Step |
Hyp |
Ref |
Expression |
1 |
|
2fveq3 |
⊢ ( 𝑤 = 𝑊 → ( Fil ‘ ( Base ‘ 𝑤 ) ) = ( Fil ‘ ( Base ‘ 𝑊 ) ) ) |
2 |
|
2fveq3 |
⊢ ( 𝑤 = 𝑊 → ( CauFilu ‘ ( UnifSt ‘ 𝑤 ) ) = ( CauFilu ‘ ( UnifSt ‘ 𝑊 ) ) ) |
3 |
2
|
eleq2d |
⊢ ( 𝑤 = 𝑊 → ( 𝑐 ∈ ( CauFilu ‘ ( UnifSt ‘ 𝑤 ) ) ↔ 𝑐 ∈ ( CauFilu ‘ ( UnifSt ‘ 𝑊 ) ) ) ) |
4 |
|
fveq2 |
⊢ ( 𝑤 = 𝑊 → ( TopOpen ‘ 𝑤 ) = ( TopOpen ‘ 𝑊 ) ) |
5 |
4
|
oveq1d |
⊢ ( 𝑤 = 𝑊 → ( ( TopOpen ‘ 𝑤 ) fLim 𝑐 ) = ( ( TopOpen ‘ 𝑊 ) fLim 𝑐 ) ) |
6 |
5
|
neeq1d |
⊢ ( 𝑤 = 𝑊 → ( ( ( TopOpen ‘ 𝑤 ) fLim 𝑐 ) ≠ ∅ ↔ ( ( TopOpen ‘ 𝑊 ) fLim 𝑐 ) ≠ ∅ ) ) |
7 |
3 6
|
imbi12d |
⊢ ( 𝑤 = 𝑊 → ( ( 𝑐 ∈ ( CauFilu ‘ ( UnifSt ‘ 𝑤 ) ) → ( ( TopOpen ‘ 𝑤 ) fLim 𝑐 ) ≠ ∅ ) ↔ ( 𝑐 ∈ ( CauFilu ‘ ( UnifSt ‘ 𝑊 ) ) → ( ( TopOpen ‘ 𝑊 ) fLim 𝑐 ) ≠ ∅ ) ) ) |
8 |
1 7
|
raleqbidv |
⊢ ( 𝑤 = 𝑊 → ( ∀ 𝑐 ∈ ( Fil ‘ ( Base ‘ 𝑤 ) ) ( 𝑐 ∈ ( CauFilu ‘ ( UnifSt ‘ 𝑤 ) ) → ( ( TopOpen ‘ 𝑤 ) fLim 𝑐 ) ≠ ∅ ) ↔ ∀ 𝑐 ∈ ( Fil ‘ ( Base ‘ 𝑊 ) ) ( 𝑐 ∈ ( CauFilu ‘ ( UnifSt ‘ 𝑊 ) ) → ( ( TopOpen ‘ 𝑊 ) fLim 𝑐 ) ≠ ∅ ) ) ) |
9 |
|
df-cusp |
⊢ CUnifSp = { 𝑤 ∈ UnifSp ∣ ∀ 𝑐 ∈ ( Fil ‘ ( Base ‘ 𝑤 ) ) ( 𝑐 ∈ ( CauFilu ‘ ( UnifSt ‘ 𝑤 ) ) → ( ( TopOpen ‘ 𝑤 ) fLim 𝑐 ) ≠ ∅ ) } |
10 |
8 9
|
elrab2 |
⊢ ( 𝑊 ∈ CUnifSp ↔ ( 𝑊 ∈ UnifSp ∧ ∀ 𝑐 ∈ ( Fil ‘ ( Base ‘ 𝑊 ) ) ( 𝑐 ∈ ( CauFilu ‘ ( UnifSt ‘ 𝑊 ) ) → ( ( TopOpen ‘ 𝑊 ) fLim 𝑐 ) ≠ ∅ ) ) ) |