢 ∧ 𝑢 ⊆ 𝑠 ∧ (𝑅 ↾t 𝑠) ∈ Comp))) → 𝑢 ∈ 𝑅) |
| 67 | | ssrab2 4080 |
. . . . . . . . . . . . . 14
⊢ {𝑣 ∈ ∪ 𝑆
∣ (𝑠 × {𝑣}) ⊆ 𝑥} ⊆ ∪ 𝑆 |
| 68 | 67 | a1i 11 |
. . . . . . . . . . . . 13
⊢
(((((𝑅 ∈
𝑛-Locally Comp ∧ 𝑆 ∈ (ran 𝑘Gen ∩ Haus)) ∧
𝑥 ∈
(𝑘Gen‘(𝑅
×t 𝑆)))
∧ 𝑦 ∈ 𝑥) ∧ ((𝑠 ∈ 𝒫 {𝑡 ∈ ∪ 𝑅 ∣ 〈𝑡, (2nd ‘𝑦)〉 ∈ 𝑥} ∧ 𝑢 ∈ 𝑅) ∧ ((1st ‘𝑦) ∈ 𝑢 ∧ 𝑢 ⊆ 𝑠 ∧ (𝑅 ↾t 𝑠) ∈ Comp))) → {𝑣 ∈ ∪ 𝑆 ∣ (𝑠 × {𝑣}) ⊆ 𝑥} ⊆ ∪ 𝑆) |
| 69 | | incom 4209 |
. . . . . . . . . . . . . . . 16
⊢ ({𝑣 ∈ ∪ 𝑆
∣ (𝑠 × {𝑣}) ⊆ 𝑥} ∩ 𝑘) = (𝑘 ∩ {𝑣 ∈ ∪ 𝑆 ∣ (𝑠 × {𝑣}) ⊆ 𝑥}) |
| 70 | | simprll 779 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢
(((((𝑅 ∈
𝑛-Locally Comp ∧ 𝑆 ∈ (ran 𝑘Gen ∩ Haus)) ∧
𝑥 ∈
(𝑘Gen‘(𝑅
×t 𝑆)))
∧ 𝑦 ∈ 𝑥) ∧ ((𝑠 ∈ 𝒫 {𝑡 ∈ ∪ 𝑅 ∣ 〈𝑡, (2nd ‘𝑦)〉 ∈ 𝑥} ∧ 𝑢 ∈ 𝑅) ∧ ((1st ‘𝑦) ∈ 𝑢 ∧ 𝑢 ⊆ 𝑠 ∧ (𝑅 ↾t 𝑠) ∈ Comp))) → 𝑠 ∈ 𝒫 {𝑡 ∈ ∪ 𝑅 ∣ 〈𝑡, (2nd ‘𝑦)〉 ∈ 𝑥}) |
| 71 | 70 | elpwid 4609 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
(((((𝑅 ∈
𝑛-Locally Comp ∧ 𝑆 ∈ (ran 𝑘Gen ∩ Haus)) ∧
𝑥 ∈
(𝑘Gen‘(𝑅
×t 𝑆)))
∧ 𝑦 ∈ 𝑥) ∧ ((𝑠 ∈ 𝒫 {𝑡 ∈ ∪ 𝑅 ∣ 〈𝑡, (2nd ‘𝑦)〉 ∈ 𝑥} ∧ 𝑢 ∈ 𝑅) ∧ ((1st ‘𝑦) ∈ 𝑢 ∧ 𝑢 ⊆ 𝑠 ∧ (𝑅 ↾t 𝑠) ∈ Comp))) → 𝑠 ⊆ {𝑡 ∈ ∪ 𝑅 ∣ 〈𝑡, (2nd ‘𝑦)〉 ∈ 𝑥}) |
| 72 | | ssrab2 4080 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ {𝑡 ∈ ∪ 𝑅
∣ 〈𝑡,
(2nd ‘𝑦)〉 ∈ 𝑥} ⊆ ∪ 𝑅 |
| 73 | 71, 72 | sstrdi 3996 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
(((((𝑅 ∈
𝑛-Locally Comp ∧ 𝑆 ∈ (ran 𝑘Gen ∩ Haus)) ∧
𝑥 ∈
(𝑘Gen‘(𝑅
×t 𝑆)))
∧ 𝑦 ∈ 𝑥) ∧ ((𝑠 ∈ 𝒫 {𝑡 ∈ ∪ 𝑅 ∣ 〈𝑡, (2nd ‘𝑦)〉 ∈ 𝑥} ∧ 𝑢 ∈ 𝑅) ∧ ((1st ‘𝑦) ∈ 𝑢 ∧ 𝑢 ⊆ 𝑠 ∧ (𝑅 ↾t 𝑠) ∈ Comp))) → 𝑠 ⊆ ∪ 𝑅) |
| 74 | 73 | adantr 480 |
. . . . . . . . . . . . . . . . . . . 20
⊢
((((((𝑅 ∈
𝑛-Locally Comp ∧ 𝑆 ∈ (ran 𝑘Gen ∩ Haus)) ∧
𝑥 ∈
(𝑘Gen‘(𝑅
×t 𝑆)))
∧ 𝑦 ∈ 𝑥) ∧ ((𝑠 ∈ 𝒫 {𝑡 ∈ ∪ 𝑅 ∣ 〈𝑡, (2nd ‘𝑦)〉 ∈ 𝑥} ∧ 𝑢 ∈ 𝑅) ∧ ((1st ‘𝑦) ∈ 𝑢 ∧ 𝑢 ⊆ 𝑠 ∧ (𝑅 ↾t 𝑠) ∈ Comp))) ∧ (𝑘 ∈ 𝒫 ∪ 𝑆
∧ (𝑆
↾t 𝑘)
∈ Comp)) → 𝑠
⊆ ∪ 𝑅) |
| 75 | | elpwi 4607 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑘 ∈ 𝒫 ∪ 𝑆
→ 𝑘 ⊆ ∪ 𝑆) |
| 76 | 75 | ad2antrl 728 |
. . . . . . . . . . . . . . . . . . . 20
⊢
((((((𝑅 ∈
𝑛-Locally Comp ∧ 𝑆 ∈ (ran 𝑘Gen ∩ Haus)) ∧
𝑥 ∈
(𝑘Gen‘(𝑅
×t 𝑆)))
∧ 𝑦 ∈ 𝑥) ∧ ((𝑠 ∈ 𝒫 {𝑡 ∈ ∪ 𝑅 ∣ 〈𝑡, (2nd ‘𝑦)〉 ∈ 𝑥} ∧ 𝑢 ∈ 𝑅) ∧ ((1st ‘𝑦) ∈ 𝑢 ∧ 𝑢 ⊆ 𝑠 ∧ (𝑅 ↾t 𝑠) ∈ Comp))) ∧ (𝑘 ∈ 𝒫 ∪ 𝑆
∧ (𝑆
↾t 𝑘)
∈ Comp)) → 𝑘
⊆ ∪ 𝑆) |
| 77 | | eldif 3961 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (𝑡 ∈ ((𝑠 × 𝑘) ∖ 𝑥) ↔ (𝑡 ∈ (𝑠 × 𝑘) ∧ ¬ 𝑡 ∈ 𝑥)) |
| 78 | 77 | anbi1i 624 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝑡 ∈ ((𝑠 × 𝑘) ∖ 𝑥) ∧ ((2nd ↾ (∪ 𝑅
× ∪ 𝑆))‘𝑡) = 𝑏) ↔ ((𝑡 ∈ (𝑠 × 𝑘) ∧ ¬ 𝑡 ∈ 𝑥) ∧ ((2nd ↾ (∪ 𝑅
× ∪ 𝑆))‘𝑡) = 𝑏)) |
| 79 | | anass 468 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (((𝑡 ∈ (𝑠 × 𝑘) ∧ ¬ 𝑡 ∈ 𝑥) ∧ ((2nd ↾ (∪ 𝑅
× ∪ 𝑆))‘𝑡) = 𝑏) ↔ (𝑡 ∈ (𝑠 × 𝑘) ∧ (¬ 𝑡 ∈ 𝑥 ∧ ((2nd ↾ (∪ 𝑅
× ∪ 𝑆))‘𝑡) = 𝑏))) |
| 80 | 78, 79 | bitri 275 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝑡 ∈ ((𝑠 × 𝑘) ∖ 𝑥) ∧ ((2nd ↾ (∪ 𝑅
× ∪ 𝑆))‘𝑡) = 𝑏) ↔ (𝑡 ∈ (𝑠 × 𝑘) ∧ (¬ 𝑡 ∈ 𝑥 ∧ ((2nd ↾ (∪ 𝑅
× ∪ 𝑆))‘𝑡) = 𝑏))) |
| 81 | 80 | rexbii2 3090 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢
(∃𝑡 ∈
((𝑠 × 𝑘) ∖ 𝑥)((2nd ↾ (∪ 𝑅
× ∪ 𝑆))‘𝑡) = 𝑏 ↔ ∃𝑡 ∈ (𝑠 × 𝑘)(¬ 𝑡 ∈ 𝑥 ∧ ((2nd ↾ (∪ 𝑅
× ∪ 𝑆))‘𝑡) = 𝑏)) |
| 82 | | ancom 460 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((¬
𝑡 ∈ 𝑥 ∧ ((2nd ↾ (∪ 𝑅
× ∪ 𝑆))‘𝑡) = 𝑏) ↔ (((2nd ↾ (∪ 𝑅
× ∪ 𝑆))‘𝑡) = 𝑏 ∧ ¬ 𝑡 ∈ 𝑥)) |
| 83 | | fveqeq2 6915 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (𝑡 = 〈𝑎, 𝑢〉 → (((2nd ↾
(∪ 𝑅 × ∪ 𝑆))‘𝑡) = 𝑏 ↔ ((2nd ↾ (∪ 𝑅
× ∪ 𝑆))‘〈𝑎, 𝑢〉) = 𝑏)) |
| 84 | | eleq1 2829 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ (𝑡 = 〈𝑎, 𝑢〉 → (𝑡 ∈ 𝑥 ↔ 〈𝑎, 𝑢〉 ∈ 𝑥)) |
| 85 | 84 | notbid 318 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (𝑡 = 〈𝑎, 𝑢〉 → (¬ 𝑡 ∈ 𝑥 ↔ ¬ 〈𝑎, 𝑢〉 ∈ 𝑥)) |
| 86 | 83, 85 | anbi12d 632 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝑡 = 〈𝑎, 𝑢〉 → ((((2nd ↾
(∪ 𝑅 × ∪ 𝑆))‘𝑡) = 𝑏 ∧ ¬ 𝑡 ∈ 𝑥) ↔ (((2nd ↾ (∪ 𝑅
× ∪ 𝑆))‘〈𝑎, 𝑢〉) = 𝑏 ∧ ¬ 〈𝑎, 𝑢〉 ∈ 𝑥))) |
| 87 | 82, 86 | bitrid 283 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑡 = 〈𝑎, 𝑢〉 → ((¬ 𝑡 ∈ 𝑥 ∧ ((2nd ↾ (∪ 𝑅
× ∪ 𝑆))‘𝑡) = 𝑏) ↔ (((2nd ↾ (∪ 𝑅
× ∪ 𝑆))‘〈𝑎, 𝑢〉) = 𝑏 ∧ ¬ 〈𝑎, 𝑢〉 ∈ 𝑥))) |
| 88 | 87 | rexxp 5853 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢
(∃𝑡 ∈
(𝑠 × 𝑘)(¬ 𝑡 ∈ 𝑥 ∧ ((2nd ↾ (∪ 𝑅
× ∪ 𝑆))‘𝑡) = 𝑏) ↔ ∃𝑎 ∈ 𝑠 ∃𝑢 ∈ 𝑘 (((2nd ↾ (∪ 𝑅
× ∪ 𝑆))‘〈𝑎, 𝑢〉) = 𝑏 ∧ ¬ 〈𝑎, 𝑢〉 ∈ 𝑥)) |
| 89 | 81, 88 | bitri 275 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢
(∃𝑡 ∈
((𝑠 × 𝑘) ∖ 𝑥)((2nd ↾ (∪ 𝑅
× ∪ 𝑆))‘𝑡) = 𝑏 ↔ ∃𝑎 ∈ 𝑠 ∃𝑢 ∈ 𝑘 (((2nd ↾ (∪ 𝑅
× ∪ 𝑆))‘〈𝑎, 𝑢〉) = 𝑏 ∧ ¬ 〈𝑎, 𝑢〉 ∈ 𝑥)) |
| 90 | | simpl 482 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
34
⊢ ((𝑠 ⊆ ∪ 𝑅
∧ 𝑘 ⊆ ∪ 𝑆)
→ 𝑠 ⊆ ∪ 𝑅) |
| 91 | 90 | sselda 3983 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
33
⊢ (((𝑠 ⊆ ∪ 𝑅
∧ 𝑘 ⊆ ∪ 𝑆)
∧ 𝑎 ∈ 𝑠) → 𝑎 ∈ ∪ 𝑅) |
| 92 | 91 | adantr 480 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
⊢ ((((𝑠 ⊆ ∪ 𝑅
∧ 𝑘 ⊆ ∪ 𝑆)
∧ 𝑎 ∈ 𝑠) ∧ 𝑢 ∈ 𝑘) → 𝑎 ∈ ∪ 𝑅) |
| 93 | | simplr 769 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
33
⊢ (((𝑠 ⊆ ∪ 𝑅
∧ 𝑘 ⊆ ∪ 𝑆)
∧ 𝑎 ∈ 𝑠) → 𝑘 ⊆ ∪ 𝑆) |
| 94 | 93 | sselda 3983 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
⊢ ((((𝑠 ⊆ ∪ 𝑅
∧ 𝑘 ⊆ ∪ 𝑆)
∧ 𝑎 ∈ 𝑠) ∧ 𝑢 ∈ 𝑘) → 𝑢 ∈ ∪ 𝑆) |
| 95 | 92, 94 | opelxpd 5724 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
⊢ ((((𝑠 ⊆ ∪ 𝑅
∧ 𝑘 ⊆ ∪ 𝑆)
∧ 𝑎 ∈ 𝑠) ∧ 𝑢 ∈ 𝑘) → 〈𝑎, 𝑢〉 ∈ (∪
𝑅 × ∪ 𝑆)) |
| 96 | 95 | fvresd 6926 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢ ((((𝑠 ⊆ ∪ 𝑅
∧ 𝑘 ⊆ ∪ 𝑆)
∧ 𝑎 ∈ 𝑠) ∧ 𝑢 ∈ 𝑘) → ((2nd ↾ (∪ 𝑅
× ∪ 𝑆))‘〈𝑎, 𝑢〉) = (2nd ‘〈𝑎, 𝑢〉)) |
| 97 | | vex 3484 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
⊢ 𝑎 ∈ V |
| 98 | | vex 3484 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
⊢ 𝑢 ∈ V |
| 99 | 97, 98 | op2nd 8023 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢
(2nd ‘〈𝑎, 𝑢〉) = 𝑢 |
| 100 | 96, 99 | eqtrdi 2793 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ ((((𝑠 ⊆ ∪ 𝑅
∧ 𝑘 ⊆ ∪ 𝑆)
∧ 𝑎 ∈ 𝑠) ∧ 𝑢 ∈ 𝑘) → ((2nd ↾ (∪ 𝑅
× ∪ 𝑆))‘〈𝑎, 𝑢〉) = 𝑢) |
| 101 | 100 | eqeq1d 2739 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ ((((𝑠 ⊆ ∪ 𝑅
∧ 𝑘 ⊆ ∪ 𝑆)
∧ 𝑎 ∈ 𝑠) ∧ 𝑢 ∈ 𝑘) → (((2nd ↾ (∪ 𝑅
× ∪ 𝑆))‘〈𝑎, 𝑢〉) = 𝑏 ↔ 𝑢 = 𝑏)) |
| 102 | 101 | anbi1d 631 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ ((((𝑠 ⊆ ∪ 𝑅
∧ 𝑘 ⊆ ∪ 𝑆)
∧ 𝑎 ∈ 𝑠) ∧ 𝑢 ∈ 𝑘) → ((((2nd ↾ (∪ 𝑅
