Step |
Hyp |
Ref |
Expression |
1 |
|
qtopbas |
⊢ ( (,) “ ( ℚ × ℚ ) ) ∈ TopBases |
2 |
|
eltg3 |
⊢ ( ( (,) “ ( ℚ × ℚ ) ) ∈ TopBases → ( 𝐴 ∈ ( topGen ‘ ( (,) “ ( ℚ × ℚ ) ) ) ↔ ∃ 𝑥 ( 𝑥 ⊆ ( (,) “ ( ℚ × ℚ ) ) ∧ 𝐴 = ∪ 𝑥 ) ) ) |
3 |
1 2
|
ax-mp |
⊢ ( 𝐴 ∈ ( topGen ‘ ( (,) “ ( ℚ × ℚ ) ) ) ↔ ∃ 𝑥 ( 𝑥 ⊆ ( (,) “ ( ℚ × ℚ ) ) ∧ 𝐴 = ∪ 𝑥 ) ) |
4 |
|
uniiun |
⊢ ∪ 𝑥 = ∪ 𝑦 ∈ 𝑥 𝑦 |
5 |
|
ssdomg |
⊢ ( ( (,) “ ( ℚ × ℚ ) ) ∈ TopBases → ( 𝑥 ⊆ ( (,) “ ( ℚ × ℚ ) ) → 𝑥 ≼ ( (,) “ ( ℚ × ℚ ) ) ) ) |
6 |
1 5
|
ax-mp |
⊢ ( 𝑥 ⊆ ( (,) “ ( ℚ × ℚ ) ) → 𝑥 ≼ ( (,) “ ( ℚ × ℚ ) ) ) |
7 |
|
omelon |
⊢ ω ∈ On |
8 |
|
qnnen |
⊢ ℚ ≈ ℕ |
9 |
|
xpen |
⊢ ( ( ℚ ≈ ℕ ∧ ℚ ≈ ℕ ) → ( ℚ × ℚ ) ≈ ( ℕ × ℕ ) ) |
10 |
8 8 9
|
mp2an |
⊢ ( ℚ × ℚ ) ≈ ( ℕ × ℕ ) |
11 |
|
xpnnen |
⊢ ( ℕ × ℕ ) ≈ ℕ |
12 |
10 11
|
entri |
⊢ ( ℚ × ℚ ) ≈ ℕ |
13 |
|
nnenom |
⊢ ℕ ≈ ω |
14 |
12 13
|
entr2i |
⊢ ω ≈ ( ℚ × ℚ ) |
15 |
|
isnumi |
⊢ ( ( ω ∈ On ∧ ω ≈ ( ℚ × ℚ ) ) → ( ℚ × ℚ ) ∈ dom card ) |
16 |
7 14 15
|
mp2an |
⊢ ( ℚ × ℚ ) ∈ dom card |
17 |
|
ioof |
⊢ (,) : ( ℝ* × ℝ* ) ⟶ 𝒫 ℝ |
18 |
|
ffun |
⊢ ( (,) : ( ℝ* × ℝ* ) ⟶ 𝒫 ℝ → Fun (,) ) |
19 |
17 18
|
ax-mp |
⊢ Fun (,) |
20 |
|
qssre |
⊢ ℚ ⊆ ℝ |
21 |
|
ressxr |
⊢ ℝ ⊆ ℝ* |
22 |
20 21
|
sstri |
⊢ ℚ ⊆ ℝ* |
23 |
|
xpss12 |
⊢ ( ( ℚ ⊆ ℝ* ∧ ℚ ⊆ ℝ* ) → ( ℚ × ℚ ) ⊆ ( ℝ* × ℝ* ) ) |
24 |
22 22 23
|
mp2an |
⊢ ( ℚ × ℚ ) ⊆ ( ℝ* × ℝ* ) |
25 |
17
|
fdmi |
⊢ dom (,) = ( ℝ* × ℝ* ) |
26 |
24 25
|
sseqtrri |
⊢ ( ℚ × ℚ ) ⊆ dom (,) |
27 |
|
fores |
⊢ ( ( Fun (,) ∧ ( ℚ × ℚ ) ⊆ dom (,) ) → ( (,) ↾ ( ℚ × ℚ ) ) : ( ℚ × ℚ ) –onto→ ( (,) “ ( ℚ × ℚ ) ) ) |
28 |
19 26 27
|
mp2an |
⊢ ( (,) ↾ ( ℚ × ℚ ) ) : ( ℚ × ℚ ) –onto→ ( (,) “ ( ℚ × ℚ ) ) |
29 |
|
fodomnum |
⊢ ( ( ℚ × ℚ ) ∈ dom card → ( ( (,) ↾ ( ℚ × ℚ ) ) : ( ℚ × ℚ ) –onto→ ( (,) “ ( ℚ × ℚ ) ) → ( (,) “ ( ℚ × ℚ ) ) ≼ ( ℚ × ℚ ) ) ) |
30 |
16 28 29
|
mp2 |
⊢ ( (,) “ ( ℚ × ℚ ) ) ≼ ( ℚ × ℚ ) |
31 |
|
domentr |
⊢ ( ( ( (,) “ ( ℚ × ℚ ) ) ≼ ( ℚ × ℚ ) ∧ ( ℚ × ℚ ) ≈ ℕ ) → ( (,) “ ( ℚ × ℚ ) ) ≼ ℕ ) |
32 |
30 12 31
|
mp2an |
