Step | Hyp | Ref
| Expression |
1 | | conntop 22314 |
. . 3
⊢ (𝑅 ∈ Conn → 𝑅 ∈ Top) |
2 | | conntop 22314 |
. . 3
⊢ (𝑆 ∈ Conn → 𝑆 ∈ Top) |
3 | | txtop 22466 |
. . 3
⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑅 ×t 𝑆) ∈ Top) |
4 | 1, 2, 3 | syl2an 599 |
. 2
⊢ ((𝑅 ∈ Conn ∧ 𝑆 ∈ Conn) → (𝑅 ×t 𝑆) ∈ Top) |
5 | | neq0 4260 |
. . . . . . 7
⊢ (¬
𝑥 = ∅ ↔
∃𝑧 𝑧 ∈ 𝑥) |
6 | | simplr 769 |
. . . . . . . . . . . 12
⊢ ((((𝑅 ∈ Conn ∧ 𝑆 ∈ Conn) ∧ 𝑥 ∈ ((𝑅 ×t 𝑆) ∩ (Clsd‘(𝑅 ×t 𝑆)))) ∧ 𝑧 ∈ 𝑥) → 𝑥 ∈ ((𝑅 ×t 𝑆) ∩ (Clsd‘(𝑅 ×t 𝑆)))) |
7 | 6 | elin1d 4112 |
. . . . . . . . . . 11
⊢ ((((𝑅 ∈ Conn ∧ 𝑆 ∈ Conn) ∧ 𝑥 ∈ ((𝑅 ×t 𝑆) ∩ (Clsd‘(𝑅 ×t 𝑆)))) ∧ 𝑧 ∈ 𝑥) → 𝑥 ∈ (𝑅 ×t 𝑆)) |
8 | | elssuni 4851 |
. . . . . . . . . . 11
⊢ (𝑥 ∈ (𝑅 ×t 𝑆) → 𝑥 ⊆ ∪ (𝑅 ×t 𝑆)) |
9 | 7, 8 | syl 17 |
. . . . . . . . . 10
⊢ ((((𝑅 ∈ Conn ∧ 𝑆 ∈ Conn) ∧ 𝑥 ∈ ((𝑅 ×t 𝑆) ∩ (Clsd‘(𝑅 ×t 𝑆)))) ∧ 𝑧 ∈ 𝑥) → 𝑥 ⊆ ∪ (𝑅 ×t 𝑆)) |
10 | | simprr 773 |
. . . . . . . . . . . . . . 15
⊢ ((((𝑅 ∈ Conn ∧ 𝑆 ∈ Conn) ∧ 𝑥 ∈ ((𝑅 ×t 𝑆) ∩ (Clsd‘(𝑅 ×t 𝑆)))) ∧ (𝑧 ∈ 𝑥 ∧ 𝑤 ∈ ∪ (𝑅 ×t 𝑆))) → 𝑤 ∈ ∪ (𝑅 ×t 𝑆)) |
11 | | simplll 775 |
. . . . . . . . . . . . . . . . 17
⊢ ((((𝑅 ∈ Conn ∧ 𝑆 ∈ Conn) ∧ 𝑥 ∈ ((𝑅 ×t 𝑆) ∩ (Clsd‘(𝑅 ×t 𝑆)))) ∧ (𝑧 ∈ 𝑥 ∧ 𝑤 ∈ ∪ (𝑅 ×t 𝑆))) → 𝑅 ∈ Conn) |
12 | 11, 1 | syl 17 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝑅 ∈ Conn ∧ 𝑆 ∈ Conn) ∧ 𝑥 ∈ ((𝑅 ×t 𝑆) ∩ (Clsd‘(𝑅 ×t 𝑆)))) ∧ (𝑧 ∈ 𝑥 ∧ 𝑤 ∈ ∪ (𝑅 ×t 𝑆))) → 𝑅 ∈ Top) |
13 | | simpllr 776 |
. . . . . . . . . . . . . . . . 17
⊢ ((((𝑅 ∈ Conn ∧ 𝑆 ∈ Conn) ∧ 𝑥 ∈ ((𝑅 ×t 𝑆) ∩ (Clsd‘(𝑅 ×t 𝑆)))) ∧ (𝑧 ∈ 𝑥 ∧ 𝑤 ∈ ∪ (𝑅 ×t 𝑆))) → 𝑆 ∈ Conn) |
14 | 13, 2 | syl 17 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝑅 ∈ Conn ∧ 𝑆 ∈ Conn) ∧ 𝑥 ∈ ((𝑅 ×t 𝑆) ∩ (Clsd‘(𝑅 ×t 𝑆)))) ∧ (𝑧 ∈ 𝑥 ∧ 𝑤 ∈ ∪ (𝑅 ×t 𝑆))) → 𝑆 ∈ Top) |
15 | | eqid 2737 |
. . . . . . . . . . . . . . . . 17
⊢ ∪ 𝑅 =
∪ 𝑅 |
16 | | eqid 2737 |
. . . . . . . . . . . . . . . . 17
⊢ ∪ 𝑆 =
∪ 𝑆 |
17 | 15, 16 | txuni 22489 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (∪ 𝑅
× ∪ 𝑆) = ∪ (𝑅 ×t 𝑆)) |
18 | 12, 14, 17 | syl2anc 587 |
. . . . . . . . . . . . . . 15