× ∪ 𝑆))‘〈𝑎, 𝑢〉) = 𝑏 ∧ ¬ 〈𝑎, 𝑢〉 ∈ 𝑥) ↔ (𝑢 = 𝑏 ∧ ¬ 〈𝑎, 𝑢〉 ∈ 𝑥))) |
| 103 | 102 | rexbidva 3177 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (((𝑠 ⊆ ∪ 𝑅
∧ 𝑘 ⊆ ∪ 𝑆)
∧ 𝑎 ∈ 𝑠) → (∃𝑢 ∈ 𝑘 (((2nd ↾ (∪ 𝑅
× ∪ 𝑆))‘〈𝑎, 𝑢〉) = 𝑏 ∧ ¬ 〈𝑎, 𝑢〉 ∈ 𝑥) ↔ ∃𝑢 ∈ 𝑘 (𝑢 = 𝑏 ∧ ¬ 〈𝑎, 𝑢〉 ∈ 𝑥))) |
| 104 | | opeq2 4874 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ (𝑢 = 𝑏 → 〈𝑎, 𝑢〉 = 〈𝑎, 𝑏〉) |
| 105 | 104 | eleq1d 2826 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ (𝑢 = 𝑏 → (〈𝑎, 𝑢〉 ∈ 𝑥 ↔ 〈𝑎, 𝑏〉 ∈ 𝑥)) |
| 106 | 105 | notbid 318 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (𝑢 = 𝑏 → (¬ 〈𝑎, 𝑢〉 ∈ 𝑥 ↔ ¬ 〈𝑎, 𝑏〉 ∈ 𝑥)) |
| 107 | 106 | ceqsrexbv 3656 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢
(∃𝑢 ∈
𝑘 (𝑢 = 𝑏 ∧ ¬ 〈𝑎, 𝑢〉 ∈ 𝑥) ↔ (𝑏 ∈ 𝑘 ∧ ¬ 〈𝑎, 𝑏〉 ∈ 𝑥)) |
| 108 | 103, 107 | bitrdi 287 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (((𝑠 ⊆ ∪ 𝑅
∧ 𝑘 ⊆ ∪ 𝑆)
∧ 𝑎 ∈ 𝑠) → (∃𝑢 ∈ 𝑘 (((2nd ↾ (∪ 𝑅
× ∪ 𝑆))‘〈𝑎, 𝑢〉) = 𝑏 ∧ ¬ 〈𝑎, 𝑢〉 ∈ 𝑥) ↔ (𝑏 ∈ 𝑘 ∧ ¬ 〈𝑎, 𝑏〉 ∈ 𝑥))) |
| 109 | 108 | rexbidva 3177 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝑠 ⊆ ∪ 𝑅
∧ 𝑘 ⊆ ∪ 𝑆)
→ (∃𝑎 ∈
𝑠 ∃𝑢 ∈ 𝑘 (((2nd ↾ (∪ 𝑅
× ∪ 𝑆))‘〈𝑎, 𝑢〉) = 𝑏 ∧ ¬ 〈𝑎, 𝑢〉 ∈ 𝑥) ↔ ∃𝑎 ∈ 𝑠 (𝑏 ∈ 𝑘 ∧ ¬ 〈𝑎, 𝑏〉 ∈ 𝑥))) |
| 110 | | r19.42v 3191 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢
(∃𝑎 ∈
𝑠 (𝑏 ∈ 𝑘 ∧ ¬ 〈𝑎, 𝑏〉 ∈ 𝑥) ↔ (𝑏 ∈ 𝑘 ∧ ∃𝑎 ∈ 𝑠 ¬ 〈𝑎, 𝑏〉 ∈ 𝑥)) |
| 111 | 109, 110 | bitrdi 287 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝑠 ⊆ ∪ 𝑅
∧ 𝑘 ⊆ ∪ 𝑆)
→ (∃𝑎 ∈
𝑠 ∃𝑢 ∈ 𝑘 (((2nd ↾ (∪ 𝑅
× ∪ 𝑆))‘〈𝑎, 𝑢〉) = 𝑏 ∧ ¬ 〈𝑎, 𝑢〉 ∈ 𝑥) ↔ (𝑏 ∈ 𝑘 ∧ ∃𝑎 ∈ 𝑠 ¬ 〈𝑎, 𝑏〉 ∈ 𝑥))) |
| 112 | 89, 111 | bitrid 283 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝑠 ⊆ ∪ 𝑅
∧ 𝑘 ⊆ ∪ 𝑆)
→ (∃𝑡 ∈
((𝑠 × 𝑘) ∖ 𝑥)((2nd ↾ (∪ 𝑅
× ∪ 𝑆))‘𝑡) = 𝑏 ↔ (𝑏 ∈ 𝑘 ∧ ∃𝑎 ∈ 𝑠 ¬ 〈𝑎, 𝑏〉 ∈ 𝑥))) |
| 113 | | f2ndres 8039 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢
(2nd ↾ (∪ 𝑅 × ∪ 𝑆)):(∪
𝑅 × ∪ 𝑆)⟶∪ 𝑆 |
| 114 | | ffn 6736 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢
((2nd ↾ (∪ 𝑅 × ∪ 𝑆)):(∪
𝑅 × ∪ 𝑆)⟶∪ 𝑆 → (2nd ↾
(∪ 𝑅 × ∪ 𝑆)) Fn (∪ 𝑅
× ∪ 𝑆)) |
| 115 | 113, 114 | ax-mp 5 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢
(2nd ↾ (∪ 𝑅 × ∪ 𝑆)) Fn (∪ 𝑅
× ∪ 𝑆) |
| 116 | | difss 4136 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝑠 × 𝑘) ∖ 𝑥) ⊆ (𝑠 × 𝑘) |
| 117 | | xpss12 5700 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝑠 ⊆ ∪ 𝑅
∧ 𝑘 ⊆ ∪ 𝑆)
→ (𝑠 × 𝑘) ⊆ (∪ 𝑅
× ∪ 𝑆)) |
| 118 | 116, 117 | sstrid 3995 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝑠 ⊆ ∪ 𝑅
∧ 𝑘 ⊆ ∪ 𝑆)
→ ((𝑠 × 𝑘) ∖ 𝑥) ⊆ (∪ 𝑅 × ∪ 𝑆)) |
| 119 | | fvelimab 6981 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢
(((2nd ↾ (∪ 𝑅 × ∪ 𝑆))
Fn (∪ 𝑅 × ∪ 𝑆) ∧ ((𝑠 × 𝑘) ∖ 𝑥) ⊆ (∪ 𝑅 × ∪ 𝑆))
→ (𝑏 ∈
((2nd ↾ (∪ 𝑅 × ∪ 𝑆)) “ ((𝑠 × 𝑘) ∖ 𝑥)) ↔ ∃𝑡 ∈ ((𝑠 × 𝑘) ∖ 𝑥)((2nd ↾ (∪ 𝑅
× ∪ 𝑆))‘𝑡) = 𝑏)) |
| 120 | 115, 118,
119 | sylancr 587 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝑠 ⊆ ∪ 𝑅
∧ 𝑘 ⊆ ∪ 𝑆)
→ (𝑏 ∈
((2nd ↾ (∪ 𝑅 × ∪ 𝑆)) “ ((𝑠 × 𝑘) ∖ 𝑥)) ↔ ∃𝑡 ∈ ((𝑠 × 𝑘) ∖ 𝑥)((2nd ↾ (∪ 𝑅
× ∪ 𝑆))‘𝑡) = 𝑏)) |
| 121 | | eldif 3961 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑏 ∈ (𝑘 ∖ {𝑣 ∈ ∪ 𝑆 ∣ (𝑠 × {𝑣}) ⊆ 𝑥}) ↔ (𝑏 ∈ 𝑘 ∧ ¬ 𝑏 ∈ {𝑣 ∈ ∪ 𝑆 ∣ (𝑠 × {𝑣}) ⊆ 𝑥})) |
| 122 | | simpr 484 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ ((𝑠 ⊆ ∪ 𝑅
∧ 𝑘 ⊆ ∪ 𝑆)
→ 𝑘 ⊆ ∪ 𝑆) |
| 123 | 122 | sselda 3983 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (((𝑠 ⊆ ∪ 𝑅
∧ 𝑘 ⊆ ∪ 𝑆)
∧ 𝑏 ∈ 𝑘) → 𝑏 ∈ ∪ 𝑆) |
| 124 | | sneq 4636 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
⊢ (𝑣 = 𝑏 → {𝑣} = {𝑏}) |
| 125 | 124 | xpeq2d 5715 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢ (𝑣 = 𝑏 → (𝑠 × {𝑣}) = (𝑠 × {𝑏})) |
| 126 | 125 | sseq1d 4015 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ (𝑣 = 𝑏 → ((𝑠 × {𝑣}) ⊆ 𝑥 ↔ (𝑠 × {𝑏}) ⊆ 𝑥)) |
| 127 | | dfss3 3972 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢ ((𝑠 × {𝑏}) ⊆ 𝑥 ↔ ∀𝑘 ∈ (𝑠 × {𝑏})𝑘 ∈ 𝑥) |
| 128 | | eleq1 2829 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
⊢ (𝑘 = 〈𝑎, 𝑡〉 → (𝑘 ∈ 𝑥 ↔ 〈𝑎, 𝑡〉 ∈ 𝑥)) |
| 129 | 128 | ralxp 5852 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢
(∀𝑘 ∈
(𝑠 × {𝑏})𝑘 ∈ 𝑥 ↔ ∀𝑎 ∈ 𝑠 ∀𝑡 ∈ {𝑏}〈𝑎, 𝑡〉 ∈ 𝑥) |
| 130 | | vex 3484 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
⊢ 𝑏 ∈ V |
| 131 | | opeq2 4874 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
33
⊢ (𝑡 = 𝑏 → 〈𝑎, 𝑡〉 = 〈𝑎, 𝑏〉) |
| 132 | 131 | eleq1d 2826 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
⊢ (𝑡 = 𝑏 → (〈𝑎, 𝑡〉 ∈ 𝑥 ↔ 〈𝑎, 𝑏〉 ∈ 𝑥)) |
| 133 | 130, 132 | ralsn 4681 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
⊢
(∀𝑡 ∈
{𝑏}〈𝑎, 𝑡〉 ∈ 𝑥 ↔ 〈𝑎, 𝑏〉 ∈ 𝑥) |
| 134 | 133 | ralbii 3093 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢
(∀𝑎 ∈
𝑠 ∀𝑡 ∈ {𝑏}〈𝑎, 𝑡〉 ∈ 𝑥 ↔ ∀𝑎 ∈ 𝑠 〈𝑎, 𝑏〉 ∈ 𝑥) |
| 135 | 127, 129,
134 | 3bitri 297 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ ((𝑠 × {𝑏}) ⊆ 𝑥 ↔ ∀𝑎 ∈ 𝑠 〈𝑎, 𝑏〉 ∈ 𝑥) |
| 136 | 126, 135 | bitrdi 287 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ (𝑣 = 𝑏 → ((𝑠 × {𝑣}) ⊆ 𝑥 ↔ ∀𝑎 ∈ 𝑠 〈𝑎, 𝑏〉 ∈ 𝑥)) |
| 137 | 136 | elrab3 3693 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (𝑏 ∈ ∪ 𝑆
→ (𝑏 ∈ {𝑣 ∈ ∪ 𝑆
∣ (𝑠 × {𝑣}) ⊆ 𝑥} ↔ ∀𝑎 ∈ 𝑠 〈𝑎, 𝑏〉 ∈ 𝑥)) |
| 138 | 123, 137 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (((𝑠 ⊆ ∪ 𝑅
∧ 𝑘 ⊆ ∪ 𝑆)
∧ 𝑏 ∈ 𝑘) → (𝑏 ∈ {𝑣 ∈ ∪ 𝑆 ∣ (𝑠 × {𝑣}) ⊆ 𝑥} ↔ ∀𝑎 ∈ 𝑠 〈𝑎, 𝑏〉 ∈ 𝑥)) |
| 139 | 138 | notbid 318 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (((𝑠 ⊆ ∪ 𝑅
∧ 𝑘 ⊆ ∪ 𝑆)
∧ 𝑏 ∈ 𝑘) → (¬ 𝑏 ∈ {𝑣 ∈ ∪ 𝑆 ∣ (𝑠 × {𝑣}) ⊆ 𝑥} ↔ ¬ ∀𝑎 ∈ 𝑠 〈𝑎, 𝑏〉 ∈ 𝑥)) |
| 140 | | rexnal 3100 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢
(∃𝑎 ∈
𝑠 ¬ 〈𝑎, 𝑏〉 ∈ 𝑥 ↔ ¬ ∀𝑎 ∈ 𝑠 〈𝑎, 𝑏〉 ∈ 𝑥) |
| 141 | 139, 140 | bitr4di 289 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (((𝑠 ⊆ ∪ 𝑅
∧ 𝑘 ⊆ ∪ 𝑆)
∧ 𝑏 ∈ 𝑘) → (¬ 𝑏 ∈ {𝑣 ∈ ∪ 𝑆 ∣ (𝑠 × {𝑣}) ⊆ 𝑥} ↔ ∃𝑎 ∈ 𝑠 ¬ 〈𝑎, 𝑏〉 ∈ 𝑥)) |
| 142 | 141 | pm5.32da 579 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝑠 ⊆ ∪ 𝑅
∧ 𝑘 ⊆ ∪ 𝑆)
→ ((𝑏 ∈ 𝑘 ∧ ¬ 𝑏 ∈ {𝑣 ∈ ∪ 𝑆 ∣ (𝑠 × {𝑣}) ⊆ 𝑥}) ↔ (𝑏 ∈ 𝑘 ∧ ∃𝑎 ∈ 𝑠 ¬ 〈𝑎, 𝑏〉 ∈ 𝑥))) |
| 143 | 121, 142 | bitrid 283 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝑠 ⊆ ∪ 𝑅
∧ 𝑘 ⊆ ∪ 𝑆)
→ (𝑏 ∈ (𝑘 ∖ {𝑣 ∈ ∪ 𝑆 ∣ (𝑠 × {𝑣}) ⊆ 𝑥}) ↔ (𝑏 ∈ 𝑘 ∧ ∃𝑎 ∈ 𝑠 ¬ 〈𝑎, 𝑏〉 ∈ 𝑥))) |
| 144 | 112, 120,
143 | 3bitr4d 311 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝑠 ⊆ ∪ 𝑅
∧ 𝑘 ⊆ ∪ 𝑆)
→ (𝑏 ∈