⊢ ( (,) “ ( ℚ × ℚ ) ) ≼ ℕ |
33 |
|
domtr |
⊢ ( ( 𝑥 ≼ ( (,) “ ( ℚ × ℚ ) ) ∧ ( (,) “ ( ℚ × ℚ ) ) ≼ ℕ ) → 𝑥 ≼ ℕ ) |
34 |
6 32 33
|
sylancl |
⊢ ( 𝑥 ⊆ ( (,) “ ( ℚ × ℚ ) ) → 𝑥 ≼ ℕ ) |
35 |
|
imassrn |
⊢ ( (,) “ ( ℚ × ℚ ) ) ⊆ ran (,) |
36 |
|
ffn |
⊢ ( (,) : ( ℝ* × ℝ* ) ⟶ 𝒫 ℝ → (,) Fn ( ℝ* × ℝ* ) ) |
37 |
17 36
|
ax-mp |
⊢ (,) Fn ( ℝ* × ℝ* ) |
38 |
|
ioombl |
⊢ ( 𝑥 (,) 𝑦 ) ∈ dom vol |
39 |
38
|
rgen2w |
⊢ ∀ 𝑥 ∈ ℝ* ∀ 𝑦 ∈ ℝ* ( 𝑥 (,) 𝑦 ) ∈ dom vol |
40 |
|
ffnov |
⊢ ( (,) : ( ℝ* × ℝ* ) ⟶ dom vol ↔ ( (,) Fn ( ℝ* × ℝ* ) ∧ ∀ 𝑥 ∈ ℝ* ∀ 𝑦 ∈ ℝ* ( 𝑥 (,) 𝑦 ) ∈ dom vol ) ) |
41 |
37 39 40
|
mpbir2an |
⊢ (,) : ( ℝ* × ℝ* ) ⟶ dom vol |
42 |
|
frn |
⊢ ( (,) : ( ℝ* × ℝ* ) ⟶ dom vol → ran (,) ⊆ dom vol ) |
43 |
41 42
|
ax-mp |
⊢ ran (,) ⊆ dom vol |
44 |
35 43
|
sstri |
⊢ ( (,) “ ( ℚ × ℚ ) ) ⊆ dom vol |
45 |
|
sstr |
⊢ ( ( 𝑥 ⊆ ( (,) “ ( ℚ × ℚ ) ) ∧ ( (,) “ ( ℚ × ℚ ) ) ⊆ dom vol ) → 𝑥 ⊆ dom vol ) |
46 |
44 45
|
mpan2 |
⊢ ( 𝑥 ⊆ ( (,) “ ( ℚ × ℚ ) ) → 𝑥 ⊆ dom vol ) |
47 |
|
dfss3 |
⊢ ( 𝑥 ⊆ dom vol ↔ ∀ 𝑦 ∈ 𝑥 𝑦 ∈ dom vol ) |
48 |
46 47
|
sylib |
⊢ ( 𝑥 ⊆ ( (,) “ ( ℚ × ℚ ) ) → ∀ 𝑦 ∈ 𝑥 𝑦 ∈ dom vol ) |
49 |
|
iunmbl2 |
⊢ ( ( 𝑥 ≼ ℕ ∧ ∀ 𝑦 ∈ 𝑥 𝑦 ∈ dom vol ) → ∪ 𝑦 ∈ 𝑥 𝑦 ∈ dom vol ) |
50 |
34 48 49
|
syl2anc |
⊢ ( 𝑥 ⊆ ( (,) “ ( ℚ × ℚ ) ) → ∪ 𝑦 ∈ 𝑥 𝑦 ∈ dom vol ) |
51 |
4 50
|
eqeltrid |
⊢ ( 𝑥 ⊆ ( (,) “ ( ℚ × ℚ ) ) → ∪ 𝑥 ∈ dom vol ) |
52 |
|
eleq1 |
⊢ ( 𝐴 = ∪ 𝑥 → ( 𝐴 ∈ dom vol ↔ ∪ 𝑥 ∈ dom vol ) ) |
53 |
51 52
|
syl5ibrcom |
⊢ ( 𝑥 ⊆ ( (,) “ ( ℚ × ℚ ) ) → ( 𝐴 = ∪ 𝑥 → 𝐴 ∈ dom vol ) ) |
54 |
53
|
imp |
⊢ ( ( 𝑥 ⊆ ( (,) “ ( ℚ × ℚ ) ) ∧ 𝐴 = ∪ 𝑥 ) → 𝐴 ∈ dom vol ) |
55 |
54
|
exlimiv |
⊢ ( ∃ 𝑥 ( 𝑥 ⊆ ( (,) “ ( ℚ × ℚ ) ) ∧ 𝐴 = ∪ 𝑥 ) → 𝐴 ∈ dom vol ) |
56 |
3 55
|
sylbi |
⊢ ( 𝐴 ∈ ( topGen ‘ ( (,) “ ( ℚ × ℚ ) ) ) → 𝐴 ∈ dom vol ) |
57 |
|
eqid |
⊢ ( topGen ‘ ( (,) “ ( ℚ × ℚ ) ) ) = ( topGen ‘ ( (,) “ ( ℚ × ℚ ) ) ) |
58 |
57
|
tgqioo |
⊢ ( topGen ‘ ran (,) ) = ( topGen ‘ ( (,) “ ( ℚ × ℚ ) ) ) |
59 |
56 58
|
eleq2s |
⊢ ( 𝐴 ∈ ( topGen ‘ ran (,) ) → 𝐴 ∈ dom vol ) |