Step |
Hyp |
Ref |
Expression |
1 |
|
cnpf2 |
⊢ ( ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) ∧ 𝐾 ∈ ( TopOn ‘ 𝑌 ) ∧ 𝐹 ∈ ( ( 𝐽 CnP 𝐾 ) ‘ 𝐴 ) ) → 𝐹 : 𝑋 ⟶ 𝑌 ) |
2 |
1
|
3expa |
⊢ ( ( ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) ∧ 𝐾 ∈ ( TopOn ‘ 𝑌 ) ) ∧ 𝐹 ∈ ( ( 𝐽 CnP 𝐾 ) ‘ 𝐴 ) ) → 𝐹 : 𝑋 ⟶ 𝑌 ) |
3 |
2
|
3adantl3 |
⊢ ( ( ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) ∧ 𝐾 ∈ ( TopOn ‘ 𝑌 ) ∧ 𝐴 ∈ 𝑋 ) ∧ 𝐹 ∈ ( ( 𝐽 CnP 𝐾 ) ‘ 𝐴 ) ) → 𝐹 : 𝑋 ⟶ 𝑌 ) |
4 |
|
cnpflfi |
⊢ ( ( 𝐴 ∈ ( 𝐽 fLim 𝑓 ) ∧ 𝐹 ∈ ( ( 𝐽 CnP 𝐾 ) ‘ 𝐴 ) ) → ( 𝐹 ‘ 𝐴 ) ∈ ( ( 𝐾 fLimf 𝑓 ) ‘ 𝐹 ) ) |
5 |
4
|
expcom |
⊢ ( 𝐹 ∈ ( ( 𝐽 CnP 𝐾 ) ‘ 𝐴 ) → ( 𝐴 ∈ ( 𝐽 fLim 𝑓 ) → ( 𝐹 ‘ 𝐴 ) ∈ ( ( 𝐾 fLimf 𝑓 ) ‘ 𝐹 ) ) ) |
6 |
5
|
ralrimivw |
⊢ ( 𝐹 ∈ ( ( 𝐽 CnP 𝐾 ) ‘ 𝐴 ) → ∀ 𝑓 ∈ ( Fil ‘ 𝑋 ) ( 𝐴 ∈ ( 𝐽 fLim 𝑓 ) → ( 𝐹 ‘ 𝐴 ) ∈ ( ( 𝐾 fLimf 𝑓 ) ‘ 𝐹 ) ) ) |
7 |
6
|
adantl |
⊢ ( ( ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) ∧ 𝐾 ∈ ( TopOn ‘ 𝑌 ) ∧ 𝐴 ∈ 𝑋 ) ∧ 𝐹 ∈ ( ( 𝐽 CnP 𝐾 ) ‘ 𝐴 ) ) → ∀ 𝑓 ∈ ( Fil ‘ 𝑋 ) ( 𝐴 ∈ ( 𝐽 fLim 𝑓 ) → ( 𝐹 ‘ 𝐴 ) ∈ ( ( 𝐾 fLimf 𝑓 ) ‘ 𝐹 ) ) ) |
8 |
3 7
|
jca |
⊢ ( ( ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) ∧ 𝐾 ∈ ( TopOn ‘ 𝑌 ) ∧ 𝐴 ∈ 𝑋 ) ∧ 𝐹 ∈ ( ( 𝐽 CnP 𝐾 ) ‘ 𝐴 ) ) → ( 𝐹 : 𝑋 ⟶ 𝑌 ∧ ∀ 𝑓 ∈ ( Fil ‘ 𝑋 ) ( 𝐴 ∈ ( 𝐽 fLim 𝑓 ) → ( 𝐹 ‘ 𝐴 ) ∈ ( ( 𝐾 fLimf 𝑓 ) ‘ 𝐹 ) ) ) ) |
9 |
8
|
ex |