((2nd ↾ (∪ 𝑅 × ∪ 𝑆)) “ ((𝑠 × 𝑘) ∖ 𝑥)) ↔ 𝑏 ∈ (𝑘 ∖ {𝑣 ∈ ∪ 𝑆 ∣ (𝑠 × {𝑣}) ⊆ 𝑥}))) |
| 145 | 144 | eqrdv 2735 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑠 ⊆ ∪ 𝑅
∧ 𝑘 ⊆ ∪ 𝑆)
→ ((2nd ↾ (∪ 𝑅 × ∪ 𝑆)) “ ((𝑠 × 𝑘) ∖ 𝑥)) = (𝑘 ∖ {𝑣 ∈ ∪ 𝑆 ∣ (𝑠 × {𝑣}) ⊆ 𝑥})) |
| 146 | 74, 76, 145 | syl2anc 584 |
. . . . . . . . . . . . . . . . . . 19
⊢
((((((𝑅 ∈
𝑛-Locally Comp ∧ 𝑆 ∈ (ran 𝑘Gen ∩ Haus)) ∧
𝑥 ∈
(𝑘Gen‘(𝑅
×t 𝑆)))
∧ 𝑦 ∈ 𝑥) ∧ ((𝑠 ∈ 𝒫 {𝑡 ∈ ∪ 𝑅 ∣ 〈𝑡, (2nd ‘𝑦)〉 ∈ 𝑥} ∧ 𝑢 ∈ 𝑅) ∧ ((1st ‘𝑦) ∈ 𝑢 ∧ 𝑢 ⊆ 𝑠 ∧ (𝑅 ↾t 𝑠) ∈ Comp))) ∧ (𝑘 ∈ 𝒫 ∪ 𝑆
∧ (𝑆
↾t 𝑘)
∈ Comp)) → ((2nd ↾ (∪
𝑅 × ∪ 𝑆))
“ ((𝑠 × 𝑘) ∖ 𝑥)) = (𝑘 ∖ {𝑣 ∈ ∪ 𝑆 ∣ (𝑠 × {𝑣}) ⊆ 𝑥})) |
| 147 | | difin 4272 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑘 ∖ (𝑘 ∩ {𝑣 ∈ ∪ 𝑆 ∣ (𝑠 × {𝑣}) ⊆ 𝑥})) = (𝑘 ∖ {𝑣 ∈ ∪ 𝑆 ∣ (𝑠 × {𝑣}) ⊆ 𝑥}) |
| 148 | 65 | adantr 480 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
((((((𝑅 ∈
𝑛-Locally Comp ∧ 𝑆 ∈ (ran 𝑘Gen ∩ Haus)) ∧
𝑥 ∈
(𝑘Gen‘(𝑅
×t 𝑆)))
∧ 𝑦 ∈ 𝑥) ∧ ((𝑠 ∈ 𝒫 {𝑡 ∈ ∪ 𝑅 ∣ 〈𝑡, (2nd ‘𝑦)〉 ∈ 𝑥} ∧ 𝑢 ∈ 𝑅) ∧ ((1st ‘𝑦) ∈ 𝑢 ∧ 𝑢 ⊆ 𝑠 ∧ (𝑅 ↾t 𝑠) ∈ Comp))) ∧ (𝑘 ∈ 𝒫 ∪ 𝑆
∧ (𝑆
↾t 𝑘)
∈ Comp)) → 𝑆
∈ Top) |
| 149 | 24 | restuni 23170 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝑆 ∈ Top ∧ 𝑘 ⊆ ∪ 𝑆)
→ 𝑘 = ∪ (𝑆
↾t 𝑘)) |
| 150 | 148, 76, 149 | syl2anc 584 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
((((((𝑅 ∈
𝑛-Locally Comp ∧ 𝑆 ∈ (ran 𝑘Gen ∩ Haus)) ∧
𝑥 ∈
(𝑘Gen‘(𝑅
×t 𝑆)))
∧ 𝑦 ∈ 𝑥) ∧ ((𝑠 ∈ 𝒫 {𝑡 ∈ ∪ 𝑅 ∣ 〈𝑡, (2nd ‘𝑦)〉 ∈ 𝑥} ∧ 𝑢 ∈ 𝑅) ∧ ((1st ‘𝑦) ∈ 𝑢 ∧ 𝑢 ⊆ 𝑠 ∧ (𝑅 ↾t 𝑠) ∈ Comp))) ∧ (𝑘 ∈ 𝒫 ∪ 𝑆
∧ (𝑆
↾t 𝑘)
∈ Comp)) → 𝑘 =
∪ (𝑆 ↾t 𝑘)) |
| 151 | 150 | difeq1d 4125 |
. . . . . . . . . . . . . . . . . . . 20
⊢
((((((𝑅 ∈
𝑛-Locally Comp ∧ 𝑆 ∈ (ran 𝑘Gen ∩ Haus)) ∧
𝑥 ∈
(𝑘Gen‘(𝑅
×t 𝑆)))
∧ 𝑦 ∈ 𝑥) ∧ ((𝑠 ∈ 𝒫 {𝑡 ∈ ∪ 𝑅 ∣ 〈𝑡, (2nd ‘𝑦)〉 ∈ 𝑥} ∧ 𝑢 ∈ 𝑅) ∧ ((1st ‘𝑦) ∈ 𝑢 ∧ 𝑢 ⊆ 𝑠 ∧ (𝑅 ↾t 𝑠) ∈ Comp))) ∧ (𝑘 ∈ 𝒫 ∪ 𝑆
∧ (𝑆
↾t 𝑘)
∈ Comp)) → (𝑘
∖ (𝑘 ∩ {𝑣 ∈ ∪ 𝑆
∣ (𝑠 × {𝑣}) ⊆ 𝑥})) = (∪ (𝑆 ↾t 𝑘) ∖ (𝑘 ∩ {𝑣 ∈ ∪ 𝑆 ∣ (𝑠 × {𝑣}) ⊆ 𝑥}))) |
| 152 | 147, 151 | eqtr3id 2791 |
. . . . . . . . . . . . . . . . . . 19
⊢
((((((𝑅 ∈
𝑛-Locally Comp ∧ 𝑆 ∈ (ran 𝑘Gen ∩ Haus)) ∧
𝑥 ∈
(𝑘Gen‘(𝑅
×t 𝑆)))
∧ 𝑦 ∈ 𝑥) ∧ ((𝑠 ∈ 𝒫 {𝑡 ∈ ∪ 𝑅 ∣ 〈𝑡, (2nd ‘𝑦)〉 ∈ 𝑥} ∧ 𝑢 ∈ 𝑅) ∧ ((1st ‘𝑦) ∈ 𝑢 ∧ 𝑢 ⊆ 𝑠 ∧ (𝑅 ↾t 𝑠) ∈ Comp))) ∧ (𝑘 ∈ 𝒫 ∪ 𝑆
∧ (𝑆
↾t 𝑘)
∈ Comp)) → (𝑘
∖ {𝑣 ∈ ∪ 𝑆
∣ (𝑠 × {𝑣}) ⊆ 𝑥}) = (∪ (𝑆 ↾t 𝑘) ∖ (𝑘 ∩ {𝑣 ∈ ∪ 𝑆 ∣ (𝑠 × {𝑣}) ⊆ 𝑥}))) |
| 153 | 146, 152 | eqtrd 2777 |
. . . . . . . . . . . . . . . . . 18
⊢
((((((𝑅 ∈
𝑛-Locally Comp ∧ 𝑆 ∈ (ran 𝑘Gen ∩ Haus)) ∧
𝑥 ∈
(𝑘Gen‘(𝑅
×t 𝑆)))
∧ 𝑦 ∈ 𝑥) ∧ ((𝑠 ∈ 𝒫 {𝑡 ∈ ∪ 𝑅 ∣ 〈𝑡, (2nd ‘𝑦)〉 ∈ 𝑥} ∧ 𝑢 ∈ 𝑅) ∧ ((1st ‘𝑦) ∈ 𝑢 ∧ 𝑢 ⊆ 𝑠 ∧ (𝑅 ↾t 𝑠) ∈ Comp))) ∧ (𝑘 ∈ 𝒫 ∪ 𝑆
∧ (𝑆
↾t 𝑘)
∈ Comp)) → ((2nd ↾ (∪
𝑅 × ∪ 𝑆))
“ ((𝑠 × 𝑘) ∖ 𝑥)) = (∪ (𝑆 ↾t 𝑘) ∖ (𝑘 ∩ {𝑣 ∈ ∪ 𝑆 ∣ (𝑠 × {𝑣}) ⊆ 𝑥}))) |
| 154 | 15 | ad2antrr 726 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
((((((𝑅 ∈
𝑛-Locally Comp ∧ 𝑆 ∈ (ran 𝑘Gen ∩ Haus)) ∧
𝑥 ∈
(𝑘Gen‘(𝑅
×t 𝑆)))
∧ 𝑦 ∈ 𝑥) ∧ ((𝑠 ∈ 𝒫 {𝑡 ∈ ∪ 𝑅 ∣ 〈𝑡, (2nd ‘𝑦)〉 ∈ 𝑥} ∧ 𝑢 ∈ 𝑅) ∧ ((1st ‘𝑦) ∈ 𝑢 ∧ 𝑢 ⊆ 𝑠 ∧ (𝑅 ↾t 𝑠) ∈ Comp))) ∧ (𝑘 ∈ 𝒫 ∪ 𝑆
∧ (𝑆
↾t 𝑘)
∈ Comp)) → 𝑆
∈ (ran 𝑘Gen ∩ Haus)) |
| 155 | 154 | elin2d 4205 |
. . . . . . . . . . . . . . . . . . . 20
⊢
((((((𝑅 ∈
𝑛-Locally Comp ∧ 𝑆 ∈ (ran 𝑘Gen ∩ Haus)) ∧
𝑥 ∈
(𝑘Gen‘(𝑅
×t 𝑆)))
∧ 𝑦 ∈ 𝑥) ∧ ((𝑠 ∈ 𝒫 {𝑡 ∈ ∪ 𝑅 ∣ 〈𝑡, (2nd ‘𝑦)〉 ∈ 𝑥} ∧ 𝑢 ∈ 𝑅) ∧ ((1st ‘𝑦) ∈ 𝑢 ∧ 𝑢 ⊆ 𝑠 ∧ (𝑅 ↾t 𝑠) ∈ Comp))) ∧ (𝑘 ∈ 𝒫 ∪ 𝑆
∧ (𝑆
↾t 𝑘)
∈ Comp)) → 𝑆
∈ Haus) |
| 156 | | df-ima 5698 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢
((2nd ↾ (∪ 𝑅 × ∪ 𝑆)) “ ((𝑠 × 𝑘) ∖ 𝑥)) = ran ((2nd ↾ (∪ 𝑅
× ∪ 𝑆)) ↾ ((𝑠 × 𝑘) ∖ 𝑥)) |
| 157 | | resres 6010 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢
((2nd ↾ (∪ 𝑅 × ∪ 𝑆)) ↾ ((𝑠 × 𝑘) ∖ 𝑥)) = (2nd ↾ ((∪ 𝑅
× ∪ 𝑆) ∩ ((𝑠 × 𝑘) ∖ 𝑥))) |
| 158 | | inss2 4238 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ ((∪ 𝑅
× ∪ 𝑆) ∩ ((𝑠 × 𝑘) ∖ 𝑥)) ⊆ ((𝑠 × 𝑘) ∖ 𝑥) |
| 159 | 158, 116 | sstri 3993 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((∪ 𝑅
× ∪ 𝑆) ∩ ((𝑠 × 𝑘) ∖ 𝑥)) ⊆ (𝑠 × 𝑘) |
| 160 | | ssres2 6022 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (((∪ 𝑅
× ∪ 𝑆) ∩ ((𝑠 × 𝑘) ∖ 𝑥)) ⊆ (𝑠 × 𝑘) → (2nd ↾ ((∪ 𝑅
× ∪ 𝑆) ∩ ((𝑠 × 𝑘) ∖ 𝑥))) ⊆ (2nd ↾ (𝑠 × 𝑘))) |
| 161 | 159, 160 | ax-mp 5 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢
(2nd ↾ ((∪ 𝑅 × ∪ 𝑆) ∩ ((𝑠 × 𝑘) ∖ 𝑥))) ⊆ (2nd ↾ (𝑠 × 𝑘)) |
| 162 | 157, 161 | eqsstri 4030 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢
((2nd ↾ (∪ 𝑅 × ∪ 𝑆)) ↾ ((𝑠 × 𝑘) ∖ 𝑥)) ⊆ (2nd ↾ (𝑠 × 𝑘)) |
| 163 | 162 | rnssi 5951 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ran
((2nd ↾ (∪ 𝑅 × ∪ 𝑆)) ↾ ((𝑠 × 𝑘) ∖ 𝑥)) ⊆ ran (2nd ↾ (𝑠 × 𝑘)) |
| 164 | 156, 163 | eqsstri 4030 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
((2nd ↾ (∪ 𝑅 × ∪ 𝑆)) “ ((𝑠 × 𝑘) ∖ 𝑥)) ⊆ ran (2nd ↾ (𝑠 × 𝑘)) |
| 165 | | f2ndres 8039 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢
(2nd ↾ (𝑠 × 𝑘)):(𝑠 × 𝑘)⟶𝑘 |
| 166 | | frn 6743 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢
((2nd ↾ (𝑠 × 𝑘)):(𝑠 × 𝑘)⟶𝑘 → ran (2nd ↾ (𝑠 × 𝑘)) ⊆ 𝑘) |
| 167 | 165, 166 | ax-mp 5 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ran
(2nd ↾ (𝑠
× 𝑘)) ⊆ 𝑘 |
| 168 | 164, 167 | sstri 3993 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
((2nd ↾ (∪ 𝑅 × ∪ 𝑆)) “ ((𝑠 × 𝑘) ∖ 𝑥)) ⊆ 𝑘 |
| 169 | 168, 76 | sstrid 3995 |
. . . . . . . . . . . . . . . . . . . 20
⊢
((((((𝑅 ∈
𝑛-Locally Comp ∧ 𝑆 ∈ (ran 𝑘Gen ∩ Haus)) ∧
𝑥 ∈
(𝑘Gen‘(𝑅
×t 𝑆)))
∧ 𝑦 ∈ 𝑥) ∧ ((𝑠 ∈ 𝒫 {𝑡 ∈ ∪ 𝑅 ∣ 〈𝑡, (2nd ‘𝑦)〉 ∈ 𝑥} ∧ 𝑢 ∈ 𝑅) ∧ ((1st ‘𝑦) ∈ 𝑢 ∧ 𝑢 ⊆ 𝑠 ∧ (𝑅 ↾t 𝑠) ∈ Comp))) ∧ (𝑘 ∈ 𝒫 ∪ 𝑆
∧ (𝑆
↾t 𝑘)
∈ Comp)) → ((2nd ↾ (∪
𝑅 × ∪ 𝑆))
“ ((𝑠 × 𝑘) ∖ 𝑥)) ⊆ ∪ 𝑆) |
| 170 | 12 | ad2antrr 726 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
((((((𝑅 ∈
𝑛-Locally Comp ∧ 𝑆 ∈ (ran 𝑘Gen ∩ Haus)) ∧
𝑥 ∈
(𝑘Gen‘(𝑅
×t 𝑆)))
∧ 𝑦 ∈ 𝑥) ∧ ((𝑠 ∈ 𝒫 {𝑡 ∈ ∪ 𝑅 ∣ 〈𝑡, (2nd ‘𝑦)〉 ∈ 𝑥} ∧ 𝑢 ∈ 𝑅) ∧ ((1st ‘𝑦) ∈ 𝑢 ∧ 𝑢 ⊆ 𝑠 ∧ (𝑅 ↾t 𝑠) ∈ Comp))) ∧ (𝑘 ∈ 𝒫 ∪ 𝑆
∧ (𝑆
↾t 𝑘)
∈ Comp)) → 𝑅
∈ (TopOn‘∪ 𝑅)) |
| 171 | 148, 17 | sylib 218 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
((((((𝑅 ∈
𝑛-Locally Comp ∧ 𝑆 ∈ (ran 𝑘Gen ∩ Haus)) ∧
𝑥 ∈
(𝑘Gen‘(𝑅
×t 𝑆)))
∧ 𝑦 ∈ 𝑥) ∧ ((𝑠 ∈ 𝒫 {𝑡 ∈ ∪ 𝑅 ∣ 〈𝑡, (2nd ‘𝑦)〉 ∈ 𝑥} ∧ 𝑢 ∈ 𝑅) ∧ ((1st ‘𝑦) ∈ 𝑢 ∧ 𝑢 ⊆ 𝑠 ∧ (𝑅 ↾t 𝑠) ∈ Comp))) ∧ (𝑘 ∈ 𝒫 ∪ 𝑆
∧ (𝑆
↾t 𝑘)
∈ Comp)) → 𝑆
∈ (TopOn‘∪ 𝑆)) |
| 172 | | tx2cn 23618 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝑅 ∈ (TopOn‘∪ 𝑅)
∧ 𝑆 ∈
