Step |
Hyp |
Ref |
Expression |
1 |
|
ptunhmeo.x |
⊢ 𝑋 = ∪ 𝐾 |
2 |
|
ptunhmeo.y |
⊢ 𝑌 = ∪ 𝐿 |
3 |
|
ptunhmeo.j |
⊢ 𝐽 = ( ∏t ‘ 𝐹 ) |
4 |
|
ptunhmeo.k |
⊢ 𝐾 = ( ∏t ‘ ( 𝐹 ↾ 𝐴 ) ) |
5 |
|
ptunhmeo.l |
⊢ 𝐿 = ( ∏t ‘ ( 𝐹 ↾ 𝐵 ) ) |
6 |
|
ptunhmeo.g |
⊢ 𝐺 = ( 𝑥 ∈ 𝑋 , 𝑦 ∈ 𝑌 ↦ ( 𝑥 ∪ 𝑦 ) ) |
7 |
|
ptunhmeo.c |
⊢ ( 𝜑 → 𝐶 ∈ 𝑉 ) |
8 |
|
ptunhmeo.f |
⊢ ( 𝜑 → 𝐹 : 𝐶 ⟶ Top ) |
9 |
|
ptunhmeo.u |
⊢ ( 𝜑 → 𝐶 = ( 𝐴 ∪ 𝐵 ) ) |
10 |
|
ptunhmeo.i |
⊢ ( 𝜑 → ( 𝐴 ∩ 𝐵 ) = ∅ ) |
11 |
|
vex |
⊢ 𝑥 ∈ V |
12 |
|
vex |
⊢ 𝑦 ∈ V |
13 |
11 12
|
op1std |
⊢ ( 𝑧 = 〈 𝑥 , 𝑦 〉 → ( 1st ‘ 𝑧 ) = 𝑥 ) |
14 |
11 12
|
op2ndd |
⊢ ( 𝑧 = 〈 𝑥 , 𝑦 〉 → ( 2nd ‘ 𝑧 ) = 𝑦 ) |
15 |
13 14
|
uneq12d |
⊢ ( 𝑧 = 〈 𝑥 , 𝑦 〉 → ( ( 1st ‘ 𝑧 ) ∪ ( 2nd ‘ 𝑧 ) ) = ( 𝑥 ∪ 𝑦 ) ) |
16 |
15
|
mpompt |
⊢ ( 𝑧 ∈ ( 𝑋 × 𝑌 ) ↦ ( ( 1st ‘ 𝑧 ) ∪ ( 2nd ‘ 𝑧 ) ) ) = ( 𝑥 ∈ 𝑋 , 𝑦 ∈ 𝑌 ↦ ( 𝑥 ∪ 𝑦 ) ) |
17 |
6 16
|
eqtr4i |
⊢ 𝐺 = ( 𝑧 ∈ ( 𝑋 × 𝑌 ) ↦ ( ( 1st ‘ 𝑧 ) ∪ ( 2nd ‘ 𝑧 ) ) ) |
18 |
|
xp1st |
⊢ ( 𝑧 ∈ ( 𝑋 × 𝑌 ) → ( 1st ‘ 𝑧 ) ∈ 𝑋 ) |
19 |
18
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( 𝑋 × 𝑌 ) ) → ( 1st ‘ 𝑧 ) ∈ 𝑋 ) |
20 |
|
ixpeq2 |
⊢ ( ∀ 𝑛 ∈ 𝐴 ∪ ( ( 𝐹 ↾ 𝐴 ) ‘ 𝑛 ) = ∪ ( 𝐹 ‘ 𝑛 ) → X 𝑛 ∈ 𝐴 ∪ ( ( 𝐹 ↾ 𝐴 ) ‘ 𝑛 ) = X 𝑛 ∈ 𝐴 ∪ ( 𝐹 ‘ 𝑛 ) ) |
21 |
|
fvres |
⊢ ( 𝑛 ∈ 𝐴 → ( ( 𝐹 ↾ 𝐴 ) ‘ 𝑛 ) = ( 𝐹 ‘ 𝑛 ) ) |
22 |
21
|
unieqd |
⊢ ( 𝑛 ∈ 𝐴 → ∪ ( ( 𝐹 ↾ 𝐴 ) ‘ 𝑛 ) = ∪ ( 𝐹 ‘ 𝑛 ) ) |
23 |
20 22
|
mprg |
⊢ X 𝑛 ∈ 𝐴 ∪ ( ( 𝐹 ↾ 𝐴 ) ‘ 𝑛 ) = X 𝑛 ∈ 𝐴 ∪ ( 𝐹 ‘ 𝑛 ) |
24 |
|
ssun1 |
⊢ 𝐴 ⊆ ( 𝐴 ∪ 𝐵 ) |
25 |
24 9
|
sseqtrrid |
⊢ ( 𝜑 → 𝐴 ⊆ 𝐶 ) |
26 |
7 25
|
ssexd |
⊢ ( 𝜑 → 𝐴 ∈ V ) |
27 |
8 25
|
fssresd |
⊢ ( 𝜑 → ( 𝐹 ↾ 𝐴 ) : 𝐴 ⟶ Top ) |
28 |
4
|
ptuni |
⊢ ( ( 𝐴 ∈ V ∧ ( 𝐹 ↾ 𝐴 ) : 𝐴 ⟶ Top ) → X 𝑛 ∈ 𝐴 ∪ ( ( 𝐹 ↾ 𝐴 ) ‘ 𝑛 ) = ∪ 𝐾 ) |
29 |
26 27 28
|
syl2anc |
⊢ ( 𝜑 → X 𝑛 ∈ 𝐴 ∪ ( ( 𝐹 ↾ 𝐴 ) ‘ 𝑛 ) = ∪ 𝐾 ) |
30 |
23 29
|
syl5eqr |
⊢ ( 𝜑 → X 𝑛 ∈ 𝐴 ∪ ( 𝐹 ‘ 𝑛 ) = ∪ 𝐾 ) |
