Step |
Hyp |
Ref |
Expression |
1 |
|
conntop |
⊢ ( 𝑅 ∈ Conn → 𝑅 ∈ Top ) |
2 |
|
conntop |
⊢ ( 𝑆 ∈ Conn → 𝑆 ∈ Top ) |
3 |
|
txtop |
⊢ ( ( 𝑅 ∈ Top ∧ 𝑆 ∈ Top ) → ( 𝑅 ×t 𝑆 ) ∈ Top ) |
4 |
1 2 3
|
syl2an |
⊢ ( ( 𝑅 ∈ Conn ∧ 𝑆 ∈ Conn ) → ( 𝑅 ×t 𝑆 ) ∈ Top ) |
5 |
|
neq0 |
⊢ ( ¬ 𝑥 = ∅ ↔ ∃ 𝑧 𝑧 ∈ 𝑥 ) |
6 |
|
simplr |
⊢ ( ( ( ( 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 |
⊢ ( ( ( ( 𝑅 ∈ Conn ∧ 𝑆 ∈ Conn ) ∧ 𝑥 ∈ ( ( 𝑅 ×t 𝑆 ) ∩ ( Clsd ‘ ( 𝑅 ×t 𝑆 ) ) ) ) ∧ 𝑧 ∈ 𝑥 ) → 𝑥 ∈ ( 𝑅 ×t 𝑆 ) ) |
8 |
|
elssuni |
⊢ ( 𝑥 ∈ ( 𝑅 ×t 𝑆 ) → 𝑥 ⊆ ∪ ( 𝑅 ×t 𝑆 ) ) |
9 |
7 8
|
syl |
⊢ ( ( ( ( 𝑅 ∈ Conn ∧ 𝑆 ∈ Conn ) ∧ 𝑥 ∈ ( ( 𝑅 ×t 𝑆 ) ∩ ( Clsd ‘ ( 𝑅 ×t 𝑆 ) ) ) ) ∧ 𝑧 ∈ 𝑥 ) → 𝑥 ⊆ ∪ ( 𝑅 ×t 𝑆 ) ) |
10 |
|
simprr |
⊢ ( ( ( ( 𝑅 ∈ Conn ∧ 𝑆 ∈ Conn ) ∧ 𝑥 ∈ ( ( 𝑅 ×t 𝑆 ) ∩ ( Clsd ‘ ( 𝑅 ×t 𝑆 ) ) ) ) ∧ ( 𝑧 ∈ 𝑥 ∧ 𝑤 ∈ ∪ ( 𝑅 ×t 𝑆 ) ) ) → 𝑤 ∈ ∪ ( 𝑅 ×t 𝑆 ) ) |
11 |
|
simplll |
⊢ ( ( ( ( 𝑅 ∈ Conn ∧ 𝑆 ∈ Conn ) ∧ 𝑥 ∈ ( ( 𝑅 ×t 𝑆 ) ∩ ( Clsd ‘ ( 𝑅 ×t 𝑆 ) ) ) ) ∧ ( 𝑧 ∈ 𝑥 ∧ 𝑤 ∈ ∪ ( 𝑅 ×t 𝑆 ) ) ) → 𝑅 ∈ Conn ) |
12 |
11 1
|
syl |
⊢ ( ( ( ( 𝑅 ∈ Conn ∧ 𝑆 ∈ Conn ) ∧ 𝑥 ∈ ( ( 𝑅 ×t 𝑆 ) ∩ ( Clsd ‘ ( 𝑅 ×t 𝑆 ) ) ) ) ∧ ( 𝑧 ∈ 𝑥 ∧ 𝑤 ∈ ∪ ( 𝑅 ×t 𝑆 ) ) ) → 𝑅 ∈ Top ) |
13 |
|
simpllr |
⊢ ( ( ( ( 𝑅 ∈ Conn ∧ 𝑆 ∈ Conn ) ∧ 𝑥 ∈ ( ( 𝑅 ×t 𝑆 ) ∩ ( Clsd ‘ ( 𝑅 ×t 𝑆 ) ) ) ) ∧ ( 𝑧 ∈ 𝑥 ∧ 𝑤 ∈ ∪ ( 𝑅 ×t 𝑆 ) ) ) → 𝑆 ∈ Conn ) |
14 |
13 2
|
syl |
⊢ ( ( ( ( 𝑅 ∈ Conn ∧ 𝑆 ∈ Conn ) ∧ 𝑥 ∈ ( ( 𝑅 ×t 𝑆 ) ∩ ( Clsd ‘ ( 𝑅 ×t 𝑆 ) ) ) ) ∧ ( 𝑧 ∈ 𝑥 ∧ 𝑤 ∈ ∪ ( 𝑅 ×t 𝑆 ) ) ) → 𝑆 ∈ Top ) |
15 |
|
eqid |
