Step |
Hyp |
Ref |
Expression |
1 |
|
tuslem.k |
⊢ 𝐾 = ( toUnifSp ‘ 𝑈 ) |
2 |
|
baseid |
⊢ Base = Slot ( Base ‘ ndx ) |
3 |
|
tsetndxnbasendx |
⊢ ( TopSet ‘ ndx ) ≠ ( Base ‘ ndx ) |
4 |
3
|
necomi |
⊢ ( Base ‘ ndx ) ≠ ( TopSet ‘ ndx ) |
5 |
2 4
|
setsnid |
⊢ ( Base ‘ { 〈 ( Base ‘ ndx ) , dom ∪ 𝑈 〉 , 〈 ( UnifSet ‘ ndx ) , 𝑈 〉 } ) = ( Base ‘ ( { 〈 ( Base ‘ ndx ) , dom ∪ 𝑈 〉 , 〈 ( UnifSet ‘ ndx ) , 𝑈 〉 } sSet 〈 ( TopSet ‘ ndx ) , ( unifTop ‘ 𝑈 ) 〉 ) ) |
6 |
|
ustbas2 |
⊢ ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) → 𝑋 = dom ∪ 𝑈 ) |
7 |
|
uniexg |
⊢ ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) → ∪ 𝑈 ∈ V ) |
8 |
|
dmexg |
⊢ ( ∪ 𝑈 ∈ V → dom ∪ 𝑈 ∈ V ) |
9 |
|
eqid |
⊢ { 〈 ( Base ‘ ndx ) , dom ∪ 𝑈 〉 , 〈 ( UnifSet ‘ ndx ) , 𝑈 〉 } = { 〈 ( Base ‘ ndx ) , dom ∪ 𝑈 〉 , 〈 ( UnifSet ‘ ndx ) , 𝑈 〉 } |
10 |
|
basendxltunifndx |
⊢ ( Base ‘ ndx ) < ( UnifSet ‘ ndx ) |
11 |
|
unifndxnn |
⊢ ( UnifSet ‘ ndx ) ∈ ℕ |
12 |
9 10 11
|
2strbas1 |
⊢ ( dom ∪ 𝑈 ∈ V → dom ∪ 𝑈 = ( Base ‘ { 〈 ( Base ‘ ndx ) , dom ∪ 𝑈 〉 , 〈 ( UnifSet ‘ ndx ) , 𝑈 〉 } ) ) |
13 |
7 8 12
|
3syl |
⊢ ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) → dom ∪ 𝑈 = ( Base ‘ { 〈 ( Base ‘ ndx ) , dom ∪ 𝑈 〉 , 〈 ( UnifSet ‘ ndx ) , 𝑈 〉 } ) ) |
14 |
6 13
|
eqtrd |
⊢ ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) → 𝑋 = ( Base ‘ { 〈 ( Base ‘ ndx ) , dom ∪ 𝑈 〉 , 〈 ( UnifSet ‘ ndx ) , 𝑈 〉 } ) ) |
15 |
|
tusval |
⊢ ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) → ( toUnifSp ‘ 𝑈 ) = ( { 〈 ( Base ‘ ndx ) , dom ∪ 𝑈 〉 , 〈 ( UnifSet ‘ ndx ) , 𝑈 〉 } sSet 〈 ( TopSet ‘ ndx ) , ( unifTop ‘ 𝑈 ) 〉 ) ) |
16 |
1 15
|
eqtrid |
⊢ ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) → 𝐾 = ( { 〈 ( Base ‘ ndx ) , dom ∪ 𝑈 〉 , 〈 ( UnifSet ‘ ndx ) , 𝑈 〉 } sSet 〈 ( TopSet ‘ ndx ) , ( unifTop ‘ 𝑈 ) 〉 ) ) |
17 |
16
|
fveq2d |
⊢ ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) → ( Base ‘ 𝐾 ) = ( Base ‘ ( { 〈 ( Base ‘ ndx ) , dom ∪ 𝑈 〉 , 〈 ( UnifSet ‘ ndx ) , 𝑈 〉 } sSet 〈 ( TopSet ‘ ndx ) , ( unifTop ‘ 𝑈 ) 〉 ) ) ) |
18 |
5 14 17
|
3eqtr4a |
⊢ ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) → 𝑋 = ( Base ‘ 𝐾 ) ) |
19 |
|
unifid |
⊢ UnifSet = Slot ( UnifSet ‘ ndx ) |
20 |
|
unifndxntsetndx |
⊢ ( UnifSet ‘ ndx ) ≠ ( TopSet ‘ ndx ) |
21 |
19 20
|
setsnid |
⊢ ( UnifSet ‘ { 〈 ( Base ‘ ndx ) , dom ∪ 𝑈 〉 , 〈 ( UnifSet ‘ ndx ) , 𝑈 〉 } ) = ( UnifSet ‘ ( { 〈 ( Base ‘ ndx ) , dom ∪ 𝑈 〉 , 〈 ( UnifSet ‘ ndx ) , 𝑈 〉 } sSet 〈 ( TopSet ‘ ndx ) , ( unifTop ‘ 𝑈 ) 〉 ) ) |
22 |
9 10 11 19
|
2strop1 |
⊢ ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) → 𝑈 = ( UnifSet ‘ { 〈 ( Base ‘ ndx ) , dom ∪ 𝑈 〉 , 〈 ( UnifSet ‘ ndx ) , 𝑈 〉 } ) ) |
