Step |
Hyp |
Ref |
Expression |
1 |
|
flfneii.x |
⊢ 𝑋 = ∪ 𝐽 |
2 |
1
|
toptopon |
⊢ ( 𝐽 ∈ Top ↔ 𝐽 ∈ ( TopOn ‘ 𝑋 ) ) |
3 |
|
flfnei |
⊢ ( ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) ∧ 𝐿 ∈ ( Fil ‘ 𝑌 ) ∧ 𝐹 : 𝑌 ⟶ 𝑋 ) → ( 𝐴 ∈ ( ( 𝐽 fLimf 𝐿 ) ‘ 𝐹 ) ↔ ( 𝐴 ∈ 𝑋 ∧ ∀ 𝑛 ∈ ( ( nei ‘ 𝐽 ) ‘ { 𝐴 } ) ∃ 𝑠 ∈ 𝐿 ( 𝐹 “ 𝑠 ) ⊆ 𝑛 ) ) ) |
4 |
2 3
|
syl3an1b |
⊢ ( ( 𝐽 ∈ Top ∧ 𝐿 ∈ ( Fil ‘ 𝑌 ) ∧ 𝐹 : 𝑌 ⟶ 𝑋 ) → ( 𝐴 ∈ ( ( 𝐽 fLimf 𝐿 ) ‘ 𝐹 ) ↔ ( 𝐴 ∈ 𝑋 ∧ ∀ 𝑛 ∈ ( ( nei ‘ 𝐽 ) ‘ { 𝐴 } ) ∃ 𝑠 ∈ 𝐿 ( 𝐹 “ 𝑠 ) ⊆ 𝑛 ) ) ) |
5 |
4
|
simplbda |
⊢ ( ( ( 𝐽 ∈ Top ∧ 𝐿 ∈ ( Fil ‘ 𝑌 ) ∧ 𝐹 : 𝑌 ⟶ 𝑋 ) ∧ 𝐴 ∈ ( ( 𝐽 fLimf 𝐿 ) ‘ 𝐹 ) ) → ∀ 𝑛 ∈ ( ( nei ‘ 𝐽 ) ‘ { 𝐴 } ) ∃ 𝑠 ∈ 𝐿 ( 𝐹 “ 𝑠 ) ⊆ 𝑛 ) |
6 |
5
|
3adant3 |
⊢ ( ( ( 𝐽 ∈ Top ∧ 𝐿 ∈ ( Fil ‘ 𝑌 ) ∧ 𝐹 : 𝑌 ⟶ 𝑋 ) ∧ 𝐴 ∈ ( ( 𝐽 fLimf 𝐿 ) ‘ 𝐹 ) ∧ 𝑁 ∈ ( ( nei ‘ 𝐽 ) ‘ { 𝐴 } ) ) → ∀ 𝑛 ∈ ( ( nei ‘ 𝐽 ) ‘ { 𝐴 } ) ∃ 𝑠 ∈ 𝐿 ( 𝐹 “ 𝑠 ) ⊆ 𝑛 ) |
7 |
|
sseq2 |
⊢ ( 𝑛 = 𝑁 → ( ( 𝐹 “ 𝑠 ) ⊆ 𝑛 ↔ ( 𝐹 “ 𝑠 ) ⊆ 𝑁 ) ) |
8 |
7
|
rexbidv |
⊢ ( 𝑛 = 𝑁 → ( ∃ 𝑠 ∈ 𝐿 ( 𝐹 “ 𝑠 ) ⊆ 𝑛 ↔ ∃ 𝑠 ∈ 𝐿 ( 𝐹 “ 𝑠 ) ⊆ 𝑁 ) ) |
9 |
8
|
rspcv |
⊢ ( 𝑁 ∈ ( ( nei ‘ 𝐽 ) ‘ { 𝐴 } ) → ( ∀ 𝑛 ∈ ( ( nei ‘ 𝐽 ) ‘ { 𝐴 } ) ∃ 𝑠 ∈ 𝐿 ( 𝐹 “ 𝑠 ) ⊆ 𝑛 → ∃ 𝑠 ∈ 𝐿 ( 𝐹 “ 𝑠 ) ⊆ 𝑁 ) ) |
10 |
9
|
3ad2ant3 |
⊢ ( ( ( 𝐽 ∈ Top ∧ 𝐿 ∈ ( Fil ‘ 𝑌 ) ∧ 𝐹 : 𝑌 ⟶ 𝑋 ) ∧ 𝐴 ∈ ( ( 𝐽 fLimf 𝐿 ) ‘ 𝐹 ) ∧ 𝑁 ∈ ( ( nei ‘ 𝐽 ) ‘ { 𝐴 } ) ) → ( ∀ 𝑛 ∈ ( ( nei ‘ 𝐽 ) ‘ { 𝐴 } ) ∃ 𝑠 ∈ 𝐿 ( 𝐹 “ 𝑠 ) ⊆ 𝑛 → ∃ 𝑠 ∈ 𝐿 ( 𝐹 “ 𝑠 ) ⊆ 𝑁 ) ) |
11 |
6 10
|
mpd |
⊢ ( ( ( 𝐽 ∈ Top ∧ 𝐿 ∈ ( Fil ‘ 𝑌 ) ∧ 𝐹 : 𝑌 ⟶ 𝑋 ) ∧ 𝐴 ∈ ( ( 𝐽 fLimf 𝐿 ) ‘ 𝐹 ) ∧ 𝑁 ∈ ( ( nei ‘ 𝐽 ) ‘ { 𝐴 } ) ) → ∃ 𝑠 ∈ 𝐿 ( 𝐹 “ 𝑠 ) ⊆ 𝑁 ) |