(TopOn‘∪ 𝑆)) → (2nd ↾ (∪ 𝑅
× ∪ 𝑆)) ∈ ((𝑅 ×t 𝑆) Cn 𝑆)) |
| 173 | 170, 171,
172 | syl2anc 584 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
((((((𝑅 ∈
𝑛-Locally Comp ∧ 𝑆 ∈ (ran 𝑘Gen ∩ Haus)) ∧
𝑥 ∈
(𝑘Gen‘(𝑅
×t 𝑆)))
∧ 𝑦 ∈ 𝑥) ∧ ((𝑠 ∈ 𝒫 {𝑡 ∈ ∪ 𝑅 ∣ 〈𝑡, (2nd ‘𝑦)〉 ∈ 𝑥} ∧ 𝑢 ∈ 𝑅) ∧ ((1st ‘𝑦) ∈ 𝑢 ∧ 𝑢 ⊆ 𝑠 ∧ (𝑅 ↾t 𝑠) ∈ Comp))) ∧ (𝑘 ∈ 𝒫 ∪ 𝑆
∧ (𝑆
↾t 𝑘)
∈ Comp)) → (2nd ↾ (∪
𝑅 × ∪ 𝑆))
∈ ((𝑅
×t 𝑆) Cn
𝑆)) |
| 174 | 27 | ad2antrr 726 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢
((((((𝑅 ∈
𝑛-Locally Comp ∧ 𝑆 ∈ (ran 𝑘Gen ∩ Haus)) ∧
𝑥 ∈
(𝑘Gen‘(𝑅
×t 𝑆)))
∧ 𝑦 ∈ 𝑥) ∧ ((𝑠 ∈ 𝒫 {𝑡 ∈ ∪ 𝑅 ∣ 〈𝑡, (2nd ‘𝑦)〉 ∈ 𝑥} ∧ 𝑢 ∈ 𝑅) ∧ ((1st ‘𝑦) ∈ 𝑢 ∧ 𝑢 ⊆ 𝑠 ∧ (𝑅 ↾t 𝑠) ∈ Comp))) ∧ (𝑘 ∈ 𝒫 ∪ 𝑆
∧ (𝑆
↾t 𝑘)
∈ Comp)) → (𝑅
×t 𝑆)
∈ Top) |
| 175 | 116 | a1i 11 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢
((((((𝑅 ∈
𝑛-Locally Comp ∧ 𝑆 ∈ (ran 𝑘Gen ∩ Haus)) ∧
𝑥 ∈
(𝑘Gen‘(𝑅
×t 𝑆)))
∧ 𝑦 ∈ 𝑥) ∧ ((𝑠 ∈ 𝒫 {𝑡 ∈ ∪ 𝑅 ∣ 〈𝑡, (2nd ‘𝑦)〉 ∈ 𝑥} ∧ 𝑢 ∈ 𝑅) ∧ ((1st ‘𝑦) ∈ 𝑢 ∧ 𝑢 ⊆ 𝑠 ∧ (𝑅 ↾t 𝑠) ∈ Comp))) ∧ (𝑘 ∈ 𝒫 ∪ 𝑆
∧ (𝑆
↾t 𝑘)
∈ Comp)) → ((𝑠
× 𝑘) ∖ 𝑥) ⊆ (𝑠 × 𝑘)) |
| 176 | | vex 3484 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ 𝑠 ∈ V |
| 177 | | vex 3484 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ 𝑘 ∈ V |
| 178 | 176, 177 | xpex 7773 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑠 × 𝑘) ∈ V |
| 179 | 178 | a1i 11 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢
((((((𝑅 ∈
𝑛-Locally Comp ∧ 𝑆 ∈ (ran 𝑘Gen ∩ Haus)) ∧
𝑥 ∈
(𝑘Gen‘(𝑅
×t 𝑆)))
∧ 𝑦 ∈ 𝑥) ∧ ((𝑠 ∈ 𝒫 {𝑡 ∈ ∪ 𝑅 ∣ 〈𝑡, (2nd ‘𝑦)〉 ∈ 𝑥} ∧ 𝑢 ∈ 𝑅) ∧ ((1st ‘𝑦) ∈ 𝑢 ∧ 𝑢 ⊆ 𝑠 ∧ (𝑅 ↾t 𝑠) ∈ Comp))) ∧ (𝑘 ∈ 𝒫 ∪ 𝑆
∧ (𝑆
↾t 𝑘)
∈ Comp)) → (𝑠
× 𝑘) ∈
V) |
| 180 | | restabs 23173 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (((𝑅 ×t 𝑆) ∈ Top ∧ ((𝑠 × 𝑘) ∖ 𝑥) ⊆ (𝑠 × 𝑘) ∧ (𝑠 × 𝑘) ∈ V) → (((𝑅 ×t 𝑆) ↾t (𝑠 × 𝑘)) ↾t ((𝑠 × 𝑘) ∖ 𝑥)) = ((𝑅 ×t 𝑆) ↾t ((𝑠 × 𝑘) ∖ 𝑥))) |
| 181 | 174, 175,
179, 180 | syl3anc 1373 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
((((((𝑅 ∈
𝑛-Locally Comp ∧ 𝑆 ∈ (ran 𝑘Gen ∩ Haus)) ∧
𝑥 ∈
(𝑘Gen‘(𝑅
×t 𝑆)))
∧ 𝑦 ∈ 𝑥) ∧ ((𝑠 ∈ 𝒫 {𝑡 ∈ ∪ 𝑅 ∣ 〈𝑡, (2nd ‘𝑦)〉 ∈ 𝑥} ∧ 𝑢 ∈ 𝑅) ∧ ((1st ‘𝑦) ∈ 𝑢 ∧ 𝑢 ⊆ 𝑠 ∧ (𝑅 ↾t 𝑠) ∈ Comp))) ∧ (𝑘 ∈ 𝒫 ∪ 𝑆
∧ (𝑆
↾t 𝑘)
∈ Comp)) → (((𝑅
×t 𝑆)
↾t (𝑠
× 𝑘))
↾t ((𝑠
× 𝑘) ∖ 𝑥)) = ((𝑅 ×t 𝑆) ↾t ((𝑠 × 𝑘) ∖ 𝑥))) |
| 182 | 64 | adantr 480 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢
((((((𝑅 ∈
𝑛-Locally Comp ∧ 𝑆 ∈ (ran 𝑘Gen ∩ Haus)) ∧
𝑥 ∈
(𝑘Gen‘(𝑅
×t 𝑆)))
∧ 𝑦 ∈ 𝑥) ∧ ((𝑠 ∈ 𝒫 {𝑡 ∈ ∪ 𝑅 ∣ 〈𝑡, (2nd ‘𝑦)〉 ∈ 𝑥} ∧ 𝑢 ∈ 𝑅) ∧ ((1st ‘𝑦) ∈ 𝑢 ∧ 𝑢 ⊆ 𝑠 ∧ (𝑅 ↾t 𝑠) ∈ Comp))) ∧ (𝑘 ∈ 𝒫 ∪ 𝑆
∧ (𝑆
↾t 𝑘)
∈ Comp)) → 𝑅
∈ Top) |
| 183 | 154, 4 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢
((((((𝑅 ∈
𝑛-Locally Comp ∧ 𝑆 ∈ (ran 𝑘Gen ∩ Haus)) ∧
𝑥 ∈
(𝑘Gen‘(𝑅
×t 𝑆)))
∧ 𝑦 ∈ 𝑥) ∧ ((𝑠 ∈ 𝒫 {𝑡 ∈ ∪ 𝑅 ∣ 〈𝑡, (2nd ‘𝑦)〉 ∈ 𝑥} ∧ 𝑢 ∈ 𝑅) ∧ ((1st ‘𝑦) ∈ 𝑢 ∧ 𝑢 ⊆ 𝑠 ∧ (𝑅 ↾t 𝑠) ∈ Comp))) ∧ (𝑘 ∈ 𝒫 ∪ 𝑆
∧ (𝑆
↾t 𝑘)
∈ Comp)) → 𝑆
∈ Top) |
| 184 | 176 | a1i 11 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢
((((((𝑅 ∈
𝑛-Locally Comp ∧ 𝑆 ∈ (ran 𝑘Gen ∩ Haus)) ∧
𝑥 ∈
(𝑘Gen‘(𝑅
×t 𝑆)))
∧ 𝑦 ∈ 𝑥) ∧ ((𝑠 ∈ 𝒫 {𝑡 ∈ ∪ 𝑅 ∣ 〈𝑡, (2nd ‘𝑦)〉 ∈ 𝑥} ∧ 𝑢 ∈ 𝑅) ∧ ((1st ‘𝑦) ∈ 𝑢 ∧ 𝑢 ⊆ 𝑠 ∧ (𝑅 ↾t 𝑠) ∈ Comp))) ∧ (𝑘 ∈ 𝒫 ∪ 𝑆
∧ (𝑆
↾t 𝑘)
∈ Comp)) → 𝑠
∈ V) |
| 185 | | simprl 771 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢
((((((𝑅 ∈
𝑛-Locally Comp ∧ 𝑆 ∈ (ran 𝑘Gen ∩ Haus)) ∧
𝑥 ∈
(𝑘Gen‘(𝑅
×t 𝑆)))
∧ 𝑦 ∈ 𝑥) ∧ ((𝑠 ∈ 𝒫 {𝑡 ∈ ∪ 𝑅 ∣ 〈𝑡, (2nd ‘𝑦)〉 ∈ 𝑥} ∧ 𝑢 ∈ 𝑅) ∧ ((1st ‘𝑦) ∈ 𝑢 ∧ 𝑢 ⊆ 𝑠 ∧ (𝑅 ↾t 𝑠) ∈ Comp))) ∧ (𝑘 ∈ 𝒫 ∪ 𝑆
∧ (𝑆
↾t 𝑘)
∈ Comp)) → 𝑘
∈ 𝒫 ∪ 𝑆) |
| 186 | | txrest 23639 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ (𝑠 ∈ V ∧ 𝑘 ∈ 𝒫 ∪ 𝑆))
→ ((𝑅
×t 𝑆)
↾t (𝑠
× 𝑘)) = ((𝑅 ↾t 𝑠) ×t (𝑆 ↾t 𝑘))) |
| 187 | 182, 183,
184, 185, 186 | syl22anc 839 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢
((((((𝑅 ∈
𝑛-Locally Comp ∧ 𝑆 ∈ (ran 𝑘Gen ∩ Haus)) ∧
𝑥 ∈
(𝑘Gen‘(𝑅
×t 𝑆)))
∧ 𝑦 ∈ 𝑥) ∧ ((𝑠 ∈ 𝒫 {𝑡 ∈ ∪ 𝑅 ∣ 〈𝑡, (2nd ‘𝑦)〉 ∈ 𝑥} ∧ 𝑢 ∈ 𝑅) ∧ ((1st ‘𝑦) ∈ 𝑢 ∧ 𝑢 ⊆ 𝑠 ∧ (𝑅 ↾t 𝑠) ∈ Comp))) ∧ (𝑘 ∈ 𝒫 ∪ 𝑆
∧ (𝑆
↾t 𝑘)
∈ Comp)) → ((𝑅
×t 𝑆)
↾t (𝑠
× 𝑘)) = ((𝑅 ↾t 𝑠) ×t (𝑆 ↾t 𝑘))) |
| 188 | | simprr3 1224 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢
(((((𝑅 ∈
𝑛-Locally Comp ∧ 𝑆 ∈ (ran 𝑘Gen ∩ Haus)) ∧
𝑥 ∈
(𝑘Gen‘(𝑅
×t 𝑆)))
∧ 𝑦 ∈ 𝑥) ∧ ((𝑠 ∈ 𝒫 {𝑡 ∈ ∪ 𝑅 ∣ 〈𝑡, (2nd ‘𝑦)〉 ∈ 𝑥} ∧ 𝑢 ∈ 𝑅) ∧ ((1st ‘𝑦) ∈ 𝑢 ∧ 𝑢 ⊆ 𝑠 ∧ (𝑅 ↾t 𝑠) ∈ Comp))) → (𝑅 ↾t 𝑠) ∈ Comp) |
| 189 | 188 | adantr 480 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢
((((((𝑅 ∈
𝑛-Locally Comp ∧ 𝑆 ∈ (ran 𝑘Gen ∩ Haus)) ∧
𝑥 ∈
(𝑘Gen‘(𝑅
×t 𝑆)))
∧ 𝑦 ∈ 𝑥) ∧ ((𝑠 ∈ 𝒫 {𝑡 ∈ ∪ 𝑅 ∣ 〈𝑡, (2nd ‘𝑦)〉 ∈ 𝑥} ∧ 𝑢 ∈ 𝑅) ∧ ((1st ‘𝑦) ∈ 𝑢 ∧ 𝑢 ⊆ 𝑠 ∧ (𝑅 ↾t 𝑠) ∈ Comp))) ∧ (𝑘 ∈ 𝒫 ∪ 𝑆
∧ (𝑆
↾t 𝑘)
∈ Comp)) → (𝑅
↾t 𝑠)
∈ Comp) |
| 190 | | simprr 773 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢
((((((𝑅 ∈
𝑛-Locally Comp ∧ 𝑆 ∈ (ran 𝑘Gen ∩ Haus)) ∧
𝑥 ∈
(𝑘Gen‘(𝑅
×t 𝑆)))
∧ 𝑦 ∈ 𝑥) ∧ ((𝑠 ∈ 𝒫 {𝑡 ∈ ∪ 𝑅 ∣ 〈𝑡, (2nd ‘𝑦)〉 ∈ 𝑥} ∧ 𝑢 ∈ 𝑅) ∧ ((1st ‘𝑦) ∈ 𝑢 ∧ 𝑢 ⊆ 𝑠 ∧ (𝑅 ↾t 𝑠) ∈ Comp))) ∧ (𝑘 ∈ 𝒫 ∪ 𝑆
∧ (𝑆
↾t 𝑘)
∈ Comp)) → (𝑆
↾t 𝑘)
∈ Comp) |
| 191 | | txcmp 23651 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (((𝑅 ↾t 𝑠) ∈ Comp ∧ (𝑆 ↾t 𝑘) ∈ Comp) → ((𝑅 ↾t 𝑠) ×t (𝑆 ↾t 𝑘)) ∈ Comp) |
| 192 | 189, 190,
191 | syl2anc 584 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢
((((((𝑅 ∈
𝑛-Locally Comp ∧ 𝑆 ∈ (ran 𝑘Gen ∩ Haus)) ∧
𝑥 ∈
(𝑘Gen‘(𝑅
×t 𝑆)))
∧ 𝑦 ∈ 𝑥) ∧ ((𝑠 ∈ 𝒫 {𝑡 ∈ ∪ 𝑅 ∣ 〈𝑡, (2nd ‘𝑦)〉 ∈ 𝑥} ∧ 𝑢 ∈ 𝑅) ∧ ((1st ‘𝑦) ∈ 𝑢 ∧ 𝑢 ⊆ 𝑠 ∧ (𝑅 ↾t 𝑠) ∈ Comp))) ∧ (𝑘 ∈ 𝒫 ∪ 𝑆
∧ (𝑆
↾t 𝑘)
∈ Comp)) → ((𝑅
↾t 𝑠)
×t (𝑆
↾t 𝑘))
∈ Comp) |
| 193 | 187, 192 | eqeltrd 2841 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢
((((((𝑅 ∈
𝑛-Locally Comp ∧ 𝑆 ∈ (ran 𝑘Gen ∩ Haus)) ∧
𝑥 ∈
(𝑘Gen‘(𝑅
×t 𝑆)))
∧ 𝑦 ∈ 𝑥) ∧ ((𝑠 ∈ 𝒫 {𝑡 ∈ ∪ 𝑅 ∣ 〈𝑡, (2nd ‘𝑦)〉 ∈ 𝑥} ∧ 𝑢 ∈ 𝑅) ∧ ((1st ‘𝑦) ∈ 𝑢 ∧ 𝑢 ⊆ 𝑠 ∧ (𝑅 ↾t 𝑠) ∈ Comp))) ∧ (𝑘 ∈ 𝒫 ∪ 𝑆
∧ (𝑆
↾t 𝑘)
∈ Comp)) → ((𝑅
×t 𝑆)
↾t (𝑠
× 𝑘)) ∈
Comp) |
| 194 | | difin 4272 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝑠 × 𝑘) ∖ ((𝑠 × 𝑘) ∩ 𝑥)) = ((𝑠 × 𝑘) ∖ 𝑥) |
| 195 | 74, 76, 117 | syl2anc 584 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢
((((((𝑅 ∈
𝑛-Locally Comp ∧ 𝑆 ∈ (ran 𝑘Gen ∩ Haus)) ∧
𝑥 ∈
(𝑘Gen‘(𝑅
×t 𝑆)))
∧ 𝑦 ∈ 𝑥) ∧ ((𝑠 ∈ 𝒫 {𝑡 ∈ ∪ 𝑅 ∣ 〈𝑡, (2nd ‘𝑦)〉 ∈ 𝑥} ∧ 𝑢 ∈ 𝑅) ∧ ((1st ‘𝑦) ∈ 𝑢 ∧ 𝑢 ⊆ 𝑠 ∧ (𝑅 ↾t 𝑠) ∈ Comp))) ∧ (𝑘 ∈ 𝒫 ∪ 𝑆
∧ (𝑆
↾t 𝑘)
∈ Comp)) → (𝑠
× 𝑘) ⊆ (∪ 𝑅
× ∪ 𝑆)) |
| 196 | 182, 148,
