| Step |
Hyp |
Ref |
Expression |
| 1 |
|
fvssunirn |
⊢ ( ∞Met ‘ 𝑋 ) ⊆ ∪ ran ∞Met |
| 2 |
1
|
sseli |
⊢ ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) → 𝐷 ∈ ∪ ran ∞Met ) |
| 3 |
|
dmeq |
⊢ ( 𝑑 = 𝐷 → dom 𝑑 = dom 𝐷 ) |
| 4 |
3
|
dmeqd |
⊢ ( 𝑑 = 𝐷 → dom dom 𝑑 = dom dom 𝐷 ) |
| 5 |
4
|
fveq2d |
⊢ ( 𝑑 = 𝐷 → ( Fil ‘ dom dom 𝑑 ) = ( Fil ‘ dom dom 𝐷 ) ) |
| 6 |
|
imaeq1 |
⊢ ( 𝑑 = 𝐷 → ( 𝑑 “ ( 𝑦 × 𝑦 ) ) = ( 𝐷 “ ( 𝑦 × 𝑦 ) ) ) |
| 7 |
6
|
sseq1d |
⊢ ( 𝑑 = 𝐷 → ( ( 𝑑 “ ( 𝑦 × 𝑦 ) ) ⊆ ( 0 [,) 𝑥 ) ↔ ( 𝐷 “ ( 𝑦 × 𝑦 ) ) ⊆ ( 0 [,) 𝑥 ) ) ) |
| 8 |
7
|
rexbidv |
⊢ ( 𝑑 = 𝐷 → ( ∃ 𝑦 ∈ 𝑓 ( 𝑑 “ ( 𝑦 × 𝑦 ) ) ⊆ ( 0 [,) 𝑥 ) ↔ ∃ 𝑦 ∈ 𝑓 ( 𝐷 “ ( 𝑦 × 𝑦 ) ) ⊆ ( 0 [,) 𝑥 ) ) ) |
| 9 |
8
|
ralbidv |
⊢ ( 𝑑 = 𝐷 → ( ∀ 𝑥 ∈ ℝ+ ∃ 𝑦 ∈ 𝑓 ( 𝑑 “ ( 𝑦 × 𝑦 ) ) ⊆ ( 0 [,) 𝑥 ) ↔ ∀ 𝑥 ∈ ℝ+ ∃ 𝑦 ∈ 𝑓 ( 𝐷 “ ( 𝑦 × 𝑦 ) ) ⊆ ( 0 [,) 𝑥 ) ) ) |
| 10 |
5 9
|
rabeqbidv |
⊢ ( 𝑑 = 𝐷 → { 𝑓 ∈ ( Fil ‘ dom dom 𝑑 ) ∣ ∀ 𝑥 ∈ ℝ+ ∃ 𝑦 ∈ 𝑓 ( 𝑑 “ ( 𝑦 × 𝑦 ) ) ⊆ ( 0 [,) 𝑥 ) } = { 𝑓 ∈ ( Fil ‘ dom dom 𝐷 ) ∣ ∀ 𝑥 ∈ ℝ+ ∃ 𝑦 ∈ 𝑓 ( 𝐷 “ ( 𝑦 × 𝑦 ) ) ⊆ ( 0 [,) 𝑥 ) } ) |
| 11 |
|
df-cfil |
⊢ CauFil = ( 𝑑 ∈ ∪ ran ∞Met ↦ { 𝑓 ∈ ( Fil ‘ dom dom 𝑑 ) ∣ ∀ 𝑥 ∈ ℝ+ ∃ 𝑦 ∈ 𝑓 ( 𝑑 “ ( 𝑦 × 𝑦 ) ) ⊆ ( 0 [,) 𝑥 ) } ) |
| 12 |
|
fvex |
⊢ ( Fil ‘ dom dom 𝐷 ) ∈ V |
| 13 |
12
|
rabex |
⊢ { 𝑓 ∈ ( Fil ‘ dom dom 𝐷 ) ∣ ∀ 𝑥 ∈ ℝ+ ∃ 𝑦 ∈ 𝑓 ( 𝐷 “ ( 𝑦 × 𝑦 ) ) ⊆ ( 0 [,) 𝑥 ) } ∈ V |
| 14 |
10 11 13
|
fvmpt |
⊢ ( 𝐷 ∈ ∪ ran ∞Met → ( CauFil ‘ 𝐷 ) = { 𝑓 ∈ ( Fil ‘ dom dom 𝐷 ) ∣ ∀ 𝑥 ∈ ℝ+ ∃ 𝑦 ∈ 𝑓 ( 𝐷 “ ( 𝑦 × 𝑦 ) ) ⊆ ( 0 [,) 𝑥 ) } ) |
| 15 |
2 14
|
syl |
⊢ ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) → ( CauFil ‘ 𝐷 ) = { 𝑓 ∈ ( Fil ‘ dom dom 𝐷 ) ∣ ∀ 𝑥 ∈ ℝ+ ∃ 𝑦 ∈ 𝑓 ( 𝐷 “ ( 𝑦 × 𝑦 ) ) ⊆ ( 0 [,) 𝑥 ) } ) |
| 16 |
|
xmetdmdm |
⊢ ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) → 𝑋 = dom dom 𝐷 ) |
| 17 |
16
|
fveq2d |
⊢ ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) → ( Fil ‘ 𝑋 ) = ( Fil ‘ dom dom 𝐷 ) ) |
| 18 |
17
|
rabeqdv |
⊢ ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) → { 𝑓 ∈ ( Fil ‘ 𝑋 ) ∣ ∀ 𝑥 ∈ ℝ+ ∃ 𝑦 ∈ 𝑓 ( 𝐷 “ ( 𝑦 × 𝑦 ) ) ⊆ ( 0 [,) 𝑥 ) } = { 𝑓 ∈ ( Fil ‘ dom dom 𝐷 ) ∣ ∀ 𝑥 ∈ ℝ+ ∃ 𝑦 ∈ 𝑓 ( 𝐷 “ ( 𝑦 × 𝑦 ) ) ⊆ ( 0 [,) 𝑥 ) } ) |
| 19 |
15 18
|
eqtr4d |
⊢ ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) → ( CauFil ‘ 𝐷 ) = { 𝑓 ∈ ( Fil ‘ 𝑋 ) ∣ ∀ 𝑥 ∈ ℝ+ ∃ 𝑦 ∈ 𝑓 ( 𝐷 “ ( 𝑦 × 𝑦 ) ) ⊆ ( 0 [,) 𝑥 ) } ) |