⊢ ((((𝑅 ∈ Conn ∧ 𝑆 ∈ Conn) ∧ 𝑥 ∈ ((𝑅 ×t 𝑆) ∩ (Clsd‘(𝑅 ×t 𝑆)))) ∧ (𝑧 ∈ 𝑥 ∧ 𝑤 ∈ ∪ (𝑅 ×t 𝑆))) → (∪ 𝑅
× ∪ 𝑆) = ∪ (𝑅 ×t 𝑆)) |
19 | 10, 18 | eleqtrrd 2841 |
. . . . . . . . . . . . . 14
⊢ ((((𝑅 ∈ Conn ∧ 𝑆 ∈ Conn) ∧ 𝑥 ∈ ((𝑅 ×t 𝑆) ∩ (Clsd‘(𝑅 ×t 𝑆)))) ∧ (𝑧 ∈ 𝑥 ∧ 𝑤 ∈ ∪ (𝑅 ×t 𝑆))) → 𝑤 ∈ (∪ 𝑅 × ∪ 𝑆)) |
20 | | 1st2nd2 7800 |
. . . . . . . . . . . . . 14
⊢ (𝑤 ∈ (∪ 𝑅
× ∪ 𝑆) → 𝑤 = 〈(1st ‘𝑤), (2nd ‘𝑤)〉) |
21 | 19, 20 | syl 17 |
. . . . . . . . . . . . 13
⊢ ((((𝑅 ∈ Conn ∧ 𝑆 ∈ Conn) ∧ 𝑥 ∈ ((𝑅 ×t 𝑆) ∩ (Clsd‘(𝑅 ×t 𝑆)))) ∧ (𝑧 ∈ 𝑥 ∧ 𝑤 ∈ ∪ (𝑅 ×t 𝑆))) → 𝑤 = 〈(1st ‘𝑤), (2nd ‘𝑤)〉) |
22 | | xp2nd 7794 |
. . . . . . . . . . . . . . . 16
⊢ (𝑤 ∈ (∪ 𝑅
× ∪ 𝑆) → (2nd ‘𝑤) ∈ ∪ 𝑆) |
23 | 19, 22 | syl 17 |
. . . . . . . . . . . . . . 15
⊢ ((((𝑅 ∈ Conn ∧ 𝑆 ∈ Conn) ∧ 𝑥 ∈ ((𝑅 ×t 𝑆) ∩ (Clsd‘(𝑅 ×t 𝑆)))) ∧ (𝑧 ∈ 𝑥 ∧ 𝑤 ∈ ∪ (𝑅 ×t 𝑆))) → (2nd
‘𝑤) ∈ ∪ 𝑆) |
24 | | eqid 2737 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑎 ∈ ∪ 𝑆
↦ 〈(1st ‘𝑤), 𝑎〉) = (𝑎 ∈ ∪ 𝑆 ↦ 〈(1st
‘𝑤), 𝑎〉) |
25 | 24 | mptpreima 6101 |
. . . . . . . . . . . . . . . . 17
⊢ (◡(𝑎 ∈ ∪ 𝑆 ↦ 〈(1st
‘𝑤), 𝑎〉) “ 𝑥) = {𝑎 ∈ ∪ 𝑆 ∣ 〈(1st
‘𝑤), 𝑎〉 ∈ 𝑥} |
26 | | toptopon2 21815 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑆 ∈ Top ↔ 𝑆 ∈ (TopOn‘∪ 𝑆)) |
27 | 14, 26 | sylib 221 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((((𝑅 ∈ Conn ∧ 𝑆 ∈ Conn) ∧ 𝑥 ∈ ((𝑅 ×t 𝑆) ∩ (Clsd‘(𝑅 ×t 𝑆)))) ∧ (𝑧 ∈ 𝑥 ∧ 𝑤 ∈ ∪ (𝑅 ×t 𝑆))) → 𝑆 ∈ (TopOn‘∪ 𝑆)) |
28 | | toptopon2 21815 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑅 ∈ Top ↔ 𝑅 ∈ (TopOn‘∪ 𝑅)) |
29 | 12, 28 | sylib 221 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((((𝑅 ∈ Conn ∧ 𝑆 ∈ Conn) ∧ 𝑥 ∈ ((𝑅 ×t 𝑆) ∩ (Clsd‘(𝑅 ×t 𝑆)))) ∧ (𝑧 ∈ 𝑥 ∧ 𝑤 ∈ ∪ (𝑅 ×t 𝑆))) → 𝑅 ∈ (TopOn‘∪ 𝑅)) |
30 | | xp1st 7793 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑤 ∈ (∪ 𝑅
× ∪ 𝑆) → (1st ‘𝑤) ∈ ∪ 𝑅) |
31 | 19, 30 | syl 17 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((((𝑅 ∈ Conn ∧ 𝑆 ∈ Conn) ∧ 𝑥 ∈ ((𝑅 ×t 𝑆) ∩ (Clsd‘(𝑅 ×t 𝑆)))) ∧ (𝑧 ∈ 𝑥 ∧ 𝑤 ∈ ∪ (𝑅 ×t 𝑆))) → (1st
‘𝑤) ∈ ∪ 𝑅) |
32 | 27, 29, 31 | cnmptc 22559 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((((𝑅 ∈ Conn ∧ 𝑆 ∈ Conn) ∧ 𝑥 ∈ ((𝑅 ×t 𝑆) ∩ (Clsd‘(𝑅 ×t 𝑆)))) ∧ (𝑧 ∈ 𝑥 ∧ 𝑤 ∈ ∪ (𝑅 ×t 𝑆))) → (𝑎 ∈ ∪ 𝑆 ↦ (1st