31 |
30 1
|
eqtr4di |
⊢ ( 𝜑 → X 𝑛 ∈ 𝐴 ∪ ( 𝐹 ‘ 𝑛 ) = 𝑋 ) |
32 |
31
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( 𝑋 × 𝑌 ) ) → X 𝑛 ∈ 𝐴 ∪ ( 𝐹 ‘ 𝑛 ) = 𝑋 ) |
33 |
19 32
|
eleqtrrd |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( 𝑋 × 𝑌 ) ) → ( 1st ‘ 𝑧 ) ∈ X 𝑛 ∈ 𝐴 ∪ ( 𝐹 ‘ 𝑛 ) ) |
34 |
|
xp2nd |
⊢ ( 𝑧 ∈ ( 𝑋 × 𝑌 ) → ( 2nd ‘ 𝑧 ) ∈ 𝑌 ) |
35 |
34
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( 𝑋 × 𝑌 ) ) → ( 2nd ‘ 𝑧 ) ∈ 𝑌 ) |
36 |
9
|
eqcomd |
⊢ ( 𝜑 → ( 𝐴 ∪ 𝐵 ) = 𝐶 ) |
37 |
|
uneqdifeq |
⊢ ( ( 𝐴 ⊆ 𝐶 ∧ ( 𝐴 ∩ 𝐵 ) = ∅ ) → ( ( 𝐴 ∪ 𝐵 ) = 𝐶 ↔ ( 𝐶 ∖ 𝐴 ) = 𝐵 ) ) |
38 |
25 10 37
|
syl2anc |
⊢ ( 𝜑 → ( ( 𝐴 ∪ 𝐵 ) = 𝐶 ↔ ( 𝐶 ∖ 𝐴 ) = 𝐵 ) ) |
39 |
36 38
|
mpbid |
⊢ ( 𝜑 → ( 𝐶 ∖ 𝐴 ) = 𝐵 ) |
40 |
39
|
ixpeq1d |
⊢ ( 𝜑 → X 𝑛 ∈ ( 𝐶 ∖ 𝐴 ) ∪ ( 𝐹 ‘ 𝑛 ) = X 𝑛 ∈ 𝐵 ∪ ( 𝐹 ‘ 𝑛 ) ) |
41 |
|
ixpeq2 |
⊢ ( ∀ 𝑛 ∈ 𝐵 ∪ ( ( 𝐹 ↾ 𝐵 ) ‘ 𝑛 ) = ∪ ( 𝐹 ‘ 𝑛 ) → X 𝑛 ∈ 𝐵 ∪ ( ( 𝐹 ↾ 𝐵 ) ‘ 𝑛 ) = X 𝑛 ∈ 𝐵 ∪ ( 𝐹 ‘ 𝑛 ) ) |
42 |
|
fvres |
⊢ ( 𝑛 ∈ 𝐵 → ( ( 𝐹 ↾ 𝐵 ) ‘ 𝑛 ) = ( 𝐹 ‘ 𝑛 ) ) |
43 |
42
|
unieqd |
⊢ ( 𝑛 ∈ 𝐵 → ∪ ( ( 𝐹 ↾ 𝐵 ) ‘ 𝑛 ) = ∪ ( 𝐹 ‘ 𝑛 ) ) |
44 |
41 43
|
mprg |
⊢ X 𝑛 ∈ 𝐵 ∪ ( ( 𝐹 ↾ 𝐵 ) ‘ 𝑛 ) = X 𝑛 ∈ 𝐵 ∪ ( 𝐹 ‘ 𝑛 ) |
45 |
|
ssun2 |
⊢ 𝐵 ⊆ ( 𝐴 ∪ 𝐵 ) |
46 |
45 9
|
sseqtrrid |
⊢ ( 𝜑 → 𝐵 ⊆ 𝐶 ) |
47 |
7 46
|
ssexd |
⊢ ( 𝜑 → 𝐵 ∈ V ) |
48 |
8 46
|
fssresd |
⊢ ( 𝜑 → ( 𝐹 ↾ 𝐵 ) : 𝐵 ⟶ Top ) |
49 |
5
|
ptuni |
⊢ ( ( 𝐵 ∈ V ∧ ( 𝐹 ↾ 𝐵 ) : 𝐵 ⟶ Top ) → X 𝑛 ∈ 𝐵 ∪ ( ( 𝐹 ↾ 𝐵 ) ‘ 𝑛 ) = ∪ 𝐿 ) |
50 |
47 48 49
|
syl2anc |
⊢ ( 𝜑 → X 𝑛 ∈ 𝐵 ∪ ( ( 𝐹 ↾ 𝐵 ) ‘ 𝑛 ) = ∪ 𝐿 ) |
51 |
44 50
|
syl5eqr |
⊢ ( 𝜑 → X 𝑛 ∈ 𝐵 ∪ ( 𝐹 ‘ 𝑛 ) = ∪ 𝐿 ) |
52 |
51 2
|
eqtr4di |
⊢ ( 𝜑 → X 𝑛 ∈ 𝐵 ∪ ( 𝐹 ‘ 𝑛 ) = 𝑌 ) |
53 |
40 52
|
eqtrd |
⊢ ( 𝜑 → X 𝑛 ∈ ( 𝐶 ∖ 𝐴 ) ∪ ( 𝐹 ‘ 𝑛 ) = 𝑌 ) |
54 |
53
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( 𝑋 × 𝑌 ) ) → X 𝑛 ∈ ( 𝐶 ∖ 𝐴 ) ∪ ( 𝐹 ‘ 𝑛 ) = 𝑌 ) |
55 |
35 54
|