⊢ ∪ 𝑅 = ∪ 𝑅 |
16 |
|
eqid |
⊢ ∪ 𝑆 = ∪ 𝑆 |
17 |
15 16
|
txuni |
⊢ ( ( 𝑅 ∈ Top ∧ 𝑆 ∈ Top ) → ( ∪ 𝑅 × ∪ 𝑆 ) = ∪ ( 𝑅 ×t 𝑆 ) ) |
18 |
12 14 17
|
syl2anc |
⊢ ( ( ( ( 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 |
⊢ ( ( ( ( 𝑅 ∈ Conn ∧ 𝑆 ∈ Conn ) ∧ 𝑥 ∈ ( ( 𝑅 ×t 𝑆 ) ∩ ( Clsd ‘ ( 𝑅 ×t 𝑆 ) ) ) ) ∧ ( 𝑧 ∈ 𝑥 ∧ 𝑤 ∈ ∪ ( 𝑅 ×t 𝑆 ) ) ) → 𝑤 ∈ ( ∪ 𝑅 × ∪ 𝑆 ) ) |
20 |
|
1st2nd2 |
⊢ ( 𝑤 ∈ ( ∪ 𝑅 × ∪ 𝑆 ) → 𝑤 = 〈 ( 1st ‘ 𝑤 ) , ( 2nd ‘ 𝑤 ) 〉 ) |
21 |
19 20
|
syl |
⊢ ( ( ( ( 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 |
⊢ ( 𝑤 ∈ ( ∪ 𝑅 × ∪ 𝑆 ) → ( 2nd ‘ 𝑤 ) ∈ ∪ 𝑆 ) |
23 |
19 22
|
syl |
⊢ ( ( ( ( 𝑅 ∈ Conn ∧ 𝑆 ∈ Conn ) ∧ 𝑥 ∈ ( ( 𝑅 ×t 𝑆 ) ∩ ( Clsd ‘ ( 𝑅 ×t 𝑆 ) ) ) ) ∧ ( 𝑧 ∈ 𝑥 ∧ 𝑤 ∈ ∪ ( 𝑅 ×t 𝑆 ) ) ) → ( 2nd ‘ 𝑤 ) ∈ ∪ 𝑆 ) |
24 |
|
eqid |
⊢ ( 𝑎 ∈ ∪ 𝑆 ↦ 〈 ( 1st ‘ 𝑤 ) , 𝑎 〉 ) = ( 𝑎 ∈ ∪ 𝑆 ↦ 〈 ( 1st ‘ 𝑤 ) , 𝑎 〉 ) |
25 |
24
|
mptpreima |
⊢ ( ◡ ( 𝑎 ∈ ∪ 𝑆 ↦ 〈 ( 1st ‘ 𝑤 ) , 𝑎 〉 ) “ 𝑥 ) = { 𝑎 ∈ ∪ 𝑆 ∣ 〈 ( 1st ‘ 𝑤 ) , 𝑎 〉 ∈ 𝑥 } |
26 |
|
toptopon2 |
⊢ ( 𝑆 ∈ Top ↔ 𝑆 ∈ ( TopOn ‘ ∪ 𝑆 ) ) |
27 |
14 26
|
sylib |
⊢ ( ( ( ( 𝑅 ∈ Conn ∧ 𝑆 ∈ Conn ) ∧ 𝑥 ∈ ( ( 𝑅 ×t 𝑆 ) ∩ ( Clsd ‘ ( 𝑅 ×t 𝑆 ) ) ) ) ∧ ( 𝑧 ∈ 𝑥 ∧ 𝑤 ∈ ∪ ( 𝑅 ×t 𝑆 ) ) ) → 𝑆 ∈ ( TopOn ‘ ∪ 𝑆 ) ) |
28 |
|
toptopon2 |
⊢ ( 𝑅 ∈ Top ↔ 𝑅 ∈ ( TopOn ‘ ∪ 𝑅 ) ) |
29 |
12 28
|
sylib |
⊢ ( ( ( ( 𝑅 ∈ Conn ∧ 𝑆 ∈ Conn ) ∧ 𝑥 ∈ ( ( 𝑅 ×t 𝑆 ) ∩ ( Clsd ‘ ( 𝑅 ×t 𝑆 ) ) ) ) ∧ ( 𝑧 ∈ 𝑥 ∧ 𝑤 ∈ ∪ ( 𝑅 ×t 𝑆 ) ) ) → 𝑅 ∈ ( TopOn ‘ ∪ 𝑅 ) ) |
30 |
|
xp1st |
⊢ ( 𝑤 ∈ ( ∪ 𝑅 × ∪ 𝑆 ) → ( 1st ‘ 𝑤 ) ∈ ∪ 𝑅 ) |
31 |
19 30
|