23 |
16
|
fveq2d |
⊢ ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) → ( UnifSet ‘ 𝐾 ) = ( UnifSet ‘ ( { 〈 ( Base ‘ ndx ) , dom ∪ 𝑈 〉 , 〈 ( UnifSet ‘ ndx ) , 𝑈 〉 } sSet 〈 ( TopSet ‘ ndx ) , ( unifTop ‘ 𝑈 ) 〉 ) ) ) |
24 |
21 22 23
|
3eqtr4a |
⊢ ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) → 𝑈 = ( UnifSet ‘ 𝐾 ) ) |
25 |
|
prex |
⊢ { 〈 ( Base ‘ ndx ) , dom ∪ 𝑈 〉 , 〈 ( UnifSet ‘ ndx ) , 𝑈 〉 } ∈ V |
26 |
|
fvex |
⊢ ( unifTop ‘ 𝑈 ) ∈ V |
27 |
|
tsetid |
⊢ TopSet = Slot ( TopSet ‘ ndx ) |
28 |
27
|
setsid |
⊢ ( ( { 〈 ( Base ‘ ndx ) , dom ∪ 𝑈 〉 , 〈 ( UnifSet ‘ ndx ) , 𝑈 〉 } ∈ V ∧ ( unifTop ‘ 𝑈 ) ∈ V ) → ( unifTop ‘ 𝑈 ) = ( TopSet ‘ ( { 〈 ( Base ‘ ndx ) , dom ∪ 𝑈 〉 , 〈 ( UnifSet ‘ ndx ) , 𝑈 〉 } sSet 〈 ( TopSet ‘ ndx ) , ( unifTop ‘ 𝑈 ) 〉 ) ) ) |
29 |
25 26 28
|
mp2an |
⊢ ( unifTop ‘ 𝑈 ) = ( TopSet ‘ ( { 〈 ( Base ‘ ndx ) , dom ∪ 𝑈 〉 , 〈 ( UnifSet ‘ ndx ) , 𝑈 〉 } sSet 〈 ( TopSet ‘ ndx ) , ( unifTop ‘ 𝑈 ) 〉 ) ) |
30 |
16
|
fveq2d |
⊢ ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) → ( TopSet ‘ 𝐾 ) = ( TopSet ‘ ( { 〈 ( Base ‘ ndx ) , dom ∪ 𝑈 〉 , 〈 ( UnifSet ‘ ndx ) , 𝑈 〉 } sSet 〈 ( TopSet ‘ ndx ) , ( unifTop ‘ 𝑈 ) 〉 ) ) ) |
31 |
29 30
|
eqtr4id |
⊢ ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) → ( unifTop ‘ 𝑈 ) = ( TopSet ‘ 𝐾 ) ) |
32 |
|
utopbas |
⊢ ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) → 𝑋 = ∪ ( unifTop ‘ 𝑈 ) ) |
33 |
31
|
unieqd |
⊢ ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) → ∪ ( unifTop ‘ 𝑈 ) = ∪ ( TopSet ‘ 𝐾 ) ) |
34 |
32 18 33
|
3eqtr3rd |
⊢ ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) → ∪ ( TopSet ‘ 𝐾 ) = ( Base ‘ 𝐾 ) ) |
35 |
34
|
oveq2d |
⊢ ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) → ( ( TopSet ‘ 𝐾 ) ↾t ∪ ( TopSet ‘ 𝐾 ) ) = ( ( TopSet ‘ 𝐾 ) ↾t ( Base ‘ 𝐾 ) ) ) |
36 |
|
fvex |
⊢ ( TopSet ‘ 𝐾 ) ∈ V |
37 |
|
eqid |
⊢ ∪ ( TopSet ‘ 𝐾 ) = ∪ ( TopSet ‘ 𝐾 ) |
38 |
37
|
restid |
⊢ ( ( TopSet ‘ 𝐾 ) ∈ V → ( ( TopSet ‘ 𝐾 ) ↾t ∪ ( TopSet ‘ 𝐾 ) ) = ( TopSet ‘ 𝐾 ) ) |
39 |
36 38
|
ax-mp |
⊢ ( ( TopSet ‘ 𝐾 ) ↾t ∪ ( TopSet ‘ 𝐾 ) ) = ( TopSet ‘ 𝐾 ) |
40 |
|
eqid |
⊢ ( Base ‘ 𝐾 ) = ( Base ‘ 𝐾 ) |
41 |
|
eqid |
⊢ ( TopSet ‘ 𝐾 ) = ( TopSet ‘ 𝐾 ) |
42 |
40 41
|
topnval |
⊢ ( ( TopSet ‘ 𝐾 ) ↾t ( Base ‘ 𝐾 ) ) = ( TopOpen ‘ 𝐾 ) |
43 |
35 39 42
|
3eqtr3g |
⊢ ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) → ( TopSet ‘ 𝐾 ) = ( TopOpen ‘ 𝐾 ) ) |
44 |
31 43
|
eqtrd |
⊢ ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) → ( unifTop ‘ 𝑈 ) = ( TopOpen ‘ 𝐾 ) ) |
45 |
18 24 44
|
3jca |
⊢ ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) → ( 𝑋 = ( Base ‘ 𝐾 ) ∧ 𝑈 = ( UnifSet ‘ 𝐾 ) ∧ ( unifTop ‘ 𝑈 ) = ( TopOpen ‘ 𝐾 ) ) ) |