eleqtrrd |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( 𝑋 × 𝑌 ) ) → ( 2nd ‘ 𝑧 ) ∈ X 𝑛 ∈ ( 𝐶 ∖ 𝐴 ) ∪ ( 𝐹 ‘ 𝑛 ) ) |
56 |
25
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( 𝑋 × 𝑌 ) ) → 𝐴 ⊆ 𝐶 ) |
57 |
|
undifixp |
⊢ ( ( ( 1st ‘ 𝑧 ) ∈ X 𝑛 ∈ 𝐴 ∪ ( 𝐹 ‘ 𝑛 ) ∧ ( 2nd ‘ 𝑧 ) ∈ X 𝑛 ∈ ( 𝐶 ∖ 𝐴 ) ∪ ( 𝐹 ‘ 𝑛 ) ∧ 𝐴 ⊆ 𝐶 ) → ( ( 1st ‘ 𝑧 ) ∪ ( 2nd ‘ 𝑧 ) ) ∈ X 𝑛 ∈ 𝐶 ∪ ( 𝐹 ‘ 𝑛 ) ) |
58 |
33 55 56 57
|
syl3anc |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( 𝑋 × 𝑌 ) ) → ( ( 1st ‘ 𝑧 ) ∪ ( 2nd ‘ 𝑧 ) ) ∈ X 𝑛 ∈ 𝐶 ∪ ( 𝐹 ‘ 𝑛 ) ) |
59 |
|
ixpfn |
⊢ ( ( ( 1st ‘ 𝑧 ) ∪ ( 2nd ‘ 𝑧 ) ) ∈ X 𝑛 ∈ 𝐶 ∪ ( 𝐹 ‘ 𝑛 ) → ( ( 1st ‘ 𝑧 ) ∪ ( 2nd ‘ 𝑧 ) ) Fn 𝐶 ) |
60 |
58 59
|
syl |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( 𝑋 × 𝑌 ) ) → ( ( 1st ‘ 𝑧 ) ∪ ( 2nd ‘ 𝑧 ) ) Fn 𝐶 ) |
61 |
|
dffn5 |
⊢ ( ( ( 1st ‘ 𝑧 ) ∪ ( 2nd ‘ 𝑧 ) ) Fn 𝐶 ↔ ( ( 1st ‘ 𝑧 ) ∪ ( 2nd ‘ 𝑧 ) ) = ( 𝑘 ∈ 𝐶 ↦ ( ( ( 1st ‘ 𝑧 ) ∪ ( 2nd ‘ 𝑧 ) ) ‘ 𝑘 ) ) ) |
62 |
60 61
|
sylib |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( 𝑋 × 𝑌 ) ) → ( ( 1st ‘ 𝑧 ) ∪ ( 2nd ‘ 𝑧 ) ) = ( 𝑘 ∈ 𝐶 ↦ ( ( ( 1st ‘ 𝑧 ) ∪ ( 2nd ‘ 𝑧 ) ) ‘ 𝑘 ) ) ) |
63 |
62
|
mpteq2dva |
⊢ ( 𝜑 → ( 𝑧 ∈ ( 𝑋 × 𝑌 ) ↦ ( ( 1st ‘ 𝑧 ) ∪ ( 2nd ‘ 𝑧 ) ) ) = ( 𝑧 ∈ ( 𝑋 × 𝑌 ) ↦ ( 𝑘 ∈ 𝐶 ↦ ( ( ( 1st ‘ 𝑧 ) ∪ ( 2nd ‘ 𝑧 ) ) ‘ 𝑘 ) ) ) ) |
64 |
17 63
|
syl5eq |
⊢ ( 𝜑 → 𝐺 = ( 𝑧 ∈ ( 𝑋 × 𝑌 ) ↦ ( 𝑘 ∈ 𝐶 ↦ ( ( ( 1st ‘ 𝑧 ) ∪ ( 2nd ‘ 𝑧 ) ) ‘ 𝑘 ) ) ) ) |
65 |
|
pttop |
⊢ ( ( 𝐴 ∈ V ∧ ( 𝐹 ↾ 𝐴 ) : 𝐴 ⟶ Top ) → ( ∏t ‘ ( 𝐹 ↾ 𝐴 ) ) ∈ Top ) |
66 |
26 27 65
|
syl2anc |
⊢ ( 𝜑 → ( ∏t ‘ ( 𝐹 ↾ 𝐴 ) ) ∈ Top ) |
67 |
4 66
|
eqeltrid |
⊢ ( 𝜑 → 𝐾 ∈ Top ) |
68 |
1
|
toptopon |
⊢ ( 𝐾 ∈ Top ↔ 𝐾 ∈ ( TopOn ‘ 𝑋 ) ) |
69 |
67 68
|
sylib |
⊢ ( 𝜑 → 𝐾 ∈ ( TopOn ‘ 𝑋 ) ) |
70 |
|
pttop |
⊢ ( ( 𝐵 ∈ V ∧ ( 𝐹 ↾ 𝐵 ) : 𝐵 ⟶ Top ) → ( ∏t ‘ ( 𝐹 ↾ 𝐵 ) ) ∈ Top ) |
71 |
47 48 70
|
syl2anc |
⊢ ( 𝜑 → ( ∏t ‘ ( 𝐹 ↾ 𝐵 ) ) ∈ Top ) |
