Step |
Hyp |
Ref |
Expression |
1 |
|
flfcntr.c |
⊢ 𝐶 = ∪ 𝐽 |
2 |
|
flfcntr.b |
⊢ 𝐵 = ∪ 𝐾 |
3 |
|
flfcntr.j |
⊢ ( 𝜑 → 𝐽 ∈ Top ) |
4 |
|
flfcntr.a |
⊢ ( 𝜑 → 𝐴 ⊆ 𝐶 ) |
5 |
|
flfcntr.1 |
⊢ ( 𝜑 → 𝐹 ∈ ( ( 𝐽 ↾t 𝐴 ) Cn 𝐾 ) ) |
6 |
|
flfcntr.y |
⊢ ( 𝜑 → 𝑋 ∈ 𝐴 ) |
7 |
|
fveq2 |
⊢ ( 𝑥 = 𝑋 → ( 𝐹 ‘ 𝑥 ) = ( 𝐹 ‘ 𝑋 ) ) |
8 |
7
|
eleq1d |
⊢ ( 𝑥 = 𝑋 → ( ( 𝐹 ‘ 𝑥 ) ∈ ( ( 𝐾 fLimf ( ( ( nei ‘ 𝐽 ) ‘ { 𝑋 } ) ↾t 𝐴 ) ) ‘ 𝐹 ) ↔ ( 𝐹 ‘ 𝑋 ) ∈ ( ( 𝐾 fLimf ( ( ( nei ‘ 𝐽 ) ‘ { 𝑋 } ) ↾t 𝐴 ) ) ‘ 𝐹 ) ) ) |
9 |
|
oveq2 |
⊢ ( 𝑎 = ( ( ( nei ‘ 𝐽 ) ‘ { 𝑋 } ) ↾t 𝐴 ) → ( ( 𝐽 ↾t 𝐴 ) fLim 𝑎 ) = ( ( 𝐽 ↾t 𝐴 ) fLim ( ( ( nei ‘ 𝐽 ) ‘ { 𝑋 } ) ↾t 𝐴 ) ) ) |
10 |
|
oveq2 |
⊢ ( 𝑎 = ( ( ( nei ‘ 𝐽 ) ‘ { 𝑋 } ) ↾t 𝐴 ) → ( 𝐾 fLimf 𝑎 ) = ( 𝐾 fLimf ( ( ( nei ‘ 𝐽 ) ‘ { 𝑋 } ) ↾t 𝐴 ) ) ) |
11 |
10
|
fveq1d |
⊢ ( 𝑎 = ( ( ( nei ‘ 𝐽 ) ‘ { 𝑋 } ) ↾t 𝐴 ) → ( ( 𝐾 fLimf 𝑎 ) ‘ 𝐹 ) = ( ( 𝐾 fLimf ( ( ( nei ‘ 𝐽 ) ‘ { 𝑋 } ) ↾t 𝐴 ) ) ‘ 𝐹 ) ) |
12 |
11
|
eleq2d |
⊢ ( 𝑎 = ( ( ( nei ‘ 𝐽 ) ‘ { 𝑋 } ) ↾t 𝐴 ) → ( ( 𝐹 ‘ 𝑥 ) ∈ ( ( 𝐾 fLimf 𝑎 ) ‘ 𝐹 ) ↔ ( 𝐹 ‘ 𝑥 ) ∈ ( ( 𝐾 fLimf ( ( ( nei ‘ 𝐽 ) ‘ { 𝑋 } ) ↾t 𝐴 ) ) ‘ 𝐹 ) ) ) |
13 |
9 12
|
raleqbidv |
⊢ ( 𝑎 = ( ( ( nei ‘ 𝐽 ) ‘ { 𝑋 } ) ↾t 𝐴 ) → ( ∀ 𝑥 ∈ ( ( 𝐽 ↾t 𝐴 ) fLim 𝑎 ) ( 𝐹 ‘ 𝑥 ) ∈ ( ( 𝐾 fLimf 𝑎 ) ‘ 𝐹 ) ↔ ∀ 𝑥 ∈ ( ( 𝐽 ↾t 𝐴 ) fLim ( ( ( nei ‘ 𝐽 ) ‘ { 𝑋 } ) ↾t 𝐴 ) ) ( 𝐹 ‘ 𝑥 ) ∈ ( ( 𝐾 fLimf ( ( ( nei ‘ 𝐽 ) ‘ { 𝑋 } ) ↾t 𝐴 ) ) ‘ 𝐹 ) ) ) |