‘𝑤)) ∈ (𝑆 Cn 𝑅)) |
33 | 27 | cnmptid 22558 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((((𝑅 ∈ Conn ∧ 𝑆 ∈ Conn) ∧ 𝑥 ∈ ((𝑅 ×t 𝑆) ∩ (Clsd‘(𝑅 ×t 𝑆)))) ∧ (𝑧 ∈ 𝑥 ∧ 𝑤 ∈ ∪ (𝑅 ×t 𝑆))) → (𝑎 ∈ ∪ 𝑆 ↦ 𝑎) ∈ (𝑆 Cn 𝑆)) |
34 | 27, 32, 33 | cnmpt1t 22562 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝑅 ∈ Conn ∧ 𝑆 ∈ Conn) ∧ 𝑥 ∈ ((𝑅 ×t 𝑆) ∩ (Clsd‘(𝑅 ×t 𝑆)))) ∧ (𝑧 ∈ 𝑥 ∧ 𝑤 ∈ ∪ (𝑅 ×t 𝑆))) → (𝑎 ∈ ∪ 𝑆 ↦ 〈(1st
‘𝑤), 𝑎〉) ∈ (𝑆 Cn (𝑅 ×t 𝑆))) |
35 | | simplr 769 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((((𝑅 ∈ Conn ∧ 𝑆 ∈ Conn) ∧ 𝑥 ∈ ((𝑅 ×t 𝑆) ∩ (Clsd‘(𝑅 ×t 𝑆)))) ∧ (𝑧 ∈ 𝑥 ∧ 𝑤 ∈ ∪ (𝑅 ×t 𝑆))) → 𝑥 ∈ ((𝑅 ×t 𝑆) ∩ (Clsd‘(𝑅 ×t 𝑆)))) |
36 | 35 | elin1d 4112 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝑅 ∈ Conn ∧ 𝑆 ∈ Conn) ∧ 𝑥 ∈ ((𝑅 ×t 𝑆) ∩ (Clsd‘(𝑅 ×t 𝑆)))) ∧ (𝑧 ∈ 𝑥 ∧ 𝑤 ∈ ∪ (𝑅 ×t 𝑆))) → 𝑥 ∈ (𝑅 ×t 𝑆)) |
37 | | cnima 22162 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝑎 ∈ ∪ 𝑆
↦ 〈(1st ‘𝑤), 𝑎〉) ∈ (𝑆 Cn (𝑅 ×t 𝑆)) ∧ 𝑥 ∈ (𝑅 ×t 𝑆)) → (◡(𝑎 ∈ ∪ 𝑆 ↦ 〈(1st
‘𝑤), 𝑎〉) “ 𝑥) ∈ 𝑆) |
38 | 34, 36, 37 | syl2anc 587 |
. . . . . . . . . . . . . . . . 17
⊢ ((((𝑅 ∈ Conn ∧ 𝑆 ∈ Conn) ∧ 𝑥 ∈ ((𝑅 ×t 𝑆) ∩ (Clsd‘(𝑅 ×t 𝑆)))) ∧ (𝑧 ∈ 𝑥 ∧ 𝑤 ∈ ∪ (𝑅 ×t 𝑆))) → (◡(𝑎 ∈ ∪ 𝑆 ↦ 〈(1st
‘𝑤), 𝑎〉) “ 𝑥) ∈ 𝑆) |
39 | 25, 38 | eqeltrrid 2843 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝑅 ∈ Conn ∧ 𝑆 ∈ Conn) ∧ 𝑥 ∈ ((𝑅 ×t 𝑆) ∩ (Clsd‘(𝑅 ×t 𝑆)))) ∧ (𝑧 ∈ 𝑥 ∧ 𝑤 ∈ ∪ (𝑅 ×t 𝑆))) → {𝑎 ∈ ∪ 𝑆 ∣ 〈(1st
‘𝑤), 𝑎〉 ∈ 𝑥} ∈ 𝑆) |
40 | | simprl 771 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((((𝑅 ∈ Conn ∧ 𝑆 ∈ Conn) ∧ 𝑥 ∈ ((𝑅 ×t 𝑆) ∩ (Clsd‘(𝑅 ×t 𝑆)))) ∧ (𝑧 ∈ 𝑥 ∧ 𝑤 ∈ ∪ (𝑅 ×t 𝑆))) → 𝑧 ∈ 𝑥) |
41 | | elunii 4824 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝑧 ∈ 𝑥 ∧ 𝑥 ∈ (𝑅 ×t 𝑆)) → 𝑧 ∈ ∪ (𝑅 ×t 𝑆)) |
42 | 40, 36, 41 | syl2anc 587 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((((𝑅 ∈ Conn ∧ 𝑆 ∈ Conn) ∧ 𝑥 ∈ ((𝑅 ×t 𝑆) ∩ (Clsd‘(𝑅 ×t 𝑆)))) ∧ (𝑧 ∈ 𝑥 ∧ 𝑤 ∈ ∪ (𝑅 ×t 𝑆))) → 𝑧 ∈ ∪ (𝑅 ×t 𝑆)) |