72 |
5 71
|
eqeltrid |
⊢ ( 𝜑 → 𝐿 ∈ Top ) |
73 |
2
|
toptopon |
⊢ ( 𝐿 ∈ Top ↔ 𝐿 ∈ ( TopOn ‘ 𝑌 ) ) |
74 |
72 73
|
sylib |
⊢ ( 𝜑 → 𝐿 ∈ ( TopOn ‘ 𝑌 ) ) |
75 |
|
txtopon |
⊢ ( ( 𝐾 ∈ ( TopOn ‘ 𝑋 ) ∧ 𝐿 ∈ ( TopOn ‘ 𝑌 ) ) → ( 𝐾 ×t 𝐿 ) ∈ ( TopOn ‘ ( 𝑋 × 𝑌 ) ) ) |
76 |
69 74 75
|
syl2anc |
⊢ ( 𝜑 → ( 𝐾 ×t 𝐿 ) ∈ ( TopOn ‘ ( 𝑋 × 𝑌 ) ) ) |
77 |
9
|
eleq2d |
⊢ ( 𝜑 → ( 𝑘 ∈ 𝐶 ↔ 𝑘 ∈ ( 𝐴 ∪ 𝐵 ) ) ) |
78 |
77
|
biimpa |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐶 ) → 𝑘 ∈ ( 𝐴 ∪ 𝐵 ) ) |
79 |
|
elun |
⊢ ( 𝑘 ∈ ( 𝐴 ∪ 𝐵 ) ↔ ( 𝑘 ∈ 𝐴 ∨ 𝑘 ∈ 𝐵 ) ) |
80 |
78 79
|
sylib |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐶 ) → ( 𝑘 ∈ 𝐴 ∨ 𝑘 ∈ 𝐵 ) ) |
81 |
|
ixpfn |
⊢ ( ( 1st ‘ 𝑧 ) ∈ X 𝑛 ∈ 𝐴 ∪ ( 𝐹 ‘ 𝑛 ) → ( 1st ‘ 𝑧 ) Fn 𝐴 ) |
82 |
33 81
|
syl |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( 𝑋 × 𝑌 ) ) → ( 1st ‘ 𝑧 ) Fn 𝐴 ) |
83 |
82
|
adantlr |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) ∧ 𝑧 ∈ ( 𝑋 × 𝑌 ) ) → ( 1st ‘ 𝑧 ) Fn 𝐴 ) |
84 |
52
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( 𝑋 × 𝑌 ) ) → X 𝑛 ∈ 𝐵 ∪ ( 𝐹 ‘ 𝑛 ) = 𝑌 ) |
85 |
35 84
|
eleqtrrd |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( 𝑋 × 𝑌 ) ) → ( 2nd ‘ 𝑧 ) ∈ X 𝑛 ∈ 𝐵 ∪ ( 𝐹 ‘ 𝑛 ) ) |
86 |
|
ixpfn |
⊢ ( ( 2nd ‘ 𝑧 ) ∈ X 𝑛 ∈ 𝐵 ∪ ( 𝐹 ‘ 𝑛 ) → ( 2nd ‘ 𝑧 ) Fn 𝐵 ) |
87 |
85 86
|
syl |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( 𝑋 × 𝑌 ) ) → ( 2nd ‘ 𝑧 ) Fn 𝐵 ) |
88 |
87
|
adantlr |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) ∧ 𝑧 ∈ ( 𝑋 × 𝑌 ) ) → ( 2nd ‘ 𝑧 ) Fn 𝐵 ) |
89 |
10
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) ∧ 𝑧 ∈ ( 𝑋 × 𝑌 ) ) → ( 𝐴 ∩ 𝐵 ) = ∅ ) |
90 |
|
simplr |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) ∧ 𝑧 ∈ ( 𝑋 × 𝑌 ) ) → 𝑘 ∈ 𝐴 ) |
91 |
|
fvun1 |