syl |
⊢ ( ( ( ( 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 |
⊢ ( ( ( ( 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 |
⊢ ( ( ( ( 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 |
⊢ ( ( ( ( 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 |
⊢ ( ( ( ( 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 |
⊢ ( ( ( ( 𝑅 ∈ Conn ∧ 𝑆 ∈ Conn ) ∧ 𝑥 ∈ ( ( 𝑅 ×t 𝑆 ) ∩ ( Clsd ‘ ( 𝑅 ×t 𝑆 ) ) ) ) ∧ ( 𝑧 ∈ 𝑥 ∧ 𝑤 ∈ ∪ ( 𝑅 ×t 𝑆 ) ) ) → 𝑥 ∈ ( 𝑅 ×t 𝑆 ) ) |
37 |
|
cnima |
⊢ ( ( ( 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 |
⊢ ( ( ( ( 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 |
⊢ ( ( ( ( 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 |
⊢ ( ( ( ( 𝑅 ∈ Conn ∧ 𝑆 ∈ Conn ) ∧ 𝑥 ∈ ( ( 𝑅 ×t 𝑆 ) ∩ ( Clsd ‘ ( 𝑅 ×t 𝑆 ) ) ) ) ∧ ( 𝑧 ∈ 𝑥 ∧ 𝑤 ∈ ∪ ( 𝑅 ×t 𝑆 ) ) ) → 𝑧 ∈ 𝑥 ) |
41 |
|
elunii |
⊢ ( ( 𝑧 ∈ 𝑥 ∧ 𝑥 ∈ ( 𝑅 ×t 𝑆 ) ) → 𝑧 ∈ ∪ ( 𝑅 ×t 𝑆 ) ) |
42 |
40 36 41
|
syl2anc |
⊢ ( ( ( ( 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 |
⊢ ( ( ( ( 𝑅 ∈ Conn ∧ 𝑆 ∈ Conn ) ∧ 𝑥 ∈ ( ( 𝑅 ×t 𝑆 ) ∩ ( Clsd ‘ ( 𝑅 ×t 𝑆 ) ) ) ) ∧ ( 𝑧 ∈ 𝑥 ∧ 𝑤 ∈ ∪ ( 𝑅 ×t 𝑆 ) ) ) → 𝑧 ∈ ( ∪ 𝑅 × ∪ 𝑆 ) ) |
44 |
|
xp2nd |
⊢ ( 𝑧 ∈ ( ∪ 𝑅 × ∪ 𝑆 ) → ( 2nd ‘ 𝑧 ) ∈ ∪ 𝑆 ) |
45 |
43 44
|
syl |
⊢ ( ( ( ( 𝑅 ∈ Conn ∧ 𝑆 ∈ Conn ) ∧ 𝑥 ∈ ( ( 𝑅 ×t 𝑆 ) ∩ ( Clsd ‘ ( 𝑅 ×t 𝑆 ) ) ) ) ∧ ( 𝑧 ∈ 𝑥 ∧ 𝑤 ∈ ∪ ( 𝑅 ×t 𝑆 ) ) ) → ( 2nd ‘ 𝑧 ) ∈ ∪ 𝑆 ) |
46 |
|
eqid |
⊢ ( 𝑎 ∈ ∪ 𝑅 ↦ 〈 𝑎 , ( 2nd ‘ 𝑧 ) 〉 ) = ( 𝑎 ∈ ∪ 𝑅 ↦ 〈 𝑎 , ( 2nd ‘ 𝑧 ) 〉 ) |
47 |
46
|