43 | 42, 18 | eleqtrrd 2841 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((((𝑅 ∈ Conn ∧ 𝑆 ∈ Conn) ∧ 𝑥 ∈ ((𝑅 ×t 𝑆) ∩ (Clsd‘(𝑅 ×t 𝑆)))) ∧ (𝑧 ∈ 𝑥 ∧ 𝑤 ∈ ∪ (𝑅 ×t 𝑆))) → 𝑧 ∈ (∪ 𝑅 × ∪ 𝑆)) |
44 | | xp2nd 7794 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑧 ∈ (∪ 𝑅
× ∪ 𝑆) → (2nd ‘𝑧) ∈ ∪ 𝑆) |
45 | 43, 44 | syl 17 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝑅 ∈ Conn ∧ 𝑆 ∈ Conn) ∧ 𝑥 ∈ ((𝑅 ×t 𝑆) ∩ (Clsd‘(𝑅 ×t 𝑆)))) ∧ (𝑧 ∈ 𝑥 ∧ 𝑤 ∈ ∪ (𝑅 ×t 𝑆))) → (2nd
‘𝑧) ∈ ∪ 𝑆) |
46 | | eqid 2737 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑎 ∈ ∪ 𝑅
↦ 〈𝑎,
(2nd ‘𝑧)〉) = (𝑎 ∈ ∪ 𝑅 ↦ 〈𝑎, (2nd ‘𝑧)〉) |
47 | 46 | mptpreima 6101 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (◡(𝑎 ∈ ∪ 𝑅 ↦ 〈𝑎, (2nd ‘𝑧)〉) “ 𝑥) = {𝑎 ∈ ∪ 𝑅 ∣ 〈𝑎, (2nd ‘𝑧)〉 ∈ 𝑥} |
48 | 29 | cnmptid 22558 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((((𝑅 ∈ Conn ∧ 𝑆 ∈ Conn) ∧ 𝑥 ∈ ((𝑅 ×t 𝑆) ∩ (Clsd‘(𝑅 ×t 𝑆)))) ∧ (𝑧 ∈ 𝑥 ∧ 𝑤 ∈ ∪ (𝑅 ×t 𝑆))) → (𝑎 ∈ ∪ 𝑅 ↦ 𝑎) ∈ (𝑅 Cn 𝑅)) |
49 | 29, 27, 45 | cnmptc 22559 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((((𝑅 ∈ Conn ∧ 𝑆 ∈ Conn) ∧ 𝑥 ∈ ((𝑅 ×t 𝑆) ∩ (Clsd‘(𝑅 ×t 𝑆)))) ∧ (𝑧 ∈ 𝑥 ∧ 𝑤 ∈ ∪ (𝑅 ×t 𝑆))) → (𝑎 ∈ ∪ 𝑅 ↦ (2nd
‘𝑧)) ∈ (𝑅 Cn 𝑆)) |
50 | 29, 48, 49 | cnmpt1t 22562 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((((𝑅 ∈ Conn ∧ 𝑆 ∈ Conn) ∧ 𝑥 ∈ ((𝑅 ×t 𝑆) ∩ (Clsd‘(𝑅 ×t 𝑆)))) ∧ (𝑧 ∈ 𝑥 ∧ 𝑤 ∈ ∪ (𝑅 ×t 𝑆))) → (𝑎 ∈ ∪ 𝑅 ↦ 〈𝑎, (2nd ‘𝑧)〉) ∈ (𝑅 Cn (𝑅 ×t 𝑆))) |
51 | | cnima 22162 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (((𝑎 ∈ ∪ 𝑅
↦ 〈𝑎,
(2nd ‘𝑧)〉) ∈ (𝑅 Cn (𝑅 ×t 𝑆)) ∧ 𝑥 ∈ (𝑅 ×t 𝑆)) → (◡(𝑎 ∈ ∪ 𝑅 ↦ 〈𝑎, (2nd ‘𝑧)〉) “ 𝑥) ∈ 𝑅) |
52 | 50, 36, 51 | syl2anc 587 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((((𝑅 ∈ Conn ∧ 𝑆 ∈ Conn) ∧ 𝑥 ∈ ((𝑅 ×t 𝑆) ∩ (Clsd‘(𝑅 ×t 𝑆)))) ∧ (𝑧 ∈ 𝑥 ∧ 𝑤 ∈ ∪ (𝑅 ×t 𝑆))) → (◡(𝑎 ∈ ∪ 𝑅 ↦ 〈𝑎, (2nd ‘𝑧)〉) “ 𝑥) ∈ 𝑅) |
53 | 47, 52 | eqeltrrid 2843 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((((𝑅 ∈ Conn ∧ 𝑆 ∈ Conn) ∧ 𝑥 ∈ ((𝑅 ×t 𝑆) ∩ (Clsd‘(𝑅 ×t 𝑆)))) ∧ (𝑧 ∈ 𝑥 ∧ 𝑤 ∈ ∪ (𝑅 ×t 𝑆))) → {𝑎 ∈ ∪ 𝑅 ∣ 〈𝑎, (2nd ‘𝑧)〉 ∈ 𝑥} ∈ 𝑅) |
54 | | xp1st 7793 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑧 ∈ (∪ 𝑅
× ∪ 𝑆) → (1st ‘𝑧) ∈ ∪ 𝑅) |