25 | syl2anc 584 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢
((((((𝑅 ∈
𝑛-Locally Comp ∧ 𝑆 ∈ (ran 𝑘Gen ∩ Haus)) ∧
𝑥 ∈
(𝑘Gen‘(𝑅
×t 𝑆)))
∧ 𝑦 ∈ 𝑥) ∧ ((𝑠 ∈ 𝒫 {𝑡 ∈ ∪ 𝑅 ∣ 〈𝑡, (2nd ‘𝑦)〉 ∈ 𝑥} ∧ 𝑢 ∈ 𝑅) ∧ ((1st ‘𝑦) ∈ 𝑢 ∧ 𝑢 ⊆ 𝑠 ∧ (𝑅 ↾t 𝑠) ∈ Comp))) ∧ (𝑘 ∈ 𝒫 ∪ 𝑆
∧ (𝑆
↾t 𝑘)
∈ Comp)) → (∪ 𝑅 × ∪ 𝑆) = ∪
(𝑅 ×t
𝑆)) |
| 197 | 195, 196 | sseqtrd 4020 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢
((((((𝑅 ∈
𝑛-Locally Comp ∧ 𝑆 ∈ (ran 𝑘Gen ∩ Haus)) ∧
𝑥 ∈
(𝑘Gen‘(𝑅
×t 𝑆)))
∧ 𝑦 ∈ 𝑥) ∧ ((𝑠 ∈ 𝒫 {𝑡 ∈ ∪ 𝑅 ∣ 〈𝑡, (2nd ‘𝑦)〉 ∈ 𝑥} ∧ 𝑢 ∈ 𝑅) ∧ ((1st ‘𝑦) ∈ 𝑢 ∧ 𝑢 ⊆ 𝑠 ∧ (𝑅 ↾t 𝑠) ∈ Comp))) ∧ (𝑘 ∈ 𝒫 ∪ 𝑆
∧ (𝑆
↾t 𝑘)
∈ Comp)) → (𝑠
× 𝑘) ⊆ ∪ (𝑅
×t 𝑆)) |
| 198 | 28 | restuni 23170 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (((𝑅 ×t 𝑆) ∈ Top ∧ (𝑠 × 𝑘) ⊆ ∪ (𝑅 ×t 𝑆)) → (𝑠 × 𝑘) = ∪ ((𝑅 ×t 𝑆) ↾t (𝑠 × 𝑘))) |
| 199 | 174, 197,
198 | syl2anc 584 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢
((((((𝑅 ∈
𝑛-Locally Comp ∧ 𝑆 ∈ (ran 𝑘Gen ∩ Haus)) ∧
𝑥 ∈
(𝑘Gen‘(𝑅
×t 𝑆)))
∧ 𝑦 ∈ 𝑥) ∧ ((𝑠 ∈ 𝒫 {𝑡 ∈ ∪ 𝑅 ∣ 〈𝑡, (2nd ‘𝑦)〉 ∈ 𝑥} ∧ 𝑢 ∈ 𝑅) ∧ ((1st ‘𝑦) ∈ 𝑢 ∧ 𝑢 ⊆ 𝑠 ∧ (𝑅 ↾t 𝑠) ∈ Comp))) ∧ (𝑘 ∈ 𝒫 ∪ 𝑆
∧ (𝑆
↾t 𝑘)
∈ Comp)) → (𝑠
× 𝑘) = ∪ ((𝑅
×t 𝑆)
↾t (𝑠
× 𝑘))) |
| 200 | 199 | difeq1d 4125 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢
((((((𝑅 ∈
𝑛-Locally Comp ∧ 𝑆 ∈ (ran 𝑘Gen ∩ Haus)) ∧
𝑥 ∈
(𝑘Gen‘(𝑅
×t 𝑆)))
∧ 𝑦 ∈ 𝑥) ∧ ((𝑠 ∈ 𝒫 {𝑡 ∈ ∪ 𝑅 ∣ 〈𝑡, (2nd ‘𝑦)〉 ∈ 𝑥} ∧ 𝑢 ∈ 𝑅) ∧ ((1st ‘𝑦) ∈ 𝑢 ∧ 𝑢 ⊆ 𝑠 ∧ (𝑅 ↾t 𝑠) ∈ Comp))) ∧ (𝑘 ∈ 𝒫 ∪ 𝑆
∧ (𝑆
↾t 𝑘)
∈ Comp)) → ((𝑠
× 𝑘) ∖ ((𝑠 × 𝑘) ∩ 𝑥)) = (∪ ((𝑅 ×t 𝑆) ↾t (𝑠 × 𝑘)) ∖ ((𝑠 × 𝑘) ∩ 𝑥))) |
| 201 | 194, 200 | eqtr3id 2791 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢
((((((𝑅 ∈
𝑛-Locally Comp ∧ 𝑆 ∈ (ran 𝑘Gen ∩ Haus)) ∧
𝑥 ∈
(𝑘Gen‘(𝑅
×t 𝑆)))
∧ 𝑦 ∈ 𝑥) ∧ ((𝑠 ∈ 𝒫 {𝑡 ∈ ∪ 𝑅 ∣ 〈𝑡, (2nd ‘𝑦)〉 ∈ 𝑥} ∧ 𝑢 ∈ 𝑅) ∧ ((1st ‘𝑦) ∈ 𝑢 ∧ 𝑢 ⊆ 𝑠 ∧ (𝑅 ↾t 𝑠) ∈ Comp))) ∧ (𝑘 ∈ 𝒫 ∪ 𝑆
∧ (𝑆
↾t 𝑘)
∈ Comp)) → ((𝑠
× 𝑘) ∖ 𝑥) = (∪ ((𝑅
×t 𝑆)
↾t (𝑠
× 𝑘)) ∖ ((𝑠 × 𝑘) ∩ 𝑥))) |
| 202 | | resttop 23168 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (((𝑅 ×t 𝑆) ∈ Top ∧ (𝑠 × 𝑘) ∈ V) → ((𝑅 ×t 𝑆) ↾t (𝑠 × 𝑘)) ∈ Top) |
| 203 | 174, 178,
202 | sylancl 586 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢
((((((𝑅 ∈
𝑛-Locally Comp ∧ 𝑆 ∈ (ran 𝑘Gen ∩ Haus)) ∧
𝑥 ∈
(𝑘Gen‘(𝑅
×t 𝑆)))
∧ 𝑦 ∈ 𝑥) ∧ ((𝑠 ∈ 𝒫 {𝑡 ∈ ∪ 𝑅 ∣ 〈𝑡, (2nd ‘𝑦)〉 ∈ 𝑥} ∧ 𝑢 ∈ 𝑅) ∧ ((1st ‘𝑦) ∈ 𝑢 ∧ 𝑢 ⊆ 𝑠 ∧ (𝑅 ↾t 𝑠) ∈ Comp))) ∧ (𝑘 ∈ 𝒫 ∪ 𝑆
∧ (𝑆
↾t 𝑘)
∈ Comp)) → ((𝑅
×t 𝑆)
↾t (𝑠
× 𝑘)) ∈
Top) |
| 204 | | incom 4209 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝑠 × 𝑘) ∩ 𝑥) = (𝑥 ∩ (𝑠 × 𝑘)) |
| 205 | 20 | ad2antrr 726 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢
((((((𝑅 ∈
𝑛-Locally Comp ∧ 𝑆 ∈ (ran 𝑘Gen ∩ Haus)) ∧
𝑥 ∈
(𝑘Gen‘(𝑅
×t 𝑆)))
∧ 𝑦 ∈ 𝑥) ∧ ((𝑠 ∈ 𝒫 {𝑡 ∈ ∪ 𝑅 ∣ 〈𝑡, (2nd ‘𝑦)〉 ∈ 𝑥} ∧ 𝑢 ∈ 𝑅) ∧ ((1st ‘𝑦) ∈ 𝑢 ∧ 𝑢 ⊆ 𝑠 ∧ (𝑅 ↾t 𝑠) ∈ Comp))) ∧ (𝑘 ∈ 𝒫 ∪ 𝑆
∧ (𝑆
↾t 𝑘)
∈ Comp)) → 𝑥
∈ (𝑘Gen‘(𝑅 ×t 𝑆))) |
| 206 | | kgeni 23545 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ ((𝑥 ∈
(𝑘Gen‘(𝑅
×t 𝑆))
∧ ((𝑅
×t 𝑆)
↾t (𝑠
× 𝑘)) ∈ Comp)
→ (𝑥 ∩ (𝑠 × 𝑘)) ∈ ((𝑅 ×t 𝑆) ↾t (𝑠 × 𝑘))) |
| 207 | 205, 193,
206 | syl2anc 584 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢
((((((𝑅 ∈
𝑛-Locally Comp ∧ 𝑆 ∈ (ran 𝑘Gen ∩ Haus)) ∧
𝑥 ∈
(𝑘Gen‘(𝑅
×t 𝑆)))
∧ 𝑦 ∈ 𝑥) ∧ ((𝑠 ∈ 𝒫 {𝑡 ∈ ∪ 𝑅 ∣ 〈𝑡, (2nd ‘𝑦)〉 ∈ 𝑥} ∧ 𝑢 ∈ 𝑅) ∧ ((1st ‘𝑦) ∈ 𝑢 ∧ 𝑢 ⊆ 𝑠 ∧ (𝑅 ↾t 𝑠) ∈ Comp))) ∧ (𝑘 ∈ 𝒫 ∪ 𝑆
∧ (𝑆
↾t 𝑘)
∈ Comp)) → (𝑥
∩ (𝑠 × 𝑘)) ∈ ((𝑅 ×t 𝑆) ↾t (𝑠 × 𝑘))) |
| 208 | 204, 207 | eqeltrid 2845 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢
((((((𝑅 ∈
𝑛-Locally Comp ∧ 𝑆 ∈ (ran 𝑘Gen ∩ Haus)) ∧
𝑥 ∈
(𝑘Gen‘(𝑅
×t 𝑆)))
∧ 𝑦 ∈ 𝑥) ∧ ((𝑠 ∈ 𝒫 {𝑡 ∈ ∪ 𝑅 ∣ 〈𝑡, (2nd ‘𝑦)〉 ∈ 𝑥} ∧ 𝑢 ∈ 𝑅) ∧ ((1st ‘𝑦) ∈ 𝑢 ∧ 𝑢 ⊆ 𝑠 ∧ (𝑅 ↾t 𝑠) ∈ Comp))) ∧ (𝑘 ∈ 𝒫 ∪ 𝑆
∧ (𝑆
↾t 𝑘)
∈ Comp)) → ((𝑠
× 𝑘) ∩ 𝑥) ∈ ((𝑅 ×t 𝑆) ↾t (𝑠 × 𝑘))) |
| 209 | | eqid 2737 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ∪ ((𝑅
×t 𝑆)
↾t (𝑠
× 𝑘)) = ∪ ((𝑅
×t 𝑆)
↾t (𝑠
× 𝑘)) |
| 210 | 209 | opncld 23041 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((((𝑅 ×t 𝑆) ↾t (𝑠 × 𝑘)) ∈ Top ∧ ((𝑠 × 𝑘) ∩ 𝑥) ∈ ((𝑅 ×t 𝑆) ↾t (𝑠 × 𝑘))) → (∪
((𝑅 ×t
𝑆) ↾t
(𝑠 × 𝑘)) ∖ ((𝑠 × 𝑘) ∩ 𝑥)) ∈ (Clsd‘((𝑅 ×t 𝑆) ↾t (𝑠 × 𝑘)))) |
| 211 | 203, 208,
210 | syl2anc 584 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢
((((((𝑅 ∈
𝑛-Locally Comp ∧ 𝑆 ∈ (ran 𝑘Gen ∩ Haus)) ∧
𝑥 ∈
(𝑘Gen‘(𝑅
×t 𝑆)))
∧ 𝑦 ∈ 𝑥) ∧ ((𝑠 ∈ 𝒫 {𝑡 ∈ ∪ 𝑅 ∣ 〈𝑡, (2nd ‘𝑦)〉 ∈ 𝑥} ∧ 𝑢 ∈ 𝑅) ∧ ((1st ‘𝑦) ∈ 𝑢 ∧ 𝑢 ⊆ 𝑠 ∧ (𝑅 ↾t 𝑠) ∈ Comp))) ∧ (𝑘 ∈ 𝒫 ∪ 𝑆
∧ (𝑆
↾t 𝑘)
∈ Comp)) → (∪ ((𝑅 ×t 𝑆) ↾t (𝑠 × 𝑘)) ∖ ((𝑠 × 𝑘) ∩ 𝑥)) ∈ (Clsd‘((𝑅 ×t 𝑆) ↾t (𝑠 × 𝑘)))) |
| 212 | 201, 211 | eqeltrd 2841 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢
((((((𝑅 ∈
𝑛-Locally Comp ∧ 𝑆 ∈ (ran 𝑘Gen ∩ Haus)) ∧
𝑥 ∈
(𝑘Gen‘(𝑅
×t 𝑆)))
∧ 𝑦 ∈ 𝑥) ∧ ((𝑠 ∈ 𝒫 {𝑡 ∈ ∪ 𝑅 ∣ 〈𝑡, (2nd ‘𝑦)〉 ∈ 𝑥} ∧ 𝑢 ∈ 𝑅) ∧ ((1st ‘𝑦) ∈ 𝑢 ∧ 𝑢 ⊆ 𝑠 ∧ (𝑅 ↾t 𝑠) ∈ Comp))) ∧ (𝑘 ∈ 𝒫 ∪ 𝑆
∧ (𝑆
↾t 𝑘)
∈ Comp)) → ((𝑠
× 𝑘) ∖ 𝑥) ∈ (Clsd‘((𝑅 ×t 𝑆) ↾t (𝑠 × 𝑘)))) |
| 213 | | cmpcld 23410 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((((𝑅 ×t 𝑆) ↾t (𝑠 × 𝑘)) ∈ Comp ∧ ((𝑠 × 𝑘) ∖ 𝑥) ∈ (Clsd‘((𝑅 ×t 𝑆) ↾t (𝑠 × 𝑘)))) → (((𝑅 ×t 𝑆) ↾t (𝑠 × 𝑘)) ↾t ((𝑠 × 𝑘) ∖ 𝑥)) ∈ Comp) |
| 214 | 193, 212,
213 | syl2anc 584 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
((((((𝑅 ∈
𝑛-Locally Comp ∧ 𝑆 ∈ (ran 𝑘Gen ∩ Haus)) ∧
𝑥 ∈
(𝑘Gen‘(𝑅
×t 𝑆)))
∧ 𝑦 ∈ 𝑥) ∧ ((𝑠 ∈ 𝒫 {𝑡 ∈ ∪ 𝑅 ∣ 〈𝑡, (2nd ‘𝑦)〉 ∈ 𝑥} ∧ 𝑢 ∈ 𝑅) ∧ ((1st ‘𝑦) ∈ 𝑢 ∧ 𝑢 ⊆ 𝑠 ∧ (𝑅 ↾t 𝑠) ∈ Comp))) ∧ (𝑘 ∈ 𝒫 ∪ 𝑆
∧ (𝑆
↾t 𝑘)
∈ Comp)) → (((𝑅
×t 𝑆)
↾t (𝑠
× 𝑘))
↾t ((𝑠
× 𝑘) ∖ 𝑥)) ∈ Comp) |
| 215 | 181, 214 | eqeltrrd 2842 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
((((((𝑅 ∈
𝑛-Locally Comp ∧ 𝑆 ∈ (ran 𝑘Gen ∩ Haus)) ∧
𝑥 ∈
(𝑘Gen‘(𝑅
×t 𝑆)))
∧ 𝑦 ∈ 𝑥) ∧ ((𝑠 ∈ 𝒫 {𝑡 ∈ ∪ 𝑅 ∣ 〈𝑡, (2nd ‘𝑦)〉 ∈ 𝑥} ∧ 𝑢 ∈ 𝑅) ∧ ((1st ‘𝑦) ∈ 𝑢 ∧ 𝑢 ⊆ 𝑠 ∧ (𝑅 ↾t 𝑠) ∈ Comp))) ∧ (𝑘 ∈ 𝒫 ∪ 𝑆
∧ (𝑆
↾t 𝑘)
∈ Comp)) → ((𝑅
×t 𝑆)
↾t ((𝑠
× 𝑘) ∖ 𝑥)) ∈ Comp) |
| 216 | | imacmp 23405 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
(((2nd ↾ (∪ 𝑅 × ∪ 𝑆))
∈ ((𝑅
×t 𝑆) Cn
𝑆) ∧ ((𝑅 ×t 𝑆) ↾t ((𝑠 × 𝑘) ∖ 𝑥)) ∈ Comp) → (𝑆 ↾t ((2nd
↾ (∪ 𝑅 × ∪ 𝑆)) “ ((𝑠 × 𝑘) ∖ 𝑥))) ∈ Comp) |
| 217 | 173, 215,
216 | syl2anc 584 |
. . . . . . . . . . . . . . . . . . . 20
⊢
((((((𝑅 ∈
𝑛-Locally Comp ∧ 𝑆 ∈ (ran 𝑘Gen ∩ Haus)) ∧
𝑥 ∈
(𝑘Gen‘(𝑅
×t 𝑆)))
∧ 𝑦 ∈ 𝑥) ∧ ((𝑠 ∈ 𝒫 {𝑡 ∈ ∪ 𝑅 ∣ 〈𝑡, (2nd ‘𝑦)〉 ∈ 𝑥} ∧ 𝑢 ∈ 𝑅) ∧ ((1st ‘𝑦) ∈ 𝑢 ∧ 𝑢 ⊆ 𝑠 ∧ (𝑅 ↾t 𝑠) ∈ Comp))) ∧ (𝑘 ∈ 𝒫 ∪ 𝑆
∧ (𝑆
↾t 𝑘)
∈ Comp)) → (𝑆