mptpreima |
⊢ ( ◡ ( 𝑎 ∈ ∪ 𝑅 ↦ 〈 𝑎 , ( 2nd ‘ 𝑧 ) 〉 ) “ 𝑥 ) = { 𝑎 ∈ ∪ 𝑅 ∣ 〈 𝑎 , ( 2nd ‘ 𝑧 ) 〉 ∈ 𝑥 } |
48 |
29
|
cnmptid |
⊢ ( ( ( ( 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 |
⊢ ( ( ( ( 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 |
⊢ ( ( ( ( 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 |
⊢ ( ( ( 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 |
⊢ ( ( ( ( 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 |
⊢ ( ( ( ( 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 |
⊢ ( 𝑧 ∈ ( ∪ 𝑅 × ∪ 𝑆 ) → ( 1st ‘ 𝑧 ) ∈ ∪ 𝑅 ) |
55 |
43 54
|
syl |
⊢ ( ( ( ( 𝑅 ∈ Conn ∧ 𝑆 ∈ Conn ) ∧ 𝑥 ∈ ( ( 𝑅 ×t 𝑆 ) ∩ ( Clsd ‘ ( 𝑅 ×t 𝑆 ) ) ) ) ∧ ( 𝑧 ∈ 𝑥 ∧ 𝑤 ∈ ∪ ( 𝑅 ×t 𝑆 ) ) ) → ( 1st ‘ 𝑧 ) ∈ ∪ 𝑅 ) |
56 |
|
1st2nd2 |
⊢ ( 𝑧 ∈ ( ∪ 𝑅 × ∪ 𝑆 ) → 𝑧 = 〈 ( 1st ‘ 𝑧 ) , ( 2nd ‘ 𝑧 ) 〉 ) |
57 |
43 56
|
syl |
⊢ ( ( ( ( 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 |
⊢ ( ( ( ( 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 |
⊢ ( 𝑎 = ( 1st ‘ 𝑧 ) → 〈 𝑎 , ( 2nd ‘ 𝑧 ) 〉 = 〈 ( 1st ‘ 𝑧 ) , ( 2nd ‘ 𝑧 ) 〉 ) |
60 |
59
|
eleq1d |
⊢ ( 𝑎 = ( 1st ‘ 𝑧 ) → ( 〈 𝑎 , ( 2nd ‘ 𝑧 ) 〉 ∈ 𝑥 ↔ 〈 ( 1st ‘ 𝑧 ) , ( 2nd ‘ 𝑧 ) 〉 ∈ 𝑥 ) ) |
61 |
60
|
rspcev |
⊢ ( ( ( 1st ‘ 𝑧 ) ∈ ∪ 𝑅 ∧ 〈 ( 1st ‘ 𝑧 ) , ( 2nd ‘ 𝑧 ) 〉 ∈ 𝑥 ) → ∃ 𝑎 ∈ ∪ 𝑅 〈 𝑎 , ( 2nd ‘ 𝑧 ) 〉 ∈ 𝑥 ) |
62 |
55 58 61
|
syl2anc |
⊢ ( ( ( ( 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 |
⊢ ( { 𝑎 ∈ ∪ 𝑅 ∣ 〈 𝑎 , ( 2nd ‘ 𝑧 ) 〉 ∈ 𝑥 } ≠ ∅ ↔ ∃ 𝑎 ∈ ∪ 𝑅 〈 𝑎 , ( 2nd ‘ 𝑧 ) 〉 ∈ 𝑥 ) |