55 | 43, 54 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((((𝑅 ∈ Conn ∧ 𝑆 ∈ Conn) ∧ 𝑥 ∈ ((𝑅 ×t 𝑆) ∩ (Clsd‘(𝑅 ×t 𝑆)))) ∧ (𝑧 ∈ 𝑥 ∧ 𝑤 ∈ ∪ (𝑅 ×t 𝑆))) → (1st
‘𝑧) ∈ ∪ 𝑅) |
56 | | 1st2nd2 7800 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑧 ∈ (∪ 𝑅
× ∪ 𝑆) → 𝑧 = 〈(1st ‘𝑧), (2nd ‘𝑧)〉) |
57 | 43, 56 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((((𝑅 ∈ Conn ∧ 𝑆 ∈ Conn) ∧ 𝑥 ∈ ((𝑅 ×t 𝑆) ∩ (Clsd‘(𝑅 ×t 𝑆)))) ∧ (𝑧 ∈ 𝑥 ∧ 𝑤 ∈ ∪ (𝑅 ×t 𝑆))) → 𝑧 = 〈(1st ‘𝑧), (2nd ‘𝑧)〉) |
58 | 57, 40 | eqeltrrd 2839 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((((𝑅 ∈ Conn ∧ 𝑆 ∈ Conn) ∧ 𝑥 ∈ ((𝑅 ×t 𝑆) ∩ (Clsd‘(𝑅 ×t 𝑆)))) ∧ (𝑧 ∈ 𝑥 ∧ 𝑤 ∈ ∪ (𝑅 ×t 𝑆))) → 〈(1st
‘𝑧), (2nd
‘𝑧)〉 ∈
𝑥) |
59 | | opeq1 4784 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑎 = (1st ‘𝑧) → 〈𝑎, (2nd ‘𝑧)〉 = 〈(1st
‘𝑧), (2nd
‘𝑧)〉) |
60 | 59 | eleq1d 2822 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑎 = (1st ‘𝑧) → (〈𝑎, (2nd ‘𝑧)〉 ∈ 𝑥 ↔ 〈(1st
‘𝑧), (2nd
‘𝑧)〉 ∈
𝑥)) |
61 | 60 | rspcev 3537 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢
(((1st ‘𝑧) ∈ ∪ 𝑅 ∧ 〈(1st
‘𝑧), (2nd
‘𝑧)〉 ∈
𝑥) → ∃𝑎 ∈ ∪ 𝑅〈𝑎, (2nd ‘𝑧)〉 ∈ 𝑥) |
62 | 55, 58, 61 | syl2anc 587 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((((𝑅 ∈ Conn ∧ 𝑆 ∈ Conn) ∧ 𝑥 ∈ ((𝑅 ×t 𝑆) ∩ (Clsd‘(𝑅 ×t 𝑆)))) ∧ (𝑧 ∈ 𝑥 ∧ 𝑤 ∈ ∪ (𝑅 ×t 𝑆))) → ∃𝑎 ∈ ∪ 𝑅〈𝑎, (2nd ‘𝑧)〉 ∈ 𝑥) |
63 | | rabn0 4300 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ({𝑎 ∈ ∪ 𝑅
∣ 〈𝑎,
(2nd ‘𝑧)〉 ∈ 𝑥} ≠ ∅ ↔ ∃𝑎 ∈ ∪ 𝑅〈𝑎, (2nd ‘𝑧)〉 ∈ 𝑥) |
64 | 62, 63 | sylibr 237 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((((𝑅 ∈ Conn ∧ 𝑆 ∈ Conn) ∧ 𝑥 ∈ ((𝑅 ×t 𝑆) ∩ (Clsd‘(𝑅 ×t 𝑆)))) ∧ (𝑧 ∈ 𝑥 ∧ 𝑤 ∈ ∪ (𝑅 ×t 𝑆))) → {𝑎 ∈ ∪ 𝑅 ∣ 〈𝑎, (2nd ‘𝑧)〉 ∈ 𝑥} ≠ ∅) |
65 | 35 | elin2d 4113 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((((𝑅 ∈ Conn ∧ 𝑆 ∈ Conn) ∧ 𝑥 ∈ ((𝑅 ×t 𝑆) ∩ (Clsd‘(𝑅 ×t 𝑆)))) ∧ (𝑧 ∈ 𝑥 ∧ 𝑤 ∈ ∪ (𝑅 ×t 𝑆))) → 𝑥 ∈ (Clsd‘(𝑅 ×t 𝑆))) |
66 | | cnclima 22165 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (((𝑎 ∈ ∪ 𝑅
↦ 〈𝑎,
(2nd ‘𝑧)〉) ∈ (𝑅 Cn (𝑅 ×t 𝑆)) ∧ 𝑥 ∈ (Clsd‘(𝑅 ×t 𝑆))) → (◡(𝑎 ∈ ∪ 𝑅 ↦ 〈𝑎, (2nd ‘𝑧)〉) “ 𝑥) ∈ (Clsd‘𝑅)) |