⊢ ( ( ( 1st ‘ 𝑧 ) Fn 𝐴 ∧ ( 2nd ‘ 𝑧 ) Fn 𝐵 ∧ ( ( 𝐴 ∩ 𝐵 ) = ∅ ∧ 𝑘 ∈ 𝐴 ) ) → ( ( ( 1st ‘ 𝑧 ) ∪ ( 2nd ‘ 𝑧 ) ) ‘ 𝑘 ) = ( ( 1st ‘ 𝑧 ) ‘ 𝑘 ) ) |
92 |
83 88 89 90 91
|
syl112anc |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) ∧ 𝑧 ∈ ( 𝑋 × 𝑌 ) ) → ( ( ( 1st ‘ 𝑧 ) ∪ ( 2nd ‘ 𝑧 ) ) ‘ 𝑘 ) = ( ( 1st ‘ 𝑧 ) ‘ 𝑘 ) ) |
93 |
92
|
mpteq2dva |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) → ( 𝑧 ∈ ( 𝑋 × 𝑌 ) ↦ ( ( ( 1st ‘ 𝑧 ) ∪ ( 2nd ‘ 𝑧 ) ) ‘ 𝑘 ) ) = ( 𝑧 ∈ ( 𝑋 × 𝑌 ) ↦ ( ( 1st ‘ 𝑧 ) ‘ 𝑘 ) ) ) |
94 |
76
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) → ( 𝐾 ×t 𝐿 ) ∈ ( TopOn ‘ ( 𝑋 × 𝑌 ) ) ) |
95 |
13
|
mpompt |
⊢ ( 𝑧 ∈ ( 𝑋 × 𝑌 ) ↦ ( 1st ‘ 𝑧 ) ) = ( 𝑥 ∈ 𝑋 , 𝑦 ∈ 𝑌 ↦ 𝑥 ) |
96 |
69
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) → 𝐾 ∈ ( TopOn ‘ 𝑋 ) ) |
97 |
74
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) → 𝐿 ∈ ( TopOn ‘ 𝑌 ) ) |
98 |
96 97
|
cnmpt1st |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) → ( 𝑥 ∈ 𝑋 , 𝑦 ∈ 𝑌 ↦ 𝑥 ) ∈ ( ( 𝐾 ×t 𝐿 ) Cn 𝐾 ) ) |
99 |
95 98
|
eqeltrid |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) → ( 𝑧 ∈ ( 𝑋 × 𝑌 ) ↦ ( 1st ‘ 𝑧 ) ) ∈ ( ( 𝐾 ×t 𝐿 ) Cn 𝐾 ) ) |
100 |
26
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) → 𝐴 ∈ V ) |
101 |
27
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) → ( 𝐹 ↾ 𝐴 ) : 𝐴 ⟶ Top ) |
102 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) → 𝑘 ∈ 𝐴 ) |
103 |
1 4
|
ptpjcn |
⊢ ( ( 𝐴 ∈ V ∧ ( 𝐹 ↾ 𝐴 ) : 𝐴 ⟶ Top ∧ 𝑘 ∈ 𝐴 ) → ( 𝑓 ∈ 𝑋 ↦ ( 𝑓 ‘ 𝑘 ) ) ∈ ( 𝐾 Cn ( ( 𝐹 ↾ 𝐴 ) ‘ 𝑘 ) ) ) |
104 |
100 101 102 103
|
syl3anc |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) → ( 𝑓 ∈ 𝑋 ↦ ( 𝑓 ‘ 𝑘 ) ) ∈ ( 𝐾 Cn ( ( 𝐹 ↾ 𝐴 ) ‘ 𝑘 ) ) ) |
105 |
|
fvres |
⊢ ( 𝑘 ∈ 𝐴 → ( ( 𝐹 ↾ 𝐴 ) ‘ 𝑘 ) = ( 𝐹 ‘ 𝑘 ) ) |
106 |
105
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) → ( ( 𝐹 ↾ 𝐴 ) ‘ 𝑘 ) = ( 𝐹 ‘ 𝑘 ) ) |
107 |
106
|