64 |
62 63
|
sylibr |
⊢ ( ( ( ( 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 |
⊢ ( ( ( ( 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 |
⊢ ( ( ( 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 |
⊢ ( ( ( ( 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 |
⊢ ( ( ( ( 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 |
⊢ ( ( ( ( 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 |
⊢ ( ( ( ( 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 |
⊢ ( 𝑎 = ( 1st ‘ 𝑤 ) → 〈 𝑎 , ( 2nd ‘ 𝑧 ) 〉 = 〈 ( 1st ‘ 𝑤 ) , ( 2nd ‘ 𝑧 ) 〉 ) |
72 |
71
|
eleq1d |
⊢ ( 𝑎 = ( 1st ‘ 𝑤 ) → ( 〈 𝑎 , ( 2nd ‘ 𝑧 ) 〉 ∈ 𝑥 ↔ 〈 ( 1st ‘ 𝑤 ) , ( 2nd ‘ 𝑧 ) 〉 ∈ 𝑥 ) ) |
73 |
72
|
elrab |
⊢ ( ( 1st ‘ 𝑤 ) ∈ { 𝑎 ∈ ∪ 𝑅 ∣ 〈 𝑎 , ( 2nd ‘ 𝑧 ) 〉 ∈ 𝑥 } ↔ ( ( 1st ‘ 𝑤 ) ∈ ∪ 𝑅 ∧ 〈 ( 1st ‘ 𝑤 ) , ( 2nd ‘ 𝑧 ) 〉 ∈ 𝑥 ) ) |
74 |
73
|
simprbi |
⊢ ( ( 1st ‘ 𝑤 ) ∈ { 𝑎 ∈ ∪ 𝑅 ∣ 〈 𝑎 , ( 2nd ‘ 𝑧 ) 〉 ∈ 𝑥 } → 〈 ( 1st ‘ 𝑤 ) , ( 2nd ‘ 𝑧 ) 〉 ∈ 𝑥 ) |
75 |
70 74
|
syl |
⊢ ( ( ( ( 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 |
⊢ ( 𝑎 = ( 2nd ‘ 𝑧 ) → 〈 ( 1st ‘ 𝑤 ) , 𝑎 〉 = 〈 ( 1st ‘ 𝑤 ) , ( 2nd ‘ 𝑧 ) 〉 ) |
77 |
76
|
eleq1d |
⊢ ( 𝑎 = ( 2nd ‘ 𝑧 ) → ( 〈 ( 1st ‘ 𝑤 ) , 𝑎 〉 ∈ 𝑥 ↔ 〈 ( 1st ‘ 𝑤 ) , ( 2nd ‘ 𝑧 ) 〉 ∈ 𝑥 ) ) |
78 |
77
|
rspcev |
⊢ ( ( ( 2nd ‘ 𝑧 ) ∈ ∪ 𝑆 ∧ 〈 ( 1st ‘ 𝑤 ) , ( 2nd ‘ 𝑧 ) 〉 ∈ 𝑥 ) → ∃ 𝑎 ∈ ∪ 𝑆 〈 ( 1st ‘ 𝑤 ) , 𝑎 〉 ∈ 𝑥 ) |
79 |
45 75 78
|
syl2anc |
⊢ ( ( ( ( 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 |