↾t ((2nd ↾ (∪
𝑅 × ∪ 𝑆))
“ ((𝑠 × 𝑘) ∖ 𝑥))) ∈ Comp) |
| 218 | 24 | hauscmp 23415 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑆 ∈ Haus ∧
((2nd ↾ (∪ 𝑅 × ∪ 𝑆)) “ ((𝑠 × 𝑘) ∖ 𝑥)) ⊆ ∪ 𝑆 ∧ (𝑆 ↾t ((2nd
↾ (∪ 𝑅 × ∪ 𝑆)) “ ((𝑠 × 𝑘) ∖ 𝑥))) ∈ Comp) → ((2nd
↾ (∪ 𝑅 × ∪ 𝑆)) “ ((𝑠 × 𝑘) ∖ 𝑥)) ∈ (Clsd‘𝑆)) |
| 219 | 155, 169,
217, 218 | syl3anc 1373 |
. . . . . . . . . . . . . . . . . . 19
⊢
((((((𝑅 ∈
𝑛-Locally Comp ∧ 𝑆 ∈ (ran 𝑘Gen ∩ Haus)) ∧
𝑥 ∈
(𝑘Gen‘(𝑅
×t 𝑆)))
∧ 𝑦 ∈ 𝑥) ∧ ((𝑠 ∈ 𝒫 {𝑡 ∈ ∪ 𝑅 ∣ 〈𝑡, (2nd ‘𝑦)〉 ∈ 𝑥} ∧ 𝑢 ∈ 𝑅) ∧ ((1st ‘𝑦) ∈ 𝑢 ∧ 𝑢 ⊆ 𝑠 ∧ (𝑅 ↾t 𝑠) ∈ Comp))) ∧ (𝑘 ∈ 𝒫 ∪ 𝑆
∧ (𝑆
↾t 𝑘)
∈ Comp)) → ((2nd ↾ (∪
𝑅 × ∪ 𝑆))
“ ((𝑠 × 𝑘) ∖ 𝑥)) ∈ (Clsd‘𝑆)) |
| 220 | 168 | a1i 11 |
. . . . . . . . . . . . . . . . . . 19
⊢
((((((𝑅 ∈
𝑛-Locally Comp ∧ 𝑆 ∈ (ran 𝑘Gen ∩ Haus)) ∧
𝑥 ∈
(𝑘Gen‘(𝑅
×t 𝑆)))
∧ 𝑦 ∈ 𝑥) ∧ ((𝑠 ∈ 𝒫 {𝑡 ∈ ∪ 𝑅 ∣ 〈𝑡, (2nd ‘𝑦)〉 ∈ 𝑥} ∧ 𝑢 ∈ 𝑅) ∧ ((1st ‘𝑦) ∈ 𝑢 ∧ 𝑢 ⊆ 𝑠 ∧ (𝑅 ↾t 𝑠) ∈ Comp))) ∧ (𝑘 ∈ 𝒫 ∪ 𝑆
∧ (𝑆
↾t 𝑘)
∈ Comp)) → ((2nd ↾ (∪
𝑅 × ∪ 𝑆))
“ ((𝑠 × 𝑘) ∖ 𝑥)) ⊆ 𝑘) |
| 221 | 24 | restcldi 23181 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑘 ⊆ ∪ 𝑆
∧ ((2nd ↾ (∪ 𝑅 × ∪ 𝑆)) “ ((𝑠 × 𝑘) ∖ 𝑥)) ∈ (Clsd‘𝑆) ∧ ((2nd ↾ (∪ 𝑅
× ∪ 𝑆)) “ ((𝑠 × 𝑘) ∖ 𝑥)) ⊆ 𝑘) → ((2nd ↾ (∪ 𝑅
× ∪ 𝑆)) “ ((𝑠 × 𝑘) ∖ 𝑥)) ∈ (Clsd‘(𝑆 ↾t 𝑘))) |
| 222 | 76, 219, 220, 221 | syl3anc 1373 |
. . . . . . . . . . . . . . . . . 18
⊢
((((((𝑅 ∈
𝑛-Locally Comp ∧ 𝑆 ∈ (ran 𝑘Gen ∩ Haus)) ∧
𝑥 ∈
(𝑘Gen‘(𝑅
×t 𝑆)))
∧ 𝑦 ∈ 𝑥) ∧ ((𝑠 ∈ 𝒫 {𝑡 ∈ ∪ 𝑅 ∣ 〈𝑡, (2nd ‘𝑦)〉 ∈ 𝑥} ∧ 𝑢 ∈ 𝑅) ∧ ((1st ‘𝑦) ∈ 𝑢 ∧ 𝑢 ⊆ 𝑠 ∧ (𝑅 ↾t 𝑠) ∈ Comp))) ∧ (𝑘 ∈ 𝒫 ∪ 𝑆
∧ (𝑆
↾t 𝑘)
∈ Comp)) → ((2nd ↾ (∪
𝑅 × ∪ 𝑆))
“ ((𝑠 × 𝑘) ∖ 𝑥)) ∈ (Clsd‘(𝑆 ↾t 𝑘))) |
| 223 | 153, 222 | eqeltrrd 2842 |
. . . . . . . . . . . . . . . . 17
⊢
((((((𝑅 ∈
𝑛-Locally Comp ∧ 𝑆 ∈ (ran 𝑘Gen ∩ Haus)) ∧
𝑥 ∈
(𝑘Gen‘(𝑅
×t 𝑆)))
∧ 𝑦 ∈ 𝑥) ∧ ((𝑠 ∈ 𝒫 {𝑡 ∈ ∪ 𝑅 ∣ 〈𝑡, (2nd ‘𝑦)〉 ∈ 𝑥} ∧ 𝑢 ∈ 𝑅) ∧ ((1st ‘𝑦) ∈ 𝑢 ∧ 𝑢 ⊆ 𝑠 ∧ (𝑅 ↾t 𝑠) ∈ Comp))) ∧ (𝑘 ∈ 𝒫 ∪ 𝑆
∧ (𝑆
↾t 𝑘)
∈ Comp)) → (∪ (𝑆 ↾t 𝑘) ∖ (𝑘 ∩ {𝑣 ∈ ∪ 𝑆 ∣ (𝑠 × {𝑣}) ⊆ 𝑥})) ∈ (Clsd‘(𝑆 ↾t 𝑘))) |
| 224 | | resttop 23168 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑆 ∈ Top ∧ 𝑘 ∈ 𝒫 ∪ 𝑆)
→ (𝑆
↾t 𝑘)
∈ Top) |
| 225 | 148, 185,
224 | syl2anc 584 |
. . . . . . . . . . . . . . . . . 18
⊢
((((((𝑅 ∈
𝑛-Locally Comp ∧ 𝑆 ∈ (ran 𝑘Gen ∩ Haus)) ∧
𝑥 ∈
(𝑘Gen‘(𝑅
×t 𝑆)))
∧ 𝑦 ∈ 𝑥) ∧ ((𝑠 ∈ 𝒫 {𝑡 ∈ ∪ 𝑅 ∣ 〈𝑡, (2nd ‘𝑦)〉 ∈ 𝑥} ∧ 𝑢 ∈ 𝑅) ∧ ((1st ‘𝑦) ∈ 𝑢 ∧ 𝑢 ⊆ 𝑠 ∧ (𝑅 ↾t 𝑠) ∈ Comp))) ∧ (𝑘 ∈ 𝒫 ∪ 𝑆
∧ (𝑆
↾t 𝑘)
∈ Comp)) → (𝑆
↾t 𝑘)
∈ Top) |
| 226 | | inss1 4237 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑘 ∩ {𝑣 ∈ ∪ 𝑆 ∣ (𝑠 × {𝑣}) ⊆ 𝑥}) ⊆ 𝑘 |
| 227 | 226, 150 | sseqtrid 4026 |
. . . . . . . . . . . . . . . . . 18
⊢
((((((𝑅 ∈
𝑛-Locally Comp ∧ 𝑆 ∈ (ran 𝑘Gen ∩ Haus)) ∧
𝑥 ∈
(𝑘Gen‘(𝑅
×t 𝑆)))
∧ 𝑦 ∈ 𝑥) ∧ ((𝑠 ∈ 𝒫 {𝑡 ∈ ∪ 𝑅 ∣ 〈𝑡, (2nd ‘𝑦)〉 ∈ 𝑥} ∧ 𝑢 ∈ 𝑅) ∧ ((1st ‘𝑦) ∈ 𝑢 ∧ 𝑢 ⊆ 𝑠 ∧ (𝑅 ↾t 𝑠) ∈ Comp))) ∧ (𝑘 ∈ 𝒫 ∪ 𝑆
∧ (𝑆
↾t 𝑘)
∈ Comp)) → (𝑘
∩ {𝑣 ∈ ∪ 𝑆
∣ (𝑠 × {𝑣}) ⊆ 𝑥}) ⊆ ∪
(𝑆 ↾t
𝑘)) |
| 228 | | eqid 2737 |
. . . . . . . . . . . . . . . . . . 19
⊢ ∪ (𝑆
↾t 𝑘) =
∪ (𝑆 ↾t 𝑘) |
| 229 | 228 | isopn2 23040 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝑆 ↾t 𝑘) ∈ Top ∧ (𝑘 ∩ {𝑣 ∈ ∪ 𝑆 ∣ (𝑠 × {𝑣}) ⊆ 𝑥}) ⊆ ∪
(𝑆 ↾t
𝑘)) → ((𝑘 ∩ {𝑣 ∈ ∪ 𝑆 ∣ (𝑠 × {𝑣}) ⊆ 𝑥}) ∈ (𝑆 ↾t 𝑘) ↔ (∪ (𝑆 ↾t 𝑘) ∖ (𝑘 ∩ {𝑣 ∈ ∪ 𝑆 ∣ (𝑠 × {𝑣}) ⊆ 𝑥})) ∈ (Clsd‘(𝑆 ↾t 𝑘)))) |
| 230 | 225, 227,
229 | syl2anc 584 |
. . . . . . . . . . . . . . . . 17
⊢
((((((𝑅 ∈
𝑛-Locally Comp ∧ 𝑆 ∈ (ran 𝑘Gen ∩ Haus)) ∧
𝑥 ∈
(𝑘Gen‘(𝑅
×t 𝑆)))
∧ 𝑦 ∈ 𝑥) ∧ ((𝑠 ∈ 𝒫 {𝑡 ∈ ∪ 𝑅 ∣ 〈𝑡, (2nd ‘𝑦)〉 ∈ 𝑥} ∧ 𝑢 ∈ 𝑅) ∧ ((1st ‘𝑦) ∈ 𝑢 ∧ 𝑢 ⊆ 𝑠 ∧ (𝑅 ↾t 𝑠) ∈ Comp))) ∧ (𝑘 ∈ 𝒫 ∪ 𝑆
∧ (𝑆
↾t 𝑘)
∈ Comp)) → ((𝑘
∩ {𝑣 ∈ ∪ 𝑆
∣ (𝑠 × {𝑣}) ⊆ 𝑥}) ∈ (𝑆 ↾t 𝑘) ↔ (∪ (𝑆 ↾t 𝑘) ∖ (𝑘 ∩ {𝑣 ∈ ∪ 𝑆 ∣ (𝑠 × {𝑣}) ⊆ 𝑥})) ∈ (Clsd‘(𝑆 ↾t 𝑘)))) |
| 231 | 223, 230 | mpbird 257 |
. . . . . . . . . . . . . . . 16
⊢
((((((𝑅 ∈
𝑛-Locally Comp ∧ 𝑆 ∈ (ran 𝑘Gen ∩ Haus)) ∧
𝑥 ∈
(𝑘Gen‘(𝑅
×t 𝑆)))
∧ 𝑦 ∈ 𝑥) ∧ ((𝑠 ∈ 𝒫 {𝑡 ∈ ∪ 𝑅 ∣ 〈𝑡, (2nd ‘𝑦)〉 ∈ 𝑥} ∧ 𝑢 ∈ 𝑅) ∧ ((1st ‘𝑦) ∈ 𝑢 ∧ 𝑢 ⊆ 𝑠 ∧ (𝑅 ↾t 𝑠) ∈ Comp))) ∧ (𝑘 ∈ 𝒫 ∪ 𝑆
∧ (𝑆
↾t 𝑘)
∈ Comp)) → (𝑘
∩ {𝑣 ∈ ∪ 𝑆
∣ (𝑠 × {𝑣}) ⊆ 𝑥}) ∈ (𝑆 ↾t 𝑘)) |
| 232 | 69, 231 | eqeltrid 2845 |
. . . . . . . . . . . . . . 15
⊢
((((((𝑅 ∈
𝑛-Locally Comp ∧ 𝑆 ∈ (ran 𝑘Gen ∩ Haus)) ∧
𝑥 ∈
(𝑘Gen‘(𝑅
×t 𝑆)))
∧ 𝑦 ∈ 𝑥) ∧ ((𝑠 ∈ 𝒫 {𝑡 ∈ ∪ 𝑅 ∣ 〈𝑡, (2nd ‘𝑦)〉 ∈ 𝑥} ∧ 𝑢 ∈ 𝑅) ∧ ((1st ‘𝑦) ∈ 𝑢 ∧ 𝑢 ⊆ 𝑠 ∧ (𝑅 ↾t 𝑠) ∈ Comp))) ∧ (𝑘 ∈ 𝒫 ∪ 𝑆
∧ (𝑆
↾t 𝑘)
∈ Comp)) → ({𝑣
∈ ∪ 𝑆 ∣ (𝑠 × {𝑣}) ⊆ 𝑥} ∩ 𝑘) ∈ (𝑆 ↾t 𝑘)) |
| 233 | 232 | expr 456 |
. . . . . . . . . . . . . 14
⊢
((((((𝑅 ∈
𝑛-Locally Comp ∧ 𝑆 ∈ (ran 𝑘Gen ∩ Haus)) ∧
𝑥 ∈
(𝑘Gen‘(𝑅
×t 𝑆)))
∧ 𝑦 ∈ 𝑥) ∧ ((𝑠 ∈ 𝒫 {𝑡 ∈ ∪ 𝑅 ∣ 〈𝑡, (2nd ‘𝑦)〉 ∈ 𝑥} ∧ 𝑢 ∈ 𝑅) ∧ ((1st ‘𝑦) ∈ 𝑢 ∧ 𝑢 ⊆ 𝑠 ∧ (𝑅 ↾t 𝑠) ∈ Comp))) ∧ 𝑘 ∈ 𝒫 ∪ 𝑆)
→ ((𝑆
↾t 𝑘)
∈ Comp → ({𝑣
∈ ∪ 𝑆 ∣ (𝑠 × {𝑣}) ⊆ 𝑥} ∩ 𝑘) ∈ (𝑆 ↾t 𝑘))) |
| 234 | 233 | ralrimiva 3146 |
. . . . . . . . . . . . 13
⊢
(((((𝑅 ∈
𝑛-Locally Comp ∧ 𝑆 ∈ (ran 𝑘Gen ∩ Haus)) ∧
𝑥 ∈
(𝑘Gen‘(𝑅
×t 𝑆)))
∧ 𝑦 ∈ 𝑥) ∧ ((𝑠 ∈ 𝒫 {𝑡 ∈ ∪ 𝑅 ∣ 〈𝑡, (2nd ‘𝑦)〉 ∈ 𝑥} ∧ 𝑢 ∈ 𝑅) ∧ ((1st ‘𝑦) ∈ 𝑢 ∧ 𝑢 ⊆ 𝑠 ∧ (𝑅 ↾t 𝑠) ∈ Comp))) → ∀𝑘 ∈ 𝒫 ∪ 𝑆((𝑆 ↾t 𝑘) ∈ Comp → ({𝑣 ∈ ∪ 𝑆 ∣ (𝑠 × {𝑣}) ⊆ 𝑥} ∩ 𝑘) ∈ (𝑆 ↾t 𝑘))) |
| 235 | 65, 17 | sylib 218 |
. . . . . . . . . . . . . 14
⊢
(((((𝑅 ∈
𝑛-Locally Comp ∧ 𝑆 ∈ (ran 𝑘Gen ∩ Haus)) ∧
𝑥 ∈
(𝑘Gen‘(𝑅
×t 𝑆)))
∧ 𝑦 ∈ 𝑥) ∧ ((𝑠 ∈ 𝒫 {𝑡 ∈ ∪ 𝑅 ∣ 〈𝑡, (2nd ‘𝑦)〉 ∈ 𝑥} ∧ 𝑢 ∈ 𝑅) ∧ ((1st ‘𝑦) ∈ 𝑢 ∧ 𝑢 ⊆ 𝑠 ∧ (𝑅 ↾t 𝑠) ∈ Comp))) → 𝑆 ∈ (TopOn‘∪ 𝑆)) |
| 236 | | elkgen 23544 |
. . . . . . . . . . . . . 14
⊢ (𝑆 ∈ (TopOn‘∪ 𝑆)
→ ({𝑣 ∈ ∪ 𝑆
∣ (𝑠 × {𝑣}) ⊆ 𝑥} ∈ (𝑘Gen‘𝑆) ↔ ({𝑣 ∈ ∪ 𝑆 ∣ (𝑠 × {𝑣}) ⊆ 𝑥} ⊆ ∪ 𝑆 ∧ ∀𝑘 ∈ 𝒫 ∪ 𝑆((𝑆 ↾t 𝑘) ∈ Comp → ({𝑣 ∈ ∪ 𝑆 ∣ (𝑠 × {𝑣}) ⊆ 𝑥} ∩ 𝑘) ∈ (𝑆 ↾t 𝑘))))) |
| 237 | 235, 236 | syl 17 |
. . . . . . . . . . . . 13
⊢
(((((𝑅 ∈
𝑛-Locally Comp ∧ 𝑆 ∈ (ran 𝑘Gen ∩ Haus)) ∧
𝑥 ∈
(𝑘Gen‘(𝑅
×t 𝑆)))
∧ 𝑦 ∈ 𝑥) ∧ ((𝑠 ∈ 𝒫 {𝑡 ∈ ∪ 𝑅 ∣ 〈𝑡, (2nd ‘𝑦)〉 ∈ 𝑥} ∧ 𝑢 ∈ 𝑅) ∧ ((1st ‘𝑦) ∈ 𝑢 ∧ 𝑢 ⊆ 𝑠 ∧ (𝑅 ↾t 𝑠) ∈ Comp))) → ({𝑣 ∈ ∪ 𝑆 ∣ (𝑠 × {𝑣}) ⊆ 𝑥} ∈ (𝑘Gen‘𝑆) ↔ ({𝑣 ∈ ∪ 𝑆 ∣ (𝑠 × {𝑣}) ⊆ 𝑥} ⊆ ∪ 𝑆 ∧ ∀𝑘 ∈ 𝒫 ∪ 𝑆((𝑆 ↾t 𝑘) ∈ Comp → ({𝑣 ∈ ∪ 𝑆 ∣ (𝑠 × {𝑣}) ⊆ 𝑥} ∩ 𝑘) ∈ (𝑆 ↾t 𝑘))))) |
| 238 | 68, 234, 237 | mpbir2and 713 |
. . . . . . . . . . . 12
⊢
(((((𝑅 ∈
𝑛-Locally Comp ∧ 𝑆 ∈ (ran 𝑘Gen ∩ Haus)) ∧
𝑥 ∈
(𝑘Gen‘(𝑅
×t 𝑆)))
∧ 𝑦 ∈ 𝑥) ∧ ((𝑠 ∈ 𝒫 {𝑡 ∈ ∪ 𝑅 ∣ 〈𝑡, (2nd ‘𝑦)〉 ∈ 𝑥} ∧ 𝑢 ∈ 𝑅) ∧ ((1st ‘𝑦) ∈ 𝑢 ∧ 𝑢 ⊆ 𝑠 ∧ (𝑅 ↾t 𝑠) ∈ Comp))) → {𝑣 ∈ ∪ 𝑆 ∣ (𝑠 × {𝑣}) ⊆ 𝑥} ∈ (𝑘Gen‘𝑆)) |
| 239 | 15 | adantr 480 |
. . . . . . . . . . . . . 14
⊢
(((((𝑅 ∈
𝑛-Locally Comp ∧ 𝑆 ∈ (ran 𝑘Gen ∩ Haus)) ∧