oveq2d |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) → ( 𝐾 Cn ( ( 𝐹 ↾ 𝐴 ) ‘ 𝑘 ) ) = ( 𝐾 Cn ( 𝐹 ‘ 𝑘 ) ) ) |
108 |
104 107
|
eleqtrd |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) → ( 𝑓 ∈ 𝑋 ↦ ( 𝑓 ‘ 𝑘 ) ) ∈ ( 𝐾 Cn ( 𝐹 ‘ 𝑘 ) ) ) |
109 |
|
fveq1 |
⊢ ( 𝑓 = ( 1st ‘ 𝑧 ) → ( 𝑓 ‘ 𝑘 ) = ( ( 1st ‘ 𝑧 ) ‘ 𝑘 ) ) |
110 |
94 99 96 108 109
|
cnmpt11 |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) → ( 𝑧 ∈ ( 𝑋 × 𝑌 ) ↦ ( ( 1st ‘ 𝑧 ) ‘ 𝑘 ) ) ∈ ( ( 𝐾 ×t 𝐿 ) Cn ( 𝐹 ‘ 𝑘 ) ) ) |
111 |
93 110
|
eqeltrd |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) → ( 𝑧 ∈ ( 𝑋 × 𝑌 ) ↦ ( ( ( 1st ‘ 𝑧 ) ∪ ( 2nd ‘ 𝑧 ) ) ‘ 𝑘 ) ) ∈ ( ( 𝐾 ×t 𝐿 ) Cn ( 𝐹 ‘ 𝑘 ) ) ) |
112 |
82
|
adantlr |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ 𝐵 ) ∧ 𝑧 ∈ ( 𝑋 × 𝑌 ) ) → ( 1st ‘ 𝑧 ) Fn 𝐴 ) |
113 |
87
|
adantlr |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ 𝐵 ) ∧ 𝑧 ∈ ( 𝑋 × 𝑌 ) ) → ( 2nd ‘ 𝑧 ) Fn 𝐵 ) |
114 |
10
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ 𝐵 ) ∧ 𝑧 ∈ ( 𝑋 × 𝑌 ) ) → ( 𝐴 ∩ 𝐵 ) = ∅ ) |
115 |
|
simplr |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ 𝐵 ) ∧ 𝑧 ∈ ( 𝑋 × 𝑌 ) ) → 𝑘 ∈ 𝐵 ) |
116 |
|
fvun2 |
⊢ ( ( ( 1st ‘ 𝑧 ) Fn 𝐴 ∧ ( 2nd ‘ 𝑧 ) Fn 𝐵 ∧ ( ( 𝐴 ∩ 𝐵 ) = ∅ ∧ 𝑘 ∈ 𝐵 ) ) → ( ( ( 1st ‘ 𝑧 ) ∪ ( 2nd ‘ 𝑧 ) ) ‘ 𝑘 ) = ( ( 2nd ‘ 𝑧 ) ‘ 𝑘 ) ) |
117 |
112 113 114 115 116
|
syl112anc |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ 𝐵 ) ∧ 𝑧 ∈ ( 𝑋 × 𝑌 ) ) → ( ( ( 1st ‘ 𝑧 ) ∪ ( 2nd ‘ 𝑧 ) ) ‘ 𝑘 ) = ( ( 2nd ‘ 𝑧 ) ‘ 𝑘 ) ) |
118 |
117
|
mpteq2dva |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐵 ) → ( 𝑧 ∈ ( 𝑋 × 𝑌 ) ↦ ( ( ( 1st ‘ 𝑧 ) ∪ ( 2nd ‘ 𝑧 ) ) ‘ 𝑘 ) ) = ( 𝑧 ∈ ( 𝑋 × 𝑌 ) ↦ ( ( 2nd ‘ 𝑧 ) ‘ 𝑘 ) ) ) |
119 |
76
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐵 ) → ( 𝐾 ×t 𝐿 ) ∈ ( TopOn ‘ ( 𝑋 × 𝑌 ) ) ) |
120 |
14
|
mpompt |