⊢ ( { 𝑎 ∈ ∪ 𝑆 ∣ 〈 ( 1st ‘ 𝑤 ) , 𝑎 〉 ∈ 𝑥 } ≠ ∅ ↔ ∃ 𝑎 ∈ ∪ 𝑆 〈 ( 1st ‘ 𝑤 ) , 𝑎 〉 ∈ 𝑥 ) |
81 |
79 80
|
sylibr |
⊢ ( ( ( ( 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 |
⊢ ( ( ( 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 |
⊢ ( ( ( ( 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 |
⊢ ( ( ( ( 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 |
⊢ ( ( ( ( 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 |
⊢ ( ( ( ( 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 |
⊢ ( 𝑎 = ( 2nd ‘ 𝑤 ) → 〈 ( 1st ‘ 𝑤 ) , 𝑎 〉 = 〈 ( 1st ‘ 𝑤 ) , ( 2nd ‘ 𝑤 ) 〉 ) |
88 |
87
|
eleq1d |
⊢ ( 𝑎 = ( 2nd ‘ 𝑤 ) → ( 〈 ( 1st ‘ 𝑤 ) , 𝑎 〉 ∈ 𝑥 ↔ 〈 ( 1st ‘ 𝑤 ) , ( 2nd ‘ 𝑤 ) 〉 ∈ 𝑥 ) ) |
89 |
88
|
elrab |
⊢ ( ( 2nd ‘ 𝑤 ) ∈ { 𝑎 ∈ ∪ 𝑆 ∣ 〈 ( 1st ‘ 𝑤 ) , 𝑎 〉 ∈ 𝑥 } ↔ ( ( 2nd ‘ 𝑤 ) ∈ ∪ 𝑆 ∧ 〈 ( 1st ‘ 𝑤 ) , ( 2nd ‘ 𝑤 ) 〉 ∈ 𝑥 ) ) |
90 |
89
|
simprbi |
⊢ ( ( 2nd ‘ 𝑤 ) ∈ { 𝑎 ∈ ∪ 𝑆 ∣ 〈 ( 1st ‘ 𝑤 ) , 𝑎 〉 ∈ 𝑥 } → 〈 ( 1st ‘ 𝑤 ) , ( 2nd ‘ 𝑤 ) 〉 ∈ 𝑥 ) |
91 |
86 90
|
syl |
⊢ ( ( ( ( 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 |
⊢ ( ( ( ( 𝑅 ∈ Conn ∧ 𝑆 ∈ Conn ) ∧ 𝑥 ∈ ( ( 𝑅 ×t 𝑆 ) ∩ ( Clsd ‘ ( 𝑅 ×t 𝑆 ) ) ) ) ∧ ( 𝑧 ∈ 𝑥 ∧ 𝑤 ∈ ∪ ( 𝑅 ×t 𝑆 ) ) ) → 𝑤 ∈ 𝑥 ) |
93 |
92
|
expr |
⊢ ( ( ( ( 𝑅 ∈ Conn ∧ 𝑆 ∈ Conn ) ∧ 𝑥 ∈ ( ( 𝑅 ×t 𝑆 ) ∩ ( Clsd ‘ ( 𝑅 ×t 𝑆 ) ) ) ) ∧ 𝑧 ∈ 𝑥 ) → ( 𝑤 ∈ ∪ ( 𝑅 ×t 𝑆 ) → 𝑤 ∈ 𝑥 ) ) |
94 |
93
|
ssrdv |
⊢ ( ( ( ( 𝑅 ∈ Conn ∧ 𝑆 ∈ Conn ) ∧ 𝑥 ∈ ( ( 𝑅 ×t 𝑆 ) ∩ ( Clsd ‘ ( 𝑅 ×t 𝑆 ) ) ) ) ∧ 𝑧 ∈ 𝑥 ) → ∪ ( 𝑅 ×t 𝑆 ) ⊆ 𝑥 ) |