⊢ ( ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) ∧ 𝐾 ∈ ( TopOn ‘ 𝑌 ) ∧ 𝐴 ∈ 𝑋 ) → ( 𝐹 ∈ ( ( 𝐽 CnP 𝐾 ) ‘ 𝐴 ) → ( 𝐹 : 𝑋 ⟶ 𝑌 ∧ ∀ 𝑓 ∈ ( Fil ‘ 𝑋 ) ( 𝐴 ∈ ( 𝐽 fLim 𝑓 ) → ( 𝐹 ‘ 𝐴 ) ∈ ( ( 𝐾 fLimf 𝑓 ) ‘ 𝐹 ) ) ) ) ) |
10 |
|
simpl1 |
⊢ ( ( ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) ∧ 𝐾 ∈ ( TopOn ‘ 𝑌 ) ∧ 𝐴 ∈ 𝑋 ) ∧ 𝐹 : 𝑋 ⟶ 𝑌 ) → 𝐽 ∈ ( TopOn ‘ 𝑋 ) ) |
11 |
|
simpl3 |
⊢ ( ( ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) ∧ 𝐾 ∈ ( TopOn ‘ 𝑌 ) ∧ 𝐴 ∈ 𝑋 ) ∧ 𝐹 : 𝑋 ⟶ 𝑌 ) → 𝐴 ∈ 𝑋 ) |
12 |
|
neiflim |
⊢ ( ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) ∧ 𝐴 ∈ 𝑋 ) → 𝐴 ∈ ( 𝐽 fLim ( ( nei ‘ 𝐽 ) ‘ { 𝐴 } ) ) ) |
13 |
10 11 12
|
syl2anc |
⊢ ( ( ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) ∧ 𝐾 ∈ ( TopOn ‘ 𝑌 ) ∧ 𝐴 ∈ 𝑋 ) ∧ 𝐹 : 𝑋 ⟶ 𝑌 ) → 𝐴 ∈ ( 𝐽 fLim ( ( nei ‘ 𝐽 ) ‘ { 𝐴 } ) ) ) |
14 |
11
|
snssd |
⊢ ( ( ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) ∧ 𝐾 ∈ ( TopOn ‘ 𝑌 ) ∧ 𝐴 ∈ 𝑋 ) ∧ 𝐹 : 𝑋 ⟶ 𝑌 ) → { 𝐴 } ⊆ 𝑋 ) |
15 |
11
|
snn0d |
⊢ ( ( ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) ∧ 𝐾 ∈ ( TopOn ‘ 𝑌 ) ∧ 𝐴 ∈ 𝑋 ) ∧ 𝐹 : 𝑋 ⟶ 𝑌 ) → { 𝐴 } ≠ ∅ ) |
16 |
|
neifil |
⊢ ( ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) ∧ { 𝐴 } ⊆ 𝑋 ∧ { 𝐴 } ≠ ∅ ) → ( ( nei ‘ 𝐽 ) ‘ { 𝐴 } ) ∈ ( Fil ‘ 𝑋 ) ) |
17 |
10 14 15 16
|
syl3anc |
⊢ ( ( ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) ∧ 𝐾 ∈ ( TopOn ‘ 𝑌 ) ∧ 𝐴 ∈ 𝑋 ) ∧ 𝐹 : 𝑋 ⟶ 𝑌 ) → ( ( nei ‘ 𝐽 ) ‘ { 𝐴 } ) ∈ ( Fil ‘ 𝑋 ) ) |
18 |
|
oveq2 |