⊢ ( 𝑧 ∈ ( 𝑋 × 𝑌 ) ↦ ( 2nd ‘ 𝑧 ) ) = ( 𝑥 ∈ 𝑋 , 𝑦 ∈ 𝑌 ↦ 𝑦 ) |
121 |
69
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐵 ) → 𝐾 ∈ ( TopOn ‘ 𝑋 ) ) |
122 |
74
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐵 ) → 𝐿 ∈ ( TopOn ‘ 𝑌 ) ) |
123 |
121 122
|
cnmpt2nd |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐵 ) → ( 𝑥 ∈ 𝑋 , 𝑦 ∈ 𝑌 ↦ 𝑦 ) ∈ ( ( 𝐾 ×t 𝐿 ) Cn 𝐿 ) ) |
124 |
120 123
|
eqeltrid |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐵 ) → ( 𝑧 ∈ ( 𝑋 × 𝑌 ) ↦ ( 2nd ‘ 𝑧 ) ) ∈ ( ( 𝐾 ×t 𝐿 ) Cn 𝐿 ) ) |
125 |
47
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐵 ) → 𝐵 ∈ V ) |
126 |
48
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐵 ) → ( 𝐹 ↾ 𝐵 ) : 𝐵 ⟶ Top ) |
127 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐵 ) → 𝑘 ∈ 𝐵 ) |
128 |
2 5
|
ptpjcn |
⊢ ( ( 𝐵 ∈ V ∧ ( 𝐹 ↾ 𝐵 ) : 𝐵 ⟶ Top ∧ 𝑘 ∈ 𝐵 ) → ( 𝑓 ∈ 𝑌 ↦ ( 𝑓 ‘ 𝑘 ) ) ∈ ( 𝐿 Cn ( ( 𝐹 ↾ 𝐵 ) ‘ 𝑘 ) ) ) |
129 |
125 126 127 128
|
syl3anc |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐵 ) → ( 𝑓 ∈ 𝑌 ↦ ( 𝑓 ‘ 𝑘 ) ) ∈ ( 𝐿 Cn ( ( 𝐹 ↾ 𝐵 ) ‘ 𝑘 ) ) ) |
130 |
|
fvres |
⊢ ( 𝑘 ∈ 𝐵 → ( ( 𝐹 ↾ 𝐵 ) ‘ 𝑘 ) = ( 𝐹 ‘ 𝑘 ) ) |
131 |
130
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐵 ) → ( ( 𝐹 ↾ 𝐵 ) ‘ 𝑘 ) = ( 𝐹 ‘ 𝑘 ) ) |
132 |
131
|
oveq2d |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐵 ) → ( 𝐿 Cn ( ( 𝐹 ↾ 𝐵 ) ‘ 𝑘 ) ) = ( 𝐿 Cn ( 𝐹 ‘ 𝑘 ) ) ) |
133 |
129 132
|
eleqtrd |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐵 ) → ( 𝑓 ∈ 𝑌 ↦ ( 𝑓 ‘ 𝑘 ) ) ∈ ( 𝐿 Cn ( 𝐹 ‘ 𝑘 ) ) ) |
134 |
|
fveq1 |
⊢ ( 𝑓 = ( 2nd ‘ 𝑧 ) → ( 𝑓 ‘ 𝑘 ) = ( ( 2nd ‘ 𝑧 ) ‘ 𝑘 ) ) |
135 |
119 124 122 133 134
|
cnmpt11 |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐵 ) → ( 𝑧 ∈ ( 𝑋 × 𝑌 ) ↦ ( ( 2nd ‘ 𝑧 ) ‘ 𝑘 ) ) ∈ ( ( 𝐾 ×t 𝐿 ) Cn ( 𝐹 ‘ 𝑘 ) ) ) |
136 |
118 135
|