14 |
1
|
toptopon |
⊢ ( 𝐽 ∈ Top ↔ 𝐽 ∈ ( TopOn ‘ 𝐶 ) ) |
15 |
3 14
|
sylib |
⊢ ( 𝜑 → 𝐽 ∈ ( TopOn ‘ 𝐶 ) ) |
16 |
|
resttopon |
⊢ ( ( 𝐽 ∈ ( TopOn ‘ 𝐶 ) ∧ 𝐴 ⊆ 𝐶 ) → ( 𝐽 ↾t 𝐴 ) ∈ ( TopOn ‘ 𝐴 ) ) |
17 |
15 4 16
|
syl2anc |
⊢ ( 𝜑 → ( 𝐽 ↾t 𝐴 ) ∈ ( TopOn ‘ 𝐴 ) ) |
18 |
|
cntop2 |
⊢ ( 𝐹 ∈ ( ( 𝐽 ↾t 𝐴 ) Cn 𝐾 ) → 𝐾 ∈ Top ) |
19 |
5 18
|
syl |
⊢ ( 𝜑 → 𝐾 ∈ Top ) |
20 |
2
|
toptopon |
⊢ ( 𝐾 ∈ Top ↔ 𝐾 ∈ ( TopOn ‘ 𝐵 ) ) |
21 |
19 20
|
sylib |
⊢ ( 𝜑 → 𝐾 ∈ ( TopOn ‘ 𝐵 ) ) |
22 |
|
cnflf |
⊢ ( ( ( 𝐽 ↾t 𝐴 ) ∈ ( TopOn ‘ 𝐴 ) ∧ 𝐾 ∈ ( TopOn ‘ 𝐵 ) ) → ( 𝐹 ∈ ( ( 𝐽 ↾t 𝐴 ) Cn 𝐾 ) ↔ ( 𝐹 : 𝐴 ⟶ 𝐵 ∧ ∀ 𝑎 ∈ ( Fil ‘ 𝐴 ) ∀ 𝑥 ∈ ( ( 𝐽 ↾t 𝐴 ) fLim 𝑎 ) ( 𝐹 ‘ 𝑥 ) ∈ ( ( 𝐾 fLimf 𝑎 ) ‘ 𝐹 ) ) ) ) |
23 |
17 21 22
|
syl2anc |
⊢ ( 𝜑 → ( 𝐹 ∈ ( ( 𝐽 ↾t 𝐴 ) Cn 𝐾 ) ↔ ( 𝐹 : 𝐴 ⟶ 𝐵 ∧ ∀ 𝑎 ∈ ( Fil ‘ 𝐴 ) ∀ 𝑥 ∈ ( ( 𝐽 ↾t 𝐴 ) fLim 𝑎 ) ( 𝐹 ‘ 𝑥 ) ∈ ( ( 𝐾 fLimf 𝑎 ) ‘ 𝐹 ) ) ) ) |
24 |
5 23
|
mpbid |
⊢ ( 𝜑 → ( 𝐹 : 𝐴 ⟶ 𝐵 ∧ ∀ 𝑎 ∈ ( Fil ‘ 𝐴 ) ∀ 𝑥 ∈ ( ( 𝐽 ↾t 𝐴 ) fLim 𝑎 ) ( 𝐹 ‘ 𝑥 ) ∈ ( ( 𝐾 fLimf 𝑎 ) ‘ 𝐹 ) ) ) |
25 |
24
|
simprd |
⊢ ( 𝜑 → ∀ 𝑎 ∈ ( Fil ‘ 𝐴 ) ∀ 𝑥 ∈ ( ( 𝐽 ↾t 𝐴 ) fLim 𝑎 ) ( 𝐹 ‘ 𝑥 ) ∈ ( ( 𝐾 fLimf 𝑎 ) ‘ 𝐹 ) ) |
26 |
1
|
sscls |
⊢ ( ( 𝐽 ∈ Top ∧ 𝐴 ⊆ 𝐶 ) → 𝐴 ⊆ ( ( cls ‘ 𝐽 ) ‘ 𝐴 ) ) |
27 |
3 4 26
|
syl2anc |
⊢ ( 𝜑 → 𝐴 ⊆ ( ( cls ‘ 𝐽 ) ‘ 𝐴 ) ) |
28 |
27 6
|
sseldd |
⊢ ( 𝜑 → 𝑋 ∈ ( ( cls ‘ 𝐽 ) ‘ 𝐴 ) ) |
29 |
4 6
|
sseldd |
⊢ ( 𝜑 → 𝑋 ∈ 𝐶 ) |
30 |
|
trnei |