95 |
9 94
|
eqssd |
⊢ ( ( ( ( 𝑅 ∈ Conn ∧ 𝑆 ∈ Conn ) ∧ 𝑥 ∈ ( ( 𝑅 ×t 𝑆 ) ∩ ( Clsd ‘ ( 𝑅 ×t 𝑆 ) ) ) ) ∧ 𝑧 ∈ 𝑥 ) → 𝑥 = ∪ ( 𝑅 ×t 𝑆 ) ) |
96 |
95
|
ex |
⊢ ( ( ( 𝑅 ∈ Conn ∧ 𝑆 ∈ Conn ) ∧ 𝑥 ∈ ( ( 𝑅 ×t 𝑆 ) ∩ ( Clsd ‘ ( 𝑅 ×t 𝑆 ) ) ) ) → ( 𝑧 ∈ 𝑥 → 𝑥 = ∪ ( 𝑅 ×t 𝑆 ) ) ) |
97 |
96
|
exlimdv |
⊢ ( ( ( 𝑅 ∈ Conn ∧ 𝑆 ∈ Conn ) ∧ 𝑥 ∈ ( ( 𝑅 ×t 𝑆 ) ∩ ( Clsd ‘ ( 𝑅 ×t 𝑆 ) ) ) ) → ( ∃ 𝑧 𝑧 ∈ 𝑥 → 𝑥 = ∪ ( 𝑅 ×t 𝑆 ) ) ) |
98 |
5 97
|
syl5bi |
⊢ ( ( ( 𝑅 ∈ Conn ∧ 𝑆 ∈ Conn ) ∧ 𝑥 ∈ ( ( 𝑅 ×t 𝑆 ) ∩ ( Clsd ‘ ( 𝑅 ×t 𝑆 ) ) ) ) → ( ¬ 𝑥 = ∅ → 𝑥 = ∪ ( 𝑅 ×t 𝑆 ) ) ) |
99 |
98
|
orrd |
⊢ ( ( ( 𝑅 ∈ Conn ∧ 𝑆 ∈ Conn ) ∧ 𝑥 ∈ ( ( 𝑅 ×t 𝑆 ) ∩ ( Clsd ‘ ( 𝑅 ×t 𝑆 ) ) ) ) → ( 𝑥 = ∅ ∨ 𝑥 = ∪ ( 𝑅 ×t 𝑆 ) ) ) |
100 |
99
|
ex |
⊢ ( ( 𝑅 ∈ Conn ∧ 𝑆 ∈ Conn ) → ( 𝑥 ∈ ( ( 𝑅 ×t 𝑆 ) ∩ ( Clsd ‘ ( 𝑅 ×t 𝑆 ) ) ) → ( 𝑥 = ∅ ∨ 𝑥 = ∪ ( 𝑅 ×t 𝑆 ) ) ) ) |
101 |
|
vex |
⊢ 𝑥 ∈ V |
102 |
101
|
elpr |
⊢ ( 𝑥 ∈ { ∅ , ∪ ( 𝑅 ×t 𝑆 ) } ↔ ( 𝑥 = ∅ ∨ 𝑥 = ∪ ( 𝑅 ×t 𝑆 ) ) ) |
103 |
100 102
|
syl6ibr |
⊢ ( ( 𝑅 ∈ Conn ∧ 𝑆 ∈ Conn ) → ( 𝑥 ∈ ( ( 𝑅 ×t 𝑆 ) ∩ ( Clsd ‘ ( 𝑅 ×t 𝑆 ) ) ) → 𝑥 ∈ { ∅ , ∪ ( 𝑅 ×t 𝑆 ) } ) ) |
104 |
103
|
ssrdv |
⊢ ( ( 𝑅 ∈ Conn ∧ 𝑆 ∈ Conn ) → ( ( 𝑅 ×t 𝑆 ) ∩ ( Clsd ‘ ( 𝑅 ×t 𝑆 ) ) ) ⊆ { ∅ , ∪ ( 𝑅 ×t 𝑆 ) } ) |
105 |
|
eqid |
⊢ ∪ ( 𝑅 ×t 𝑆 ) = ∪ ( 𝑅 ×t 𝑆 ) |
106 |
105
|
isconn2 |
⊢ ( ( 𝑅 ×t 𝑆 ) ∈ Conn ↔ ( ( 𝑅 ×t 𝑆 ) ∈ Top ∧ ( ( 𝑅 ×t 𝑆 ) ∩ ( Clsd ‘ ( 𝑅 ×t 𝑆 ) ) ) ⊆ { ∅ , ∪ ( 𝑅 ×t 𝑆 ) } ) ) |
107 |
4 104 106
|
sylanbrc |
⊢ ( ( 𝑅 ∈ Conn ∧ 𝑆 ∈ Conn ) → ( 𝑅 ×t 𝑆 ) ∈ Conn ) |