eqeltrd |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐵 ) → ( 𝑧 ∈ ( 𝑋 × 𝑌 ) ↦ ( ( ( 1st ‘ 𝑧 ) ∪ ( 2nd ‘ 𝑧 ) ) ‘ 𝑘 ) ) ∈ ( ( 𝐾 ×t 𝐿 ) Cn ( 𝐹 ‘ 𝑘 ) ) ) |
137 |
111 136
|
jaodan |
⊢ ( ( 𝜑 ∧ ( 𝑘 ∈ 𝐴 ∨ 𝑘 ∈ 𝐵 ) ) → ( 𝑧 ∈ ( 𝑋 × 𝑌 ) ↦ ( ( ( 1st ‘ 𝑧 ) ∪ ( 2nd ‘ 𝑧 ) ) ‘ 𝑘 ) ) ∈ ( ( 𝐾 ×t 𝐿 ) Cn ( 𝐹 ‘ 𝑘 ) ) ) |
138 |
80 137
|
syldan |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐶 ) → ( 𝑧 ∈ ( 𝑋 × 𝑌 ) ↦ ( ( ( 1st ‘ 𝑧 ) ∪ ( 2nd ‘ 𝑧 ) ) ‘ 𝑘 ) ) ∈ ( ( 𝐾 ×t 𝐿 ) Cn ( 𝐹 ‘ 𝑘 ) ) ) |
139 |
3 76 7 8 138
|
ptcn |
⊢ ( 𝜑 → ( 𝑧 ∈ ( 𝑋 × 𝑌 ) ↦ ( 𝑘 ∈ 𝐶 ↦ ( ( ( 1st ‘ 𝑧 ) ∪ ( 2nd ‘ 𝑧 ) ) ‘ 𝑘 ) ) ) ∈ ( ( 𝐾 ×t 𝐿 ) Cn 𝐽 ) ) |
140 |
64 139
|
eqeltrd |
⊢ ( 𝜑 → 𝐺 ∈ ( ( 𝐾 ×t 𝐿 ) Cn 𝐽 ) ) |
141 |
1 2 3 4 5 6 7 8 9 10
|
ptuncnv |
⊢ ( 𝜑 → ◡ 𝐺 = ( 𝑧 ∈ ∪ 𝐽 ↦ 〈 ( 𝑧 ↾ 𝐴 ) , ( 𝑧 ↾ 𝐵 ) 〉 ) ) |
142 |
|
pttop |
⊢ ( ( 𝐶 ∈ 𝑉 ∧ 𝐹 : 𝐶 ⟶ Top ) → ( ∏t ‘ 𝐹 ) ∈ Top ) |
143 |
7 8 142
|
syl2anc |
⊢ ( 𝜑 → ( ∏t ‘ 𝐹 ) ∈ Top ) |
144 |
3 143
|
eqeltrid |
⊢ ( 𝜑 → 𝐽 ∈ Top ) |
145 |
|
eqid |
⊢ ∪ 𝐽 = ∪ 𝐽 |
146 |
145
|
toptopon |
⊢ ( 𝐽 ∈ Top ↔ 𝐽 ∈ ( TopOn ‘ ∪ 𝐽 ) ) |
147 |
144 146
|
sylib |
⊢ ( 𝜑 → 𝐽 ∈ ( TopOn ‘ ∪ 𝐽 ) ) |
148 |
145 3 4
|
ptrescn |
⊢ ( ( 𝐶 ∈ 𝑉 ∧ 𝐹 : 𝐶 ⟶ Top ∧ 𝐴 ⊆ 𝐶 ) → ( 𝑧 ∈ ∪ 𝐽 ↦ ( 𝑧 ↾ 𝐴 ) ) ∈ ( 𝐽 Cn 𝐾 ) ) |
149 |
7 8 25 148
|
syl3anc |
⊢ ( 𝜑 → ( 𝑧 ∈ ∪ 𝐽 ↦ ( 𝑧 ↾ 𝐴 ) ) ∈ ( 𝐽 Cn 𝐾 ) ) |
150 |
145 3 5
|
ptrescn |
⊢ ( ( 𝐶 ∈ 𝑉 ∧ 𝐹 : 𝐶 ⟶ Top ∧ 𝐵 ⊆ 𝐶 ) → ( 𝑧 ∈ ∪ 𝐽 ↦ ( 𝑧 ↾ 𝐵 ) ) ∈ ( 𝐽 Cn 𝐿 ) ) |
151 |
7 8 46 150
|
syl3anc |
⊢ ( 𝜑 → ( 𝑧 ∈ ∪ 𝐽 ↦ ( 𝑧 ↾ 𝐵 ) ) ∈ ( 𝐽 Cn 𝐿 ) ) |
152 |
147 149 151
|
cnmpt1t |
⊢ ( 𝜑 → ( 𝑧 ∈ ∪ 𝐽 ↦ 〈 ( 𝑧 ↾ 𝐴 ) , ( 𝑧 ↾ 𝐵 ) 〉 ) ∈ ( 𝐽 Cn ( 𝐾 ×t 𝐿 ) ) ) |
153 |
141 152
|
eqeltrd |
⊢ ( 𝜑 → ◡ 𝐺 ∈ ( 𝐽 Cn ( 𝐾 ×t 𝐿 ) ) ) |
154 |
|
ishmeo |
⊢ ( 𝐺 ∈ ( ( 𝐾 ×t 𝐿 ) Homeo 𝐽 ) ↔ ( 𝐺 ∈ ( ( 𝐾 ×t 𝐿 ) Cn 𝐽 ) ∧ ◡ 𝐺 ∈ ( 𝐽 Cn ( 𝐾 ×t 𝐿 ) ) ) ) |
155 |
140 153 154
|
sylanbrc |
⊢ ( 𝜑 → 𝐺 ∈ ( ( 𝐾 ×t 𝐿 ) Homeo 𝐽 ) ) |