⊢ ( 𝑓 = ( ( nei ‘ 𝐽 ) ‘ { 𝐴 } ) → ( 𝐽 fLim 𝑓 ) = ( 𝐽 fLim ( ( nei ‘ 𝐽 ) ‘ { 𝐴 } ) ) ) |
19 |
18
|
eleq2d |
⊢ ( 𝑓 = ( ( nei ‘ 𝐽 ) ‘ { 𝐴 } ) → ( 𝐴 ∈ ( 𝐽 fLim 𝑓 ) ↔ 𝐴 ∈ ( 𝐽 fLim ( ( nei ‘ 𝐽 ) ‘ { 𝐴 } ) ) ) ) |
20 |
|
oveq2 |
⊢ ( 𝑓 = ( ( nei ‘ 𝐽 ) ‘ { 𝐴 } ) → ( 𝐾 fLimf 𝑓 ) = ( 𝐾 fLimf ( ( nei ‘ 𝐽 ) ‘ { 𝐴 } ) ) ) |
21 |
20
|
fveq1d |
⊢ ( 𝑓 = ( ( nei ‘ 𝐽 ) ‘ { 𝐴 } ) → ( ( 𝐾 fLimf 𝑓 ) ‘ 𝐹 ) = ( ( 𝐾 fLimf ( ( nei ‘ 𝐽 ) ‘ { 𝐴 } ) ) ‘ 𝐹 ) ) |
22 |
21
|
eleq2d |
⊢ ( 𝑓 = ( ( nei ‘ 𝐽 ) ‘ { 𝐴 } ) → ( ( 𝐹 ‘ 𝐴 ) ∈ ( ( 𝐾 fLimf 𝑓 ) ‘ 𝐹 ) ↔ ( 𝐹 ‘ 𝐴 ) ∈ ( ( 𝐾 fLimf ( ( nei ‘ 𝐽 ) ‘ { 𝐴 } ) ) ‘ 𝐹 ) ) ) |
23 |
19 22
|
imbi12d |
⊢ ( 𝑓 = ( ( nei ‘ 𝐽 ) ‘ { 𝐴 } ) → ( ( 𝐴 ∈ ( 𝐽 fLim 𝑓 ) → ( 𝐹 ‘ 𝐴 ) ∈ ( ( 𝐾 fLimf 𝑓 ) ‘ 𝐹 ) ) ↔ ( 𝐴 ∈ ( 𝐽 fLim ( ( nei ‘ 𝐽 ) ‘ { 𝐴 } ) ) → ( 𝐹 ‘ 𝐴 ) ∈ ( ( 𝐾 fLimf ( ( nei ‘ 𝐽 ) ‘ { 𝐴 } ) ) ‘ 𝐹 ) ) ) ) |
24 |
23
|
rspcv |
⊢ ( ( ( nei ‘ 𝐽 ) ‘ { 𝐴 } ) ∈ ( Fil ‘ 𝑋 ) → ( ∀ 𝑓 ∈ ( Fil ‘ 𝑋 ) ( 𝐴 ∈ ( 𝐽 fLim 𝑓 ) → ( 𝐹 ‘ 𝐴 ) ∈ ( ( 𝐾 fLimf 𝑓 ) ‘ 𝐹 ) ) → ( 𝐴 ∈ ( 𝐽 fLim ( ( nei ‘ 𝐽 ) ‘ { 𝐴 } ) ) → ( 𝐹 ‘ 𝐴 ) ∈ ( ( 𝐾 fLimf ( ( nei ‘ 𝐽 ) ‘ { 𝐴 } ) ) ‘ 𝐹 ) ) ) ) |
25 |
17 24
|
syl |
⊢ ( ( ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) ∧ 𝐾 ∈ ( TopOn ‘ 𝑌 ) ∧ 𝐴 ∈ 𝑋 ) ∧ 𝐹 : 𝑋 ⟶ 𝑌 ) → ( ∀ 𝑓 ∈ ( Fil ‘ 𝑋 ) ( 𝐴 ∈ ( 𝐽 fLim 𝑓 ) → ( 𝐹 ‘ 𝐴 ) ∈ ( ( 𝐾 fLimf 𝑓 ) ‘ 𝐹 ) ) → ( 𝐴 ∈ ( 𝐽 fLim ( ( nei ‘ 𝐽 ) ‘ { 𝐴 } ) ) → ( 𝐹 ‘ 𝐴 ) ∈ ( ( 𝐾 fLimf ( ( nei ‘ 𝐽 ) ‘ { 𝐴 } ) ) ‘ 𝐹 ) ) ) ) |