67 | 50, 65, 66 | syl2anc 587 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((((𝑅 ∈ Conn ∧ 𝑆 ∈ Conn) ∧ 𝑥 ∈ ((𝑅 ×t 𝑆) ∩ (Clsd‘(𝑅 ×t 𝑆)))) ∧ (𝑧 ∈ 𝑥 ∧ 𝑤 ∈ ∪ (𝑅 ×t 𝑆))) → (◡(𝑎 ∈ ∪ 𝑅 ↦ 〈𝑎, (2nd ‘𝑧)〉) “ 𝑥) ∈ (Clsd‘𝑅)) |
68 | 47, 67 | eqeltrrid 2843 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((((𝑅 ∈ Conn ∧ 𝑆 ∈ Conn) ∧ 𝑥 ∈ ((𝑅 ×t 𝑆) ∩ (Clsd‘(𝑅 ×t 𝑆)))) ∧ (𝑧 ∈ 𝑥 ∧ 𝑤 ∈ ∪ (𝑅 ×t 𝑆))) → {𝑎 ∈ ∪ 𝑅 ∣ 〈𝑎, (2nd ‘𝑧)〉 ∈ 𝑥} ∈ (Clsd‘𝑅)) |
69 | 15, 11, 53, 64, 68 | connclo 22312 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((((𝑅 ∈ Conn ∧ 𝑆 ∈ Conn) ∧ 𝑥 ∈ ((𝑅 ×t 𝑆) ∩ (Clsd‘(𝑅 ×t 𝑆)))) ∧ (𝑧 ∈ 𝑥 ∧ 𝑤 ∈ ∪ (𝑅 ×t 𝑆))) → {𝑎 ∈ ∪ 𝑅 ∣ 〈𝑎, (2nd ‘𝑧)〉 ∈ 𝑥} = ∪
𝑅) |
70 | 31, 69 | eleqtrrd 2841 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((((𝑅 ∈ Conn ∧ 𝑆 ∈ Conn) ∧ 𝑥 ∈ ((𝑅 ×t 𝑆) ∩ (Clsd‘(𝑅 ×t 𝑆)))) ∧ (𝑧 ∈ 𝑥 ∧ 𝑤 ∈ ∪ (𝑅 ×t 𝑆))) → (1st
‘𝑤) ∈ {𝑎 ∈ ∪ 𝑅
∣ 〈𝑎,
(2nd ‘𝑧)〉 ∈ 𝑥}) |
71 | | opeq1 4784 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑎 = (1st ‘𝑤) → 〈𝑎, (2nd ‘𝑧)〉 = 〈(1st
‘𝑤), (2nd
‘𝑧)〉) |
72 | 71 | eleq1d 2822 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑎 = (1st ‘𝑤) → (〈𝑎, (2nd ‘𝑧)〉 ∈ 𝑥 ↔ 〈(1st
‘𝑤), (2nd
‘𝑧)〉 ∈
𝑥)) |
73 | 72 | elrab 3602 |
. . . . . . . . . . . . . . . . . . . 20
⊢
((1st ‘𝑤) ∈ {𝑎 ∈ ∪ 𝑅 ∣ 〈𝑎, (2nd ‘𝑧)〉 ∈ 𝑥} ↔ ((1st
‘𝑤) ∈ ∪ 𝑅
∧ 〈(1st ‘𝑤), (2nd ‘𝑧)〉 ∈ 𝑥)) |
74 | 73 | simprbi 500 |
. . . . . . . . . . . . . . . . . . 19
⊢
((1st ‘𝑤) ∈ {𝑎 ∈ ∪ 𝑅 ∣ 〈𝑎, (2nd ‘𝑧)〉 ∈ 𝑥} → 〈(1st
‘𝑤), (2nd
‘𝑧)〉 ∈
𝑥) |
75 | 70, 74 | syl 17 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝑅 ∈ Conn ∧ 𝑆 ∈ Conn) ∧ 𝑥 ∈ ((𝑅 ×t 𝑆) ∩ (Clsd‘(𝑅 ×t 𝑆)))) ∧ (𝑧 ∈ 𝑥 ∧ 𝑤 ∈ ∪ (𝑅 ×t 𝑆))) → 〈(1st
‘𝑤), (2nd
‘𝑧)〉 ∈
𝑥) |
76 | | opeq2 4785 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑎 = (2nd ‘𝑧) → 〈(1st
‘𝑤), 𝑎〉 = 〈(1st
‘𝑤), (2nd
‘𝑧)〉) |
77 | 76 | eleq1d 2822 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑎 = (2nd ‘𝑧) → (〈(1st
‘𝑤), 𝑎〉 ∈ 𝑥 ↔ 〈(1st ‘𝑤), (2nd ‘𝑧)〉 ∈ 𝑥)) |
78 | 77 | rspcev 3537 |
. . . . . . . . . . . . . . . . . 18
⊢
(((2nd ‘𝑧) ∈ ∪ 𝑆 ∧ 〈(1st
‘𝑤), (2nd
‘𝑧)〉 ∈
𝑥) → ∃𝑎 ∈ ∪ 𝑆〈(1st ‘𝑤), 𝑎〉 ∈ 𝑥) |