⊢ ( ( 𝐽 ∈ ( TopOn ‘ 𝐶 ) ∧ 𝐴 ⊆ 𝐶 ∧ 𝑋 ∈ 𝐶 ) → ( 𝑋 ∈ ( ( cls ‘ 𝐽 ) ‘ 𝐴 ) ↔ ( ( ( nei ‘ 𝐽 ) ‘ { 𝑋 } ) ↾t 𝐴 ) ∈ ( Fil ‘ 𝐴 ) ) ) |
31 |
15 4 29 30
|
syl3anc |
⊢ ( 𝜑 → ( 𝑋 ∈ ( ( cls ‘ 𝐽 ) ‘ 𝐴 ) ↔ ( ( ( nei ‘ 𝐽 ) ‘ { 𝑋 } ) ↾t 𝐴 ) ∈ ( Fil ‘ 𝐴 ) ) ) |
32 |
28 31
|
mpbid |
⊢ ( 𝜑 → ( ( ( nei ‘ 𝐽 ) ‘ { 𝑋 } ) ↾t 𝐴 ) ∈ ( Fil ‘ 𝐴 ) ) |
33 |
13 25 32
|
rspcdva |
⊢ ( 𝜑 → ∀ 𝑥 ∈ ( ( 𝐽 ↾t 𝐴 ) fLim ( ( ( nei ‘ 𝐽 ) ‘ { 𝑋 } ) ↾t 𝐴 ) ) ( 𝐹 ‘ 𝑥 ) ∈ ( ( 𝐾 fLimf ( ( ( nei ‘ 𝐽 ) ‘ { 𝑋 } ) ↾t 𝐴 ) ) ‘ 𝐹 ) ) |
34 |
|
neiflim |
⊢ ( ( ( 𝐽 ↾t 𝐴 ) ∈ ( TopOn ‘ 𝐴 ) ∧ 𝑋 ∈ 𝐴 ) → 𝑋 ∈ ( ( 𝐽 ↾t 𝐴 ) fLim ( ( nei ‘ ( 𝐽 ↾t 𝐴 ) ) ‘ { 𝑋 } ) ) ) |
35 |
17 6 34
|
syl2anc |
⊢ ( 𝜑 → 𝑋 ∈ ( ( 𝐽 ↾t 𝐴 ) fLim ( ( nei ‘ ( 𝐽 ↾t 𝐴 ) ) ‘ { 𝑋 } ) ) ) |
36 |
6
|
snssd |
⊢ ( 𝜑 → { 𝑋 } ⊆ 𝐴 ) |
37 |
1
|
neitr |
⊢ ( ( 𝐽 ∈ Top ∧ 𝐴 ⊆ 𝐶 ∧ { 𝑋 } ⊆ 𝐴 ) → ( ( nei ‘ ( 𝐽 ↾t 𝐴 ) ) ‘ { 𝑋 } ) = ( ( ( nei ‘ 𝐽 ) ‘ { 𝑋 } ) ↾t 𝐴 ) ) |
38 |
3 4 36 37
|
syl3anc |
⊢ ( 𝜑 → ( ( nei ‘ ( 𝐽 ↾t 𝐴 ) ) ‘ { 𝑋 } ) = ( ( ( nei ‘ 𝐽 ) ‘ { 𝑋 } ) ↾t 𝐴 ) ) |
39 |
38
|
oveq2d |
⊢ ( 𝜑 → ( ( 𝐽 ↾t 𝐴 ) fLim ( ( nei ‘ ( 𝐽 ↾t 𝐴 ) ) ‘ { 𝑋 } ) ) = ( ( 𝐽 ↾t 𝐴 ) fLim ( ( ( nei ‘ 𝐽 ) ‘ { 𝑋 } ) ↾t 𝐴 ) ) ) |
40 |
35 39
|
eleqtrd |
⊢ ( 𝜑 → 𝑋 ∈ ( ( 𝐽 ↾t 𝐴 ) fLim ( ( ( nei ‘ 𝐽 ) ‘ { 𝑋 } ) ↾t 𝐴 ) ) ) |
41 |
8 33 40
|
rspcdva |
⊢ ( 𝜑 → ( 𝐹 ‘ 𝑋 ) ∈ ( ( 𝐾 fLimf ( ( ( nei ‘ 𝐽 ) ‘ { 𝑋 } ) ↾t 𝐴 ) ) ‘ 𝐹 ) ) |