26 |
13 25
|
mpid |
⊢ ( ( ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) ∧ 𝐾 ∈ ( TopOn ‘ 𝑌 ) ∧ 𝐴 ∈ 𝑋 ) ∧ 𝐹 : 𝑋 ⟶ 𝑌 ) → ( ∀ 𝑓 ∈ ( Fil ‘ 𝑋 ) ( 𝐴 ∈ ( 𝐽 fLim 𝑓 ) → ( 𝐹 ‘ 𝐴 ) ∈ ( ( 𝐾 fLimf 𝑓 ) ‘ 𝐹 ) ) → ( 𝐹 ‘ 𝐴 ) ∈ ( ( 𝐾 fLimf ( ( nei ‘ 𝐽 ) ‘ { 𝐴 } ) ) ‘ 𝐹 ) ) ) |
27 |
26
|
imdistanda |
⊢ ( ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) ∧ 𝐾 ∈ ( TopOn ‘ 𝑌 ) ∧ 𝐴 ∈ 𝑋 ) → ( ( 𝐹 : 𝑋 ⟶ 𝑌 ∧ ∀ 𝑓 ∈ ( Fil ‘ 𝑋 ) ( 𝐴 ∈ ( 𝐽 fLim 𝑓 ) → ( 𝐹 ‘ 𝐴 ) ∈ ( ( 𝐾 fLimf 𝑓 ) ‘ 𝐹 ) ) ) → ( 𝐹 : 𝑋 ⟶ 𝑌 ∧ ( 𝐹 ‘ 𝐴 ) ∈ ( ( 𝐾 fLimf ( ( nei ‘ 𝐽 ) ‘ { 𝐴 } ) ) ‘ 𝐹 ) ) ) ) |
28 |
|
eqid |
⊢ ( ( nei ‘ 𝐽 ) ‘ { 𝐴 } ) = ( ( nei ‘ 𝐽 ) ‘ { 𝐴 } ) |
29 |
28
|
cnpflf2 |
⊢ ( ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) ∧ 𝐾 ∈ ( TopOn ‘ 𝑌 ) ∧ 𝐴 ∈ 𝑋 ) → ( 𝐹 ∈ ( ( 𝐽 CnP 𝐾 ) ‘ 𝐴 ) ↔ ( 𝐹 : 𝑋 ⟶ 𝑌 ∧ ( 𝐹 ‘ 𝐴 ) ∈ ( ( 𝐾 fLimf ( ( nei ‘ 𝐽 ) ‘ { 𝐴 } ) ) ‘ 𝐹 ) ) ) ) |
30 |
27 29
|
sylibrd |
⊢ ( ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) ∧ 𝐾 ∈ ( TopOn ‘ 𝑌 ) ∧ 𝐴 ∈ 𝑋 ) → ( ( 𝐹 : 𝑋 ⟶ 𝑌 ∧ ∀ 𝑓 ∈ ( Fil ‘ 𝑋 ) ( 𝐴 ∈ ( 𝐽 fLim 𝑓 ) → ( 𝐹 ‘ 𝐴 ) ∈ ( ( 𝐾 fLimf 𝑓 ) ‘ 𝐹 ) ) ) → 𝐹 ∈ ( ( 𝐽 CnP 𝐾 ) ‘ 𝐴 ) ) ) |
31 |
9 30
|
impbid |
⊢ ( ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) ∧ 𝐾 ∈ ( TopOn ‘ 𝑌 ) ∧ 𝐴 ∈ 𝑋 ) → ( 𝐹 ∈ ( ( 𝐽 CnP 𝐾 ) ‘ 𝐴 ) ↔ ( 𝐹 : 𝑋 ⟶ 𝑌 ∧ ∀ 𝑓 ∈ ( Fil ‘ 𝑋 ) ( 𝐴 ∈ ( 𝐽 fLim 𝑓 ) → ( 𝐹 ‘ 𝐴 ) ∈ ( ( 𝐾 fLimf 𝑓 ) ‘ 𝐹 ) ) ) ) ) |