79 | 45, 75, 78 | syl2anc 587 |
. . . . . . . . . . . . . . . . 17
⊢ ((((𝑅 ∈ Conn ∧ 𝑆 ∈ Conn) ∧ 𝑥 ∈ ((𝑅 ×t 𝑆) ∩ (Clsd‘(𝑅 ×t 𝑆)))) ∧ (𝑧 ∈ 𝑥 ∧ 𝑤 ∈ ∪ (𝑅 ×t 𝑆))) → ∃𝑎 ∈ ∪ 𝑆〈(1st ‘𝑤), 𝑎〉 ∈ 𝑥) |
80 | | rabn0 4300 |
. . . . . . . . . . . . . . . . 17
⊢ ({𝑎 ∈ ∪ 𝑆
∣ 〈(1st ‘𝑤), 𝑎〉 ∈ 𝑥} ≠ ∅ ↔ ∃𝑎 ∈ ∪ 𝑆〈(1st ‘𝑤), 𝑎〉 ∈ 𝑥) |
81 | 79, 80 | sylibr 237 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝑅 ∈ Conn ∧ 𝑆 ∈ Conn) ∧ 𝑥 ∈ ((𝑅 ×t 𝑆) ∩ (Clsd‘(𝑅 ×t 𝑆)))) ∧ (𝑧 ∈ 𝑥 ∧ 𝑤 ∈ ∪ (𝑅 ×t 𝑆))) → {𝑎 ∈ ∪ 𝑆 ∣ 〈(1st
‘𝑤), 𝑎〉 ∈ 𝑥} ≠ ∅) |
82 | | cnclima 22165 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝑎 ∈ ∪ 𝑆
↦ 〈(1st ‘𝑤), 𝑎〉) ∈ (𝑆 Cn (𝑅 ×t 𝑆)) ∧ 𝑥 ∈ (Clsd‘(𝑅 ×t 𝑆))) → (◡(𝑎 ∈ ∪ 𝑆 ↦ 〈(1st
‘𝑤), 𝑎〉) “ 𝑥) ∈ (Clsd‘𝑆)) |
83 | 34, 65, 82 | syl2anc 587 |
. . . . . . . . . . . . . . . . 17
⊢ ((((𝑅 ∈ Conn ∧ 𝑆 ∈ Conn) ∧ 𝑥 ∈ ((𝑅 ×t 𝑆) ∩ (Clsd‘(𝑅 ×t 𝑆)))) ∧ (𝑧 ∈ 𝑥 ∧ 𝑤 ∈ ∪ (𝑅 ×t 𝑆))) → (◡(𝑎 ∈ ∪ 𝑆 ↦ 〈(1st
‘𝑤), 𝑎〉) “ 𝑥) ∈ (Clsd‘𝑆)) |
84 | 25, 83 | eqeltrrid 2843 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝑅 ∈ Conn ∧ 𝑆 ∈ Conn) ∧ 𝑥 ∈ ((𝑅 ×t 𝑆) ∩ (Clsd‘(𝑅 ×t 𝑆)))) ∧ (𝑧 ∈ 𝑥 ∧ 𝑤 ∈ ∪ (𝑅 ×t 𝑆))) → {𝑎 ∈ ∪ 𝑆 ∣ 〈(1st
‘𝑤), 𝑎〉 ∈ 𝑥} ∈ (Clsd‘𝑆)) |
85 | 16, 13, 39, 81, 84 | connclo 22312 |
. . . . . . . . . . . . . . 15
⊢ ((((𝑅 ∈ Conn ∧ 𝑆 ∈ Conn) ∧ 𝑥 ∈ ((𝑅 ×t 𝑆) ∩ (Clsd‘(𝑅 ×t 𝑆)))) ∧ (𝑧 ∈ 𝑥 ∧ 𝑤 ∈ ∪ (𝑅 ×t 𝑆))) → {𝑎 ∈ ∪ 𝑆 ∣ 〈(1st
‘𝑤), 𝑎〉 ∈ 𝑥} = ∪ 𝑆) |
86 | 23, 85 | eleqtrrd 2841 |
. . . . . . . . . . . . . 14
⊢ ((((𝑅 ∈ Conn ∧ 𝑆 ∈ Conn) ∧ 𝑥 ∈ ((𝑅 ×t 𝑆) ∩ (Clsd‘(𝑅 ×t 𝑆)))) ∧ (𝑧 ∈ 𝑥 ∧ 𝑤 ∈ ∪ (𝑅 ×t 𝑆))) → (2nd
‘𝑤) ∈ {𝑎 ∈ ∪ 𝑆
∣ 〈(1st ‘𝑤), 𝑎〉 ∈ 𝑥}) |
87 | | opeq2 4785 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑎 = (2nd ‘𝑤) → 〈(1st
‘𝑤), 𝑎〉 = 〈(1st
‘𝑤), (2nd
‘𝑤)〉) |
88 | 87 | eleq1d 2822 |
. . . . . . . . . . . . . . . 16
⊢ (𝑎 = (2nd ‘𝑤) → (〈(1st
‘𝑤), 𝑎〉 ∈ 𝑥 ↔ 〈(1st ‘𝑤), (2nd ‘𝑤)〉 ∈ 𝑥)) |
89 | 88 | elrab 3602 |
. . . . . . . . . . . . . . 15
⊢
((2nd ‘𝑤) ∈ {𝑎 ∈ ∪ 𝑆 ∣ 〈(1st
‘𝑤), 𝑎〉 ∈ 𝑥} ↔ ((2nd ‘𝑤) ∈ ∪ 𝑆
∧ 〈(1st ‘𝑤), (2nd ‘𝑤)〉 ∈ 𝑥)) |
90 | 89 | simprbi 500 |
. . . . . . . . . . . . . 14
⊢
((2nd ‘𝑤) ∈ {𝑎 ∈ ∪ 𝑆 ∣ 〈(1st