𝑥 ∈
(𝑘Gen‘(𝑅
×t 𝑆)))
∧ 𝑦 ∈ 𝑥) ∧ ((𝑠 ∈ 𝒫 {𝑡 ∈ ∪ 𝑅 ∣ 〈𝑡, (2nd ‘𝑦)〉 ∈ 𝑥} ∧ 𝑢 ∈ 𝑅) ∧ ((1st ‘𝑦) ∈ 𝑢 ∧ 𝑢 ⊆ 𝑠 ∧ (𝑅 ↾t 𝑠) ∈ Comp))) → 𝑆 ∈ (ran 𝑘Gen ∩
Haus)) |
| 240 | 239, 2 | syl 17 |
. . . . . . . . . . . . 13
⊢
(((((𝑅 ∈
𝑛-Locally Comp ∧ 𝑆 ∈ (ran 𝑘Gen ∩ Haus)) ∧
𝑥 ∈
(𝑘Gen‘(𝑅
×t 𝑆)))
∧ 𝑦 ∈ 𝑥) ∧ ((𝑠 ∈ 𝒫 {𝑡 ∈ ∪ 𝑅 ∣ 〈𝑡, (2nd ‘𝑦)〉 ∈ 𝑥} ∧ 𝑢 ∈ 𝑅) ∧ ((1st ‘𝑦) ∈ 𝑢 ∧ 𝑢 ⊆ 𝑠 ∧ (𝑅 ↾t 𝑠) ∈ Comp))) → 𝑆 ∈ ran 𝑘Gen) |
| 241 | | kgenidm 23555 |
. . . . . . . . . . . . 13
⊢ (𝑆 ∈ ran 𝑘Gen →
(𝑘Gen‘𝑆) =
𝑆) |
| 242 | 240, 241 | syl 17 |
. . . . . . . . . . . 12
⊢
(((((𝑅 ∈
𝑛-Locally Comp ∧ 𝑆 ∈ (ran 𝑘Gen ∩ Haus)) ∧
𝑥 ∈
(𝑘Gen‘(𝑅
×t 𝑆)))
∧ 𝑦 ∈ 𝑥) ∧ ((𝑠 ∈ 𝒫 {𝑡 ∈ ∪ 𝑅 ∣ 〈𝑡, (2nd ‘𝑦)〉 ∈ 𝑥} ∧ 𝑢 ∈ 𝑅) ∧ ((1st ‘𝑦) ∈ 𝑢 ∧ 𝑢 ⊆ 𝑠 ∧ (𝑅 ↾t 𝑠) ∈ Comp))) →
(𝑘Gen‘𝑆) =
𝑆) |
| 243 | 238, 242 | eleqtrd 2843 |
. . . . . . . . . . 11
⊢
(((((𝑅 ∈
𝑛-Locally Comp ∧ 𝑆 ∈ (ran 𝑘Gen ∩ Haus)) ∧
𝑥 ∈
(𝑘Gen‘(𝑅
×t 𝑆)))
∧ 𝑦 ∈ 𝑥) ∧ ((𝑠 ∈ 𝒫 {𝑡 ∈ ∪ 𝑅 ∣ 〈𝑡, (2nd ‘𝑦)〉 ∈ 𝑥} ∧ 𝑢 ∈ 𝑅) ∧ ((1st ‘𝑦) ∈ 𝑢 ∧ 𝑢 ⊆ 𝑠 ∧ (𝑅 ↾t 𝑠) ∈ Comp))) → {𝑣 ∈ ∪ 𝑆 ∣ (𝑠 × {𝑣}) ⊆ 𝑥} ∈ 𝑆) |
| 244 | | txopn 23610 |
. . . . . . . . . . 11
⊢ (((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ (𝑢 ∈ 𝑅 ∧ {𝑣 ∈ ∪ 𝑆 ∣ (𝑠 × {𝑣}) ⊆ 𝑥} ∈ 𝑆)) → (𝑢 × {𝑣 ∈ ∪ 𝑆 ∣ (𝑠 × {𝑣}) ⊆ 𝑥}) ∈ (𝑅 ×t 𝑆)) |
| 245 | 64, 65, 66, 243, 244 | syl22anc 839 |
. . . . . . . . . 10
⊢
(((((𝑅 ∈
𝑛-Locally Comp ∧ 𝑆 ∈ (ran 𝑘Gen ∩ Haus)) ∧
𝑥 ∈
(𝑘Gen‘(𝑅
×t 𝑆)))
∧ 𝑦 ∈ 𝑥) ∧ ((𝑠 ∈ 𝒫 {𝑡 ∈ ∪ 𝑅 ∣ 〈𝑡, (2nd ‘𝑦)〉 ∈ 𝑥} ∧ 𝑢 ∈ 𝑅) ∧ ((1st ‘𝑦) ∈ 𝑢 ∧ 𝑢 ⊆ 𝑠 ∧ (𝑅 ↾t 𝑠) ∈ Comp))) → (𝑢 × {𝑣 ∈ ∪ 𝑆 ∣ (𝑠 × {𝑣}) ⊆ 𝑥}) ∈ (𝑅 ×t 𝑆)) |
| 246 | 59 | adantr 480 |
. . . . . . . . . . 11
⊢
(((((𝑅 ∈
𝑛-Locally Comp ∧ 𝑆 ∈ (ran 𝑘Gen ∩ Haus)) ∧
𝑥 ∈
(𝑘Gen‘(𝑅
×t 𝑆)))
∧ 𝑦 ∈ 𝑥) ∧ ((𝑠 ∈ 𝒫 {𝑡 ∈ ∪ 𝑅 ∣ 〈𝑡, (2nd ‘𝑦)〉 ∈ 𝑥} ∧ 𝑢 ∈ 𝑅) ∧ ((1st ‘𝑦) ∈ 𝑢 ∧ 𝑢 ⊆ 𝑠 ∧ (𝑅 ↾t 𝑠) ∈ Comp))) → 𝑦 = 〈(1st ‘𝑦), (2nd ‘𝑦)〉) |
| 247 | | simprr1 1222 |
. . . . . . . . . . . 12
⊢
(((((𝑅 ∈
𝑛-Locally Comp ∧ 𝑆 ∈ (ran 𝑘Gen ∩ Haus)) ∧
𝑥 ∈
(𝑘Gen‘(𝑅
×t 𝑆)))
∧ 𝑦 ∈ 𝑥) ∧ ((𝑠 ∈ 𝒫 {𝑡 ∈ ∪ 𝑅 ∣ 〈𝑡, (2nd ‘𝑦)〉 ∈ 𝑥} ∧ 𝑢 ∈ 𝑅) ∧ ((1st ‘𝑦) ∈ 𝑢 ∧ 𝑢 ⊆ 𝑠 ∧ (𝑅 ↾t 𝑠) ∈ Comp))) → (1st
‘𝑦) ∈ 𝑢) |
| 248 | | sneq 4636 |
. . . . . . . . . . . . . . 15
⊢ (𝑣 = (2nd ‘𝑦) → {𝑣} = {(2nd ‘𝑦)}) |
| 249 | 248 | xpeq2d 5715 |
. . . . . . . . . . . . . 14
⊢ (𝑣 = (2nd ‘𝑦) → (𝑠 × {𝑣}) = (𝑠 × {(2nd ‘𝑦)})) |
| 250 | 249 | sseq1d 4015 |
. . . . . . . . . . . . 13
⊢ (𝑣 = (2nd ‘𝑦) → ((𝑠 × {𝑣}) ⊆ 𝑥 ↔ (𝑠 × {(2nd ‘𝑦)}) ⊆ 𝑥)) |
| 251 | 34 | adantr 480 |
. . . . . . . . . . . . 13
⊢
(((((𝑅 ∈
𝑛-Locally Comp ∧ 𝑆 ∈ (ran 𝑘Gen ∩ Haus)) ∧
𝑥 ∈
(𝑘Gen‘(𝑅
×t 𝑆)))
∧ 𝑦 ∈ 𝑥) ∧ ((𝑠 ∈ 𝒫 {𝑡 ∈ ∪ 𝑅 ∣ 〈𝑡, (2nd ‘𝑦)〉 ∈ 𝑥} ∧ 𝑢 ∈ 𝑅) ∧ ((1st ‘𝑦) ∈ 𝑢 ∧ 𝑢 ⊆ 𝑠 ∧ (𝑅 ↾t 𝑠) ∈ Comp))) → (2nd
‘𝑦) ∈ ∪ 𝑆) |
| 252 | | relxp 5703 |
. . . . . . . . . . . . . . 15
⊢ Rel
(𝑠 × {(2nd
‘𝑦)}) |
| 253 | 252 | a1i 11 |
. . . . . . . . . . . . . 14
⊢
(((((𝑅 ∈
𝑛-Locally Comp ∧ 𝑆 ∈ (ran 𝑘Gen ∩ Haus)) ∧
𝑥 ∈
(𝑘Gen‘(𝑅
×t 𝑆)))
∧ 𝑦 ∈ 𝑥) ∧ ((𝑠 ∈ 𝒫 {𝑡 ∈ ∪ 𝑅 ∣ 〈𝑡, (2nd ‘𝑦)〉 ∈ 𝑥} ∧ 𝑢 ∈ 𝑅) ∧ ((1st ‘𝑦) ∈ 𝑢 ∧ 𝑢 ⊆ 𝑠 ∧ (𝑅 ↾t 𝑠) ∈ Comp))) → Rel (𝑠 × {(2nd
‘𝑦)})) |
| 254 | | opelxp 5721 |
. . . . . . . . . . . . . . 15
⊢
(〈𝑎, 𝑏〉 ∈ (𝑠 × {(2nd
‘𝑦)}) ↔ (𝑎 ∈ 𝑠 ∧ 𝑏 ∈ {(2nd ‘𝑦)})) |
| 255 | 71 | sselda 3983 |
. . . . . . . . . . . . . . . . . 18
⊢
((((((𝑅 ∈
𝑛-Locally Comp ∧ 𝑆 ∈ (ran 𝑘Gen ∩ Haus)) ∧
𝑥 ∈
(𝑘Gen‘(𝑅
×t 𝑆)))
∧ 𝑦 ∈ 𝑥) ∧ ((𝑠 ∈ 𝒫 {𝑡 ∈ ∪ 𝑅 ∣ 〈𝑡, (2nd ‘𝑦)〉 ∈ 𝑥} ∧ 𝑢 ∈ 𝑅) ∧ ((1st ‘𝑦) ∈ 𝑢 ∧ 𝑢 ⊆ 𝑠 ∧ (𝑅 ↾t 𝑠) ∈ Comp))) ∧ 𝑎 ∈ 𝑠) → 𝑎 ∈ {𝑡 ∈ ∪ 𝑅 ∣ 〈𝑡, (2nd ‘𝑦)〉 ∈ 𝑥}) |
| 256 | | opeq1 4873 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑡 = 𝑎 → 〈𝑡, (2nd ‘𝑦)〉 = 〈𝑎, (2nd ‘𝑦)〉) |
| 257 | 256 | eleq1d 2826 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑡 = 𝑎 → (〈𝑡, (2nd ‘𝑦)〉 ∈ 𝑥 ↔ 〈𝑎, (2nd ‘𝑦)〉 ∈ 𝑥)) |
| 258 | 257 | elrab 3692 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑎 ∈ {𝑡 ∈ ∪ 𝑅 ∣ 〈𝑡, (2nd ‘𝑦)〉 ∈ 𝑥} ↔ (𝑎 ∈ ∪ 𝑅 ∧ 〈𝑎, (2nd ‘𝑦)〉 ∈ 𝑥)) |
| 259 | 258 | simprbi 496 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑎 ∈ {𝑡 ∈ ∪ 𝑅 ∣ 〈𝑡, (2nd ‘𝑦)〉 ∈ 𝑥} → 〈𝑎, (2nd ‘𝑦)〉 ∈ 𝑥) |
| 260 | 255, 259 | syl 17 |
. . . . . . . . . . . . . . . . 17
⊢
((((((𝑅 ∈
𝑛-Locally Comp ∧ 𝑆 ∈ (ran 𝑘Gen ∩ Haus)) ∧
𝑥 ∈
(𝑘Gen‘(𝑅
×t 𝑆)))
∧ 𝑦 ∈ 𝑥) ∧ ((𝑠 ∈ 𝒫 {𝑡 ∈ ∪ 𝑅 ∣ 〈𝑡, (2nd ‘𝑦)〉 ∈ 𝑥} ∧ 𝑢 ∈ 𝑅) ∧ ((1st ‘𝑦) ∈ 𝑢 ∧ 𝑢 ⊆ 𝑠 ∧ (𝑅 ↾t 𝑠) ∈ Comp))) ∧ 𝑎 ∈ 𝑠) → 〈𝑎, (2nd ‘𝑦)〉 ∈ 𝑥) |
| 261 | | elsni 4643 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑏 ∈ {(2nd
‘𝑦)} → 𝑏 = (2nd ‘𝑦)) |
| 262 | 261 | opeq2d 4880 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑏 ∈ {(2nd
‘𝑦)} →
〈𝑎, 𝑏〉 = 〈𝑎, (2nd ‘𝑦)〉) |
| 263 | 262 | eleq1d 2826 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑏 ∈ {(2nd
‘𝑦)} →
(〈𝑎, 𝑏〉 ∈ 𝑥 ↔ 〈𝑎, (2nd ‘𝑦)〉 ∈ 𝑥)) |
| 264 | 260, 263 | syl5ibrcom 247 |
. . . . . . . . . . . . . . . 16
⊢
((((((𝑅 ∈
𝑛-Locally Comp ∧ 𝑆 ∈ (ran 𝑘Gen ∩ Haus)) ∧
𝑥 ∈
(𝑘Gen‘(𝑅
×t 𝑆)))
∧ 𝑦 ∈ 𝑥) ∧ ((𝑠 ∈ 𝒫 {𝑡 ∈ ∪ 𝑅 ∣ 〈𝑡, (2nd ‘𝑦)〉 ∈ 𝑥} ∧ 𝑢 ∈ 𝑅) ∧ ((1st ‘𝑦) ∈ 𝑢 ∧ 𝑢 ⊆ 𝑠 ∧ (𝑅 ↾t 𝑠) ∈ Comp))) ∧ 𝑎 ∈ 𝑠) → (𝑏 ∈ {(2nd ‘𝑦)} → 〈𝑎, 𝑏〉 ∈ 𝑥)) |
| 265 | 264 | expimpd 453 |
. . . . . . . . . . . . . . 15
⊢
(((((𝑅 ∈
𝑛-Locally Comp ∧ 𝑆 ∈ (ran 𝑘Gen ∩ Haus)) ∧
𝑥 ∈
(𝑘Gen‘(𝑅
×t 𝑆)))
∧ 𝑦 ∈ 𝑥) ∧ ((𝑠 ∈ 𝒫 {𝑡 ∈ ∪ 𝑅 ∣ 〈𝑡, (2nd ‘𝑦)〉 ∈ 𝑥} ∧ 𝑢 ∈ 𝑅) ∧ ((1st ‘𝑦) ∈ 𝑢 ∧ 𝑢 ⊆ 𝑠 ∧ (𝑅 ↾t 𝑠) ∈ Comp))) → ((𝑎 ∈ 𝑠 ∧ 𝑏 ∈ {(2nd ‘𝑦)}) → 〈𝑎, 𝑏〉 ∈ 𝑥)) |
| 266 | 254, 265 | biimtrid 242 |
. . . . . . . . . . . . . 14
⊢
(((((𝑅 ∈
𝑛-Locally Comp ∧ 𝑆 ∈ (ran 𝑘Gen ∩ Haus)) ∧
𝑥 ∈
(𝑘Gen‘(𝑅
×t 𝑆)))
∧ 𝑦 ∈ 𝑥) ∧ ((𝑠 ∈ 𝒫 {𝑡 ∈ ∪ 𝑅 ∣ 〈𝑡, (2nd ‘𝑦)〉 ∈ 𝑥} ∧ 𝑢 ∈ 𝑅) ∧ ((1st ‘𝑦) ∈ 𝑢 ∧ 𝑢 ⊆ 𝑠 ∧ (𝑅 ↾t 𝑠) ∈ Comp))) → (〈𝑎, 𝑏〉 ∈ (𝑠 × {(2nd ‘𝑦)}) → 〈𝑎, 𝑏〉 ∈ 𝑥)) |
| 267 | 253, 266 | relssdv 5798 |
. . . . . . . . . . . . 13
⊢
(((((𝑅 ∈
𝑛-Locally Comp ∧ 𝑆 ∈ (ran 𝑘Gen ∩ Haus)) ∧
𝑥 ∈
(𝑘Gen‘(𝑅
×t 𝑆)))
∧ 𝑦 ∈ 𝑥) ∧ ((𝑠 ∈ 𝒫 {𝑡 ∈ ∪ 𝑅 ∣ 〈𝑡, (2nd ‘𝑦)〉 ∈ 𝑥} ∧ 𝑢 ∈ 𝑅) ∧ ((1st ‘𝑦) ∈ 𝑢 ∧ 𝑢 ⊆ 𝑠 ∧ (𝑅 ↾t 𝑠) ∈ Comp))) → (𝑠 × {(2nd ‘𝑦)}) ⊆ 𝑥) |
| 268 | 250, 251,
267 | elrabd 3694 |
. . . . . . . . . . . 12
⊢
(((((𝑅 ∈
𝑛-Locally Comp ∧ 𝑆 ∈ (ran 𝑘Gen ∩ Haus)) ∧
𝑥 ∈
(𝑘Gen‘(𝑅
×t 𝑆)))
∧ 𝑦 ∈ 𝑥) ∧ ((𝑠 ∈ 𝒫 {𝑡 ∈ ∪ 𝑅 ∣ 〈𝑡, (2nd ‘𝑦)〉 ∈ 𝑥} ∧ 𝑢 ∈ 𝑅) ∧ ((1st ‘𝑦) ∈ 𝑢 ∧ 𝑢 ⊆ 𝑠 ∧ (𝑅 ↾t 𝑠) ∈ Comp))) → (2nd
‘𝑦) ∈ {𝑣 ∈ ∪ 𝑆
∣ (𝑠 × {𝑣}) ⊆ 𝑥}) |
| 269 | 247, 268 | opelxpd 5724 |
. . . . . . . . . . 11
⊢
(((((𝑅 ∈
𝑛-Locally Comp ∧ 𝑆 ∈ (ran 𝑘Gen ∩ Haus)) ∧
𝑥 ∈
(𝑘Gen‘(𝑅
×t 𝑆)))
∧ 𝑦 ∈ 𝑥) ∧ ((𝑠 ∈ 𝒫 {𝑡 ∈ ∪ 𝑅 ∣ 〈𝑡, (2nd ‘𝑦)〉 ∈ 𝑥} ∧ 𝑢 ∈ 𝑅) ∧ ((1st ‘𝑦) ∈ 𝑢 ∧ 𝑢 ⊆ 𝑠 ∧ (𝑅 ↾t 𝑠) ∈ Comp))) → 〈(1st