‘𝑤), 𝑎〉 ∈ 𝑥} → 〈(1st ‘𝑤), (2nd ‘𝑤)〉 ∈ 𝑥) |
91 | 86, 90 | syl 17 |
. . . . . . . . . . . . 13
⊢ ((((𝑅 ∈ Conn ∧ 𝑆 ∈ Conn) ∧ 𝑥 ∈ ((𝑅 ×t 𝑆) ∩ (Clsd‘(𝑅 ×t 𝑆)))) ∧ (𝑧 ∈ 𝑥 ∧ 𝑤 ∈ ∪ (𝑅 ×t 𝑆))) → 〈(1st
‘𝑤), (2nd
‘𝑤)〉 ∈
𝑥) |
92 | 21, 91 | eqeltrd 2838 |
. . . . . . . . . . . 12
⊢ ((((𝑅 ∈ Conn ∧ 𝑆 ∈ Conn) ∧ 𝑥 ∈ ((𝑅 ×t 𝑆) ∩ (Clsd‘(𝑅 ×t 𝑆)))) ∧ (𝑧 ∈ 𝑥 ∧ 𝑤 ∈ ∪ (𝑅 ×t 𝑆))) → 𝑤 ∈ 𝑥) |
93 | 92 | expr 460 |
. . . . . . . . . . 11
⊢ ((((𝑅 ∈ Conn ∧ 𝑆 ∈ Conn) ∧ 𝑥 ∈ ((𝑅 ×t 𝑆) ∩ (Clsd‘(𝑅 ×t 𝑆)))) ∧ 𝑧 ∈ 𝑥) → (𝑤 ∈ ∪ (𝑅 ×t 𝑆) → 𝑤 ∈ 𝑥)) |
94 | 93 | ssrdv 3907 |
. . . . . . . . . 10
⊢ ((((𝑅 ∈ Conn ∧ 𝑆 ∈ Conn) ∧ 𝑥 ∈ ((𝑅 ×t 𝑆) ∩ (Clsd‘(𝑅 ×t 𝑆)))) ∧ 𝑧 ∈ 𝑥) → ∪ (𝑅 ×t 𝑆) ⊆ 𝑥) |
95 | 9, 94 | eqssd 3918 |
. . . . . . . . 9
⊢ ((((𝑅 ∈ Conn ∧ 𝑆 ∈ Conn) ∧ 𝑥 ∈ ((𝑅 ×t 𝑆) ∩ (Clsd‘(𝑅 ×t 𝑆)))) ∧ 𝑧 ∈ 𝑥) → 𝑥 = ∪ (𝑅 ×t 𝑆)) |
96 | 95 | ex 416 |
. . . . . . . 8
⊢ (((𝑅 ∈ Conn ∧ 𝑆 ∈ Conn) ∧ 𝑥 ∈ ((𝑅 ×t 𝑆) ∩ (Clsd‘(𝑅 ×t 𝑆)))) → (𝑧 ∈ 𝑥 → 𝑥 = ∪ (𝑅 ×t 𝑆))) |
97 | 96 | exlimdv 1941 |
. . . . . . 7
⊢ (((𝑅 ∈ Conn ∧ 𝑆 ∈ Conn) ∧ 𝑥 ∈ ((𝑅 ×t 𝑆) ∩ (Clsd‘(𝑅 ×t 𝑆)))) → (∃𝑧 𝑧 ∈ 𝑥 → 𝑥 = ∪ (𝑅 ×t 𝑆))) |
98 | 5, 97 | syl5bi 245 |
. . . . . 6
⊢ (((𝑅 ∈ Conn ∧ 𝑆 ∈ Conn) ∧ 𝑥 ∈ ((𝑅 ×t 𝑆) ∩ (Clsd‘(𝑅 ×t 𝑆)))) → (¬ 𝑥 = ∅ → 𝑥 = ∪ (𝑅 ×t 𝑆))) |
99 | 98 | orrd 863 |
. . . . 5
⊢ (((𝑅 ∈ Conn ∧ 𝑆 ∈ Conn) ∧ 𝑥 ∈ ((𝑅 ×t 𝑆) ∩ (Clsd‘(𝑅 ×t 𝑆)))) → (𝑥 = ∅ ∨ 𝑥 = ∪ (𝑅 ×t 𝑆))) |
100 | 99 | ex 416 |
. . . 4
⊢ ((𝑅 ∈ Conn ∧ 𝑆 ∈ Conn) → (𝑥 ∈ ((𝑅 ×t 𝑆) ∩ (Clsd‘(𝑅 ×t 𝑆))) → (𝑥 = ∅ ∨ 𝑥 = ∪ (𝑅 ×t 𝑆)))) |
101 | | vex 3412 |
. . . . 5
⊢ 𝑥 ∈ V |
102 | 101 | elpr 4564 |
. . . 4
⊢ (𝑥 ∈ {∅, ∪ (𝑅
×t 𝑆)}
↔ (𝑥 = ∅ ∨
𝑥 = ∪ (𝑅
×t 𝑆))) |
103 | 100, 102 | syl6ibr 255 |
. . 3
⊢ ((𝑅 ∈ Conn ∧ 𝑆 ∈ Conn) → (𝑥 ∈ ((𝑅 ×t 𝑆) ∩ (Clsd‘(𝑅 ×t 𝑆))) → 𝑥 ∈ {∅, ∪ (𝑅
×t 𝑆)})) |
104 | 103 | ssrdv 3907 |
. 2
⊢ ((𝑅 ∈ Conn ∧ 𝑆 ∈ Conn) → ((𝑅 ×t 𝑆) ∩ (Clsd‘(𝑅 ×t 𝑆))) ⊆ {∅, ∪ (𝑅
×t 𝑆)}) |
105 | | eqid 2737 |
. . 3
⊢ ∪ (𝑅
×t 𝑆) =
∪ (𝑅 ×t 𝑆) |
106 | 105 | isconn2 22311 |
. 2
⊢ ((𝑅 ×t 𝑆) ∈ Conn ↔ ((𝑅 ×t 𝑆) ∈ Top ∧ ((𝑅 ×t 𝑆) ∩ (Clsd‘(𝑅 ×t 𝑆))) ⊆ {∅, ∪ (𝑅
×t 𝑆)})) |
107 | 4, 104, 106 | sylanbrc 586 |
1
⊢ ((𝑅 ∈ Conn ∧ 𝑆 ∈ Conn) → (𝑅 ×t 𝑆) ∈ Conn) |