‘𝑦), (2nd
‘𝑦)〉 ∈
(𝑢 × {𝑣 ∈ ∪ 𝑆
∣ (𝑠 × {𝑣}) ⊆ 𝑥})) |
| 270 | 246, 269 | eqeltrd 2841 |
. . . . . . . . . 10
⊢
(((((𝑅 ∈
𝑛-Locally Comp ∧ 𝑆 ∈ (ran 𝑘Gen ∩ Haus)) ∧
𝑥 ∈
(𝑘Gen‘(𝑅
×t 𝑆)))
∧ 𝑦 ∈ 𝑥) ∧ ((𝑠 ∈ 𝒫 {𝑡 ∈ ∪ 𝑅 ∣ 〈𝑡, (2nd ‘𝑦)〉 ∈ 𝑥} ∧ 𝑢 ∈ 𝑅) ∧ ((1st ‘𝑦) ∈ 𝑢 ∧ 𝑢 ⊆ 𝑠 ∧ (𝑅 ↾t 𝑠) ∈ Comp))) → 𝑦 ∈ (𝑢 × {𝑣 ∈ ∪ 𝑆 ∣ (𝑠 × {𝑣}) ⊆ 𝑥})) |
| 271 | | relxp 5703 |
. . . . . . . . . . . 12
⊢ Rel
(𝑢 × {𝑣 ∈ ∪ 𝑆
∣ (𝑠 × {𝑣}) ⊆ 𝑥}) |
| 272 | 271 | a1i 11 |
. . . . . . . . . . 11
⊢
(((((𝑅 ∈
𝑛-Locally Comp ∧ 𝑆 ∈ (ran 𝑘Gen ∩ Haus)) ∧
𝑥 ∈
(𝑘Gen‘(𝑅
×t 𝑆)))
∧ 𝑦 ∈ 𝑥) ∧ ((𝑠 ∈ 𝒫 {𝑡 ∈ ∪ 𝑅 ∣ 〈𝑡, (2nd ‘𝑦)〉 ∈ 𝑥} ∧ 𝑢 ∈ 𝑅) ∧ ((1st ‘𝑦) ∈ 𝑢 ∧ 𝑢 ⊆ 𝑠 ∧ (𝑅 ↾t 𝑠) ∈ Comp))) → Rel (𝑢 × {𝑣 ∈ ∪ 𝑆 ∣ (𝑠 × {𝑣}) ⊆ 𝑥})) |
| 273 | | opelxp 5721 |
. . . . . . . . . . . 12
⊢
(〈𝑎, 𝑏〉 ∈ (𝑢 × {𝑣 ∈ ∪ 𝑆 ∣ (𝑠 × {𝑣}) ⊆ 𝑥}) ↔ (𝑎 ∈ 𝑢 ∧ 𝑏 ∈ {𝑣 ∈ ∪ 𝑆 ∣ (𝑠 × {𝑣}) ⊆ 𝑥})) |
| 274 | 126 | elrab 3692 |
. . . . . . . . . . . . . . 15
⊢ (𝑏 ∈ {𝑣 ∈ ∪ 𝑆 ∣ (𝑠 × {𝑣}) ⊆ 𝑥} ↔ (𝑏 ∈ ∪ 𝑆 ∧ (𝑠 × {𝑏}) ⊆ 𝑥)) |
| 275 | 274 | simprbi 496 |
. . . . . . . . . . . . . 14
⊢ (𝑏 ∈ {𝑣 ∈ ∪ 𝑆 ∣ (𝑠 × {𝑣}) ⊆ 𝑥} → (𝑠 × {𝑏}) ⊆ 𝑥) |
| 276 | | simprr2 1223 |
. . . . . . . . . . . . . . . 16
⊢
(((((𝑅 ∈
𝑛-Locally Comp ∧ 𝑆 ∈ (ran 𝑘Gen ∩ Haus)) ∧
𝑥 ∈
(𝑘Gen‘(𝑅
×t 𝑆)))
∧ 𝑦 ∈ 𝑥) ∧ ((𝑠 ∈ 𝒫 {𝑡 ∈ ∪ 𝑅 ∣ 〈𝑡, (2nd ‘𝑦)〉 ∈ 𝑥} ∧ 𝑢 ∈ 𝑅) ∧ ((1st ‘𝑦) ∈ 𝑢 ∧ 𝑢 ⊆ 𝑠 ∧ (𝑅 ↾t 𝑠) ∈ Comp))) → 𝑢 ⊆ 𝑠) |
| 277 | 276 | sselda 3983 |
. . . . . . . . . . . . . . 15
⊢
((((((𝑅 ∈
𝑛-Locally Comp ∧ 𝑆 ∈ (ran 𝑘Gen ∩ Haus)) ∧
𝑥 ∈
(𝑘Gen‘(𝑅
×t 𝑆)))
∧ 𝑦 ∈ 𝑥) ∧ ((𝑠 ∈ 𝒫 {𝑡 ∈ ∪ 𝑅 ∣ 〈𝑡, (2nd ‘𝑦)〉 ∈ 𝑥} ∧ 𝑢 ∈ 𝑅) ∧ ((1st ‘𝑦) ∈ 𝑢 ∧ 𝑢 ⊆ 𝑠 ∧ (𝑅 ↾t 𝑠) ∈ Comp))) ∧ 𝑎 ∈ 𝑢) → 𝑎 ∈ 𝑠) |
| 278 | | vsnid 4663 |
. . . . . . . . . . . . . . 15
⊢ 𝑏 ∈ {𝑏} |
| 279 | | opelxpi 5722 |
. . . . . . . . . . . . . . 15
⊢ ((𝑎 ∈ 𝑠 ∧ 𝑏 ∈ {𝑏}) → 〈𝑎, 𝑏〉 ∈ (𝑠 × {𝑏})) |
| 280 | 277, 278,
279 | sylancl 586 |
. . . . . . . . . . . . . 14
⊢
((((((𝑅 ∈
𝑛-Locally Comp ∧ 𝑆 ∈ (ran 𝑘Gen ∩ Haus)) ∧
𝑥 ∈
(𝑘Gen‘(𝑅
×t 𝑆)))
∧ 𝑦 ∈ 𝑥) ∧ ((𝑠 ∈ 𝒫 {𝑡 ∈ ∪ 𝑅 ∣ 〈𝑡, (2nd ‘𝑦)〉 ∈ 𝑥} ∧ 𝑢 ∈ 𝑅) ∧ ((1st ‘𝑦) ∈ 𝑢 ∧ 𝑢 ⊆ 𝑠 ∧ (𝑅 ↾t 𝑠) ∈ Comp))) ∧ 𝑎 ∈ 𝑢) → 〈𝑎, 𝑏〉 ∈ (𝑠 × {𝑏})) |
| 281 | | ssel 3977 |
. . . . . . . . . . . . . 14
⊢ ((𝑠 × {𝑏}) ⊆ 𝑥 → (〈𝑎, 𝑏〉 ∈ (𝑠 × {𝑏}) → 〈𝑎, 𝑏〉 ∈ 𝑥)) |
| 282 | 275, 280,
281 | syl2imc 41 |
. . . . . . . . . . . . 13
⊢
((((((𝑅 ∈
𝑛-Locally Comp ∧ 𝑆 ∈ (ran 𝑘Gen ∩ Haus)) ∧
𝑥 ∈
(𝑘Gen‘(𝑅
×t 𝑆)))
∧ 𝑦 ∈ 𝑥) ∧ ((𝑠 ∈ 𝒫 {𝑡 ∈ ∪ 𝑅 ∣ 〈𝑡, (2nd ‘𝑦)〉 ∈ 𝑥} ∧ 𝑢 ∈ 𝑅) ∧ ((1st ‘𝑦) ∈ 𝑢 ∧ 𝑢 ⊆ 𝑠 ∧ (𝑅 ↾t 𝑠) ∈ Comp))) ∧ 𝑎 ∈ 𝑢) → (𝑏 ∈ {𝑣 ∈ ∪ 𝑆 ∣ (𝑠 × {𝑣}) ⊆ 𝑥} → 〈𝑎, 𝑏〉 ∈ 𝑥)) |
| 283 | 282 | expimpd 453 |
. . . . . . . . . . . 12
⊢
(((((𝑅 ∈
𝑛-Locally Comp ∧ 𝑆 ∈ (ran 𝑘Gen ∩ Haus)) ∧
𝑥 ∈
(𝑘Gen‘(𝑅
×t 𝑆)))
∧ 𝑦 ∈ 𝑥) ∧ ((𝑠 ∈ 𝒫 {𝑡 ∈ ∪ 𝑅 ∣ 〈𝑡, (2nd ‘𝑦)〉 ∈ 𝑥} ∧ 𝑢 ∈ 𝑅) ∧ ((1st ‘𝑦) ∈ 𝑢 ∧ 𝑢 ⊆ 𝑠 ∧ (𝑅 ↾t 𝑠) ∈ Comp))) → ((𝑎 ∈ 𝑢 ∧ 𝑏 ∈ {𝑣 ∈ ∪ 𝑆 ∣ (𝑠 × {𝑣}) ⊆ 𝑥}) → 〈𝑎, 𝑏〉 ∈ 𝑥)) |
| 284 | 273, 283 | biimtrid 242 |
. . . . . . . . . . 11
⊢
(((((𝑅 ∈
𝑛-Locally Comp ∧ 𝑆 ∈ (ran 𝑘Gen ∩ Haus)) ∧
𝑥 ∈
(𝑘Gen‘(𝑅
×t 𝑆)))
∧ 𝑦 ∈ 𝑥) ∧ ((𝑠 ∈ 𝒫 {𝑡 ∈ ∪ 𝑅 ∣ 〈𝑡, (2nd ‘𝑦)〉 ∈ 𝑥} ∧ 𝑢 ∈ 𝑅) ∧ ((1st ‘𝑦) ∈ 𝑢 ∧ 𝑢 ⊆ 𝑠 ∧ (𝑅 ↾t 𝑠) ∈ Comp))) → (〈𝑎, 𝑏〉 ∈ (𝑢 × {𝑣 ∈ ∪ 𝑆 ∣ (𝑠 × {𝑣}) ⊆ 𝑥}) → 〈𝑎, 𝑏〉 ∈ 𝑥)) |
| 285 | 272, 284 | relssdv 5798 |
. . . . . . . . . 10
⊢
(((((𝑅 ∈
𝑛-Locally Comp ∧ 𝑆 ∈ (ran 𝑘Gen ∩ Haus)) ∧
𝑥 ∈
(𝑘Gen‘(𝑅
×t 𝑆)))
∧ 𝑦 ∈ 𝑥) ∧ ((𝑠 ∈ 𝒫 {𝑡 ∈ ∪ 𝑅 ∣ 〈𝑡, (2nd ‘𝑦)〉 ∈ 𝑥} ∧ 𝑢 ∈ 𝑅) ∧ ((1st ‘𝑦) ∈ 𝑢 ∧ 𝑢 ⊆ 𝑠 ∧ (𝑅 ↾t 𝑠) ∈ Comp))) → (𝑢 × {𝑣 ∈ ∪ 𝑆 ∣ (𝑠 × {𝑣}) ⊆ 𝑥}) ⊆ 𝑥) |
| 286 | | eleq2 2830 |
. . . . . . . . . . . 12
⊢ (𝑡 = (𝑢 × {𝑣 ∈ ∪ 𝑆 ∣ (𝑠 × {𝑣}) ⊆ 𝑥}) → (𝑦 ∈ 𝑡 ↔ 𝑦 ∈ (𝑢 × {𝑣 ∈ ∪ 𝑆 ∣ (𝑠 × {𝑣}) ⊆ 𝑥}))) |
| 287 | | sseq1 4009 |
. . . . . . . . . . . 12
⊢ (𝑡 = (𝑢 × {𝑣 ∈ ∪ 𝑆 ∣ (𝑠 × {𝑣}) ⊆ 𝑥}) → (𝑡 ⊆ 𝑥 ↔ (𝑢 × {𝑣 ∈ ∪ 𝑆 ∣ (𝑠 × {𝑣}) ⊆ 𝑥}) ⊆ 𝑥)) |
| 288 | 286, 287 | anbi12d 632 |
. . . . . . . . . . 11
⊢ (𝑡 = (𝑢 × {𝑣 ∈ ∪ 𝑆 ∣ (𝑠 × {𝑣}) ⊆ 𝑥}) → ((𝑦 ∈ 𝑡 ∧ 𝑡 ⊆ 𝑥) ↔ (𝑦 ∈ (𝑢 × {𝑣 ∈ ∪ 𝑆 ∣ (𝑠 × {𝑣}) ⊆ 𝑥}) ∧ (𝑢 × {𝑣 ∈ ∪ 𝑆 ∣ (𝑠 × {𝑣}) ⊆ 𝑥}) ⊆ 𝑥))) |
| 289 | 288 | rspcev 3622 |
. . . . . . . . . 10
⊢ (((𝑢 × {𝑣 ∈ ∪ 𝑆 ∣ (𝑠 × {𝑣}) ⊆ 𝑥}) ∈ (𝑅 ×t 𝑆) ∧ (𝑦 ∈ (𝑢 × {𝑣 ∈ ∪ 𝑆 ∣ (𝑠 × {𝑣}) ⊆ 𝑥}) ∧ (𝑢 × {𝑣 ∈ ∪ 𝑆 ∣ (𝑠 × {𝑣}) ⊆ 𝑥}) ⊆ 𝑥)) → ∃𝑡 ∈ (𝑅 ×t 𝑆)(𝑦 ∈ 𝑡 ∧ 𝑡 ⊆ 𝑥)) |
| 290 | 245, 270,
285, 289 | syl12anc 837 |
. . . . . . . . 9
⊢
(((((𝑅 ∈
𝑛-Locally Comp ∧ 𝑆 ∈ (ran 𝑘Gen ∩ Haus)) ∧
𝑥 ∈
(𝑘Gen‘(𝑅
×t 𝑆)))
∧ 𝑦 ∈ 𝑥) ∧ ((𝑠 ∈ 𝒫 {𝑡 ∈ ∪ 𝑅 ∣ 〈𝑡, (2nd ‘𝑦)〉 ∈ 𝑥} ∧ 𝑢 ∈ 𝑅) ∧ ((1st ‘𝑦) ∈ 𝑢 ∧ 𝑢 ⊆ 𝑠 ∧ (𝑅 ↾t 𝑠) ∈ Comp))) → ∃𝑡 ∈ (𝑅 ×t 𝑆)(𝑦 ∈ 𝑡 ∧ 𝑡 ⊆ 𝑥)) |
| 291 | 290 | expr 456 |
. . . . . . . 8
⊢
(((((𝑅 ∈
𝑛-Locally Comp ∧ 𝑆 ∈ (ran 𝑘Gen ∩ Haus)) ∧
𝑥 ∈
(𝑘Gen‘(𝑅
×t 𝑆)))
∧ 𝑦 ∈ 𝑥) ∧ (𝑠 ∈ 𝒫 {𝑡 ∈ ∪ 𝑅 ∣ 〈𝑡, (2nd ‘𝑦)〉 ∈ 𝑥} ∧ 𝑢 ∈ 𝑅)) → (((1st ‘𝑦) ∈ 𝑢 ∧ 𝑢 ⊆ 𝑠 ∧ (𝑅 ↾t 𝑠) ∈ Comp) → ∃𝑡 ∈ (𝑅 ×t 𝑆)(𝑦 ∈ 𝑡 ∧ 𝑡 ⊆ 𝑥))) |
| 292 | 291 | rexlimdvva 3213 |
. . . . . . 7
⊢ ((((𝑅 ∈ 𝑛-Locally Comp
∧ 𝑆 ∈ (ran
𝑘Gen ∩ Haus)) ∧ 𝑥 ∈ (𝑘Gen‘(𝑅 ×t 𝑆))) ∧ 𝑦 ∈ 𝑥) → (∃𝑠 ∈ 𝒫 {𝑡 ∈ ∪ 𝑅 ∣ 〈𝑡, (2nd ‘𝑦)〉 ∈ 𝑥}∃𝑢 ∈ 𝑅 ((1st ‘𝑦) ∈ 𝑢 ∧ 𝑢 ⊆ 𝑠 ∧ (𝑅 ↾t 𝑠) ∈ Comp) → ∃𝑡 ∈ (𝑅 ×t 𝑆)(𝑦 ∈ 𝑡 ∧ 𝑡 ⊆ 𝑥))) |
| 293 | 63, 292 | mpd 15 |
. . . . . 6
⊢ ((((𝑅 ∈ 𝑛-Locally Comp
∧ 𝑆 ∈ (ran
𝑘Gen ∩ Haus)) ∧ 𝑥 ∈ (𝑘Gen‘(𝑅 ×t 𝑆))) ∧ 𝑦 ∈ 𝑥) → ∃𝑡 ∈ (𝑅 ×t 𝑆)(𝑦 ∈ 𝑡 ∧ 𝑡 ⊆ 𝑥)) |
| 294 | 293 | ralrimiva 3146 |
. . . . 5
⊢ (((𝑅 ∈ 𝑛-Locally Comp
∧ 𝑆 ∈ (ran
𝑘Gen ∩ Haus)) ∧ 𝑥 ∈ (𝑘Gen‘(𝑅 ×t 𝑆))) → ∀𝑦 ∈ 𝑥 ∃𝑡 ∈ (𝑅 ×t 𝑆)(𝑦 ∈ 𝑡 ∧ 𝑡 ⊆ 𝑥)) |
| 295 | 6 | adantr 480 |
. . . . . 6
⊢ (((𝑅 ∈ 𝑛-Locally Comp
∧ 𝑆 ∈ (ran
𝑘Gen ∩ Haus)) ∧ 𝑥 ∈ (𝑘Gen‘(𝑅 ×t 𝑆))) → (𝑅 ×t 𝑆) ∈ Top) |
| 296 | | eltop2 22982 |
. . . . . 6
⊢ ((𝑅 ×t 𝑆) ∈ Top → (𝑥 ∈ (𝑅 ×t 𝑆) ↔ ∀𝑦 ∈ 𝑥 ∃𝑡 ∈ (𝑅 ×t 𝑆)(𝑦 ∈ 𝑡 ∧ 𝑡 ⊆ 𝑥))) |
| 297 | 295, 296 | syl 17 |
. . . . 5
⊢ (((𝑅 ∈ 𝑛-Locally Comp
∧ 𝑆 ∈ (ran
𝑘Gen ∩ Haus)) ∧ 𝑥 ∈ (𝑘Gen‘(𝑅 ×t 𝑆))) → (𝑥 ∈ (𝑅 ×t 𝑆) ↔ ∀𝑦 ∈ 𝑥 ∃𝑡 ∈ (𝑅 ×t 𝑆)(𝑦 ∈ 𝑡 ∧ 𝑡 ⊆ 𝑥))) |
| 298 | 294, 297 | mpbird 257 |
. . . 4
⊢ (((𝑅 ∈ 𝑛-Locally Comp
∧ 𝑆 ∈ (ran
𝑘Gen ∩ Haus)) ∧ 𝑥 ∈ (𝑘Gen‘(𝑅 ×t 𝑆))) → 𝑥 ∈ (𝑅 ×t 𝑆)) |
| 299 | 298 | ex 412 |
. . 3
⊢ ((𝑅 ∈ 𝑛-Locally Comp
∧ 𝑆 ∈ (ran
𝑘Gen ∩ Haus)) → (𝑥 ∈ (𝑘Gen‘(𝑅 ×t 𝑆)) → 𝑥 ∈ (𝑅 ×t 𝑆))) |
| 300 | 299 | ssrdv 3989 |
. 2
⊢ ((𝑅 ∈ 𝑛-Locally Comp
∧ 𝑆 ∈ (ran
𝑘Gen ∩ Haus)) → (𝑘Gen‘(𝑅 ×t 𝑆)) ⊆ (𝑅 ×t 𝑆)) |
| 301 | | iskgen2 23556 |
. 2
⊢ ((𝑅 ×t 𝑆) ∈ ran 𝑘Gen ↔
((𝑅 ×t
𝑆) ∈ Top ∧
(𝑘Gen‘(𝑅
×t 𝑆))
⊆ (𝑅
×t 𝑆))) |
| 302 | 6, 300, 301 | sylanbrc 583 |
1
⊢ ((𝑅 ∈ 𝑛-Locally Comp
∧ 𝑆 ∈ (ran
𝑘Gen ∩ Haus)) → (𝑅 ×t 𝑆) ∈ ran 𝑘Gen) |