Step |
Hyp |
Ref |
Expression |
1 |
|
resstopn.1 |
⊢ 𝐻 = ( 𝐾 ↾s 𝐴 ) |
2 |
|
resstopn.2 |
⊢ 𝐽 = ( TopOpen ‘ 𝐾 ) |
3 |
|
fvex |
⊢ ( TopSet ‘ 𝐾 ) ∈ V |
4 |
|
fvex |
⊢ ( Base ‘ 𝐾 ) ∈ V |
5 |
|
restco |
⊢ ( ( ( TopSet ‘ 𝐾 ) ∈ V ∧ ( Base ‘ 𝐾 ) ∈ V ∧ 𝐴 ∈ V ) → ( ( ( TopSet ‘ 𝐾 ) ↾t ( Base ‘ 𝐾 ) ) ↾t 𝐴 ) = ( ( TopSet ‘ 𝐾 ) ↾t ( ( Base ‘ 𝐾 ) ∩ 𝐴 ) ) ) |
6 |
3 4 5
|
mp3an12 |
⊢ ( 𝐴 ∈ V → ( ( ( TopSet ‘ 𝐾 ) ↾t ( Base ‘ 𝐾 ) ) ↾t 𝐴 ) = ( ( TopSet ‘ 𝐾 ) ↾t ( ( Base ‘ 𝐾 ) ∩ 𝐴 ) ) ) |
7 |
|
eqid |
⊢ ( TopSet ‘ 𝐾 ) = ( TopSet ‘ 𝐾 ) |
8 |
1 7
|
resstset |
⊢ ( 𝐴 ∈ V → ( TopSet ‘ 𝐾 ) = ( TopSet ‘ 𝐻 ) ) |
9 |
|
incom |
⊢ ( ( Base ‘ 𝐾 ) ∩ 𝐴 ) = ( 𝐴 ∩ ( Base ‘ 𝐾 ) ) |
10 |
|
eqid |
⊢ ( Base ‘ 𝐾 ) = ( Base ‘ 𝐾 ) |
11 |
1 10
|
ressbas |
⊢ ( 𝐴 ∈ V → ( 𝐴 ∩ ( Base ‘ 𝐾 ) ) = ( Base ‘ 𝐻 ) ) |
12 |
9 11
|
syl5eq |
⊢ ( 𝐴 ∈ V → ( ( Base ‘ 𝐾 ) ∩ 𝐴 ) = ( Base ‘ 𝐻 ) ) |
13 |
8 12
|
oveq12d |
⊢ ( 𝐴 ∈ V → ( ( TopSet ‘ 𝐾 ) ↾t ( ( Base ‘ 𝐾 ) ∩ 𝐴 ) ) = ( ( TopSet ‘ 𝐻 ) ↾t ( Base ‘ 𝐻 ) ) ) |
14 |
6 13
|
eqtrd |
⊢ ( 𝐴 ∈ V → ( ( ( TopSet ‘ 𝐾 ) ↾t ( Base ‘ 𝐾 ) ) ↾t 𝐴 ) = ( ( TopSet ‘ 𝐻 ) ↾t ( Base ‘ 𝐻 ) ) ) |
15 |
10 7
|
topnval |
⊢ ( ( TopSet ‘ 𝐾 ) ↾t ( Base ‘ 𝐾 ) ) = ( TopOpen ‘ 𝐾 ) |
16 |
15 2
|
eqtr4i |
⊢ ( ( TopSet ‘ 𝐾 ) ↾t ( Base ‘ 𝐾 ) ) = 𝐽 |
17 |
16
|
oveq1i |
⊢ ( ( ( TopSet ‘ 𝐾 ) ↾t ( Base ‘ 𝐾 ) ) ↾t 𝐴 ) = ( 𝐽 ↾t 𝐴 ) |
18 |
|
eqid |
⊢ ( Base ‘ 𝐻 ) = ( Base ‘ 𝐻 ) |
19 |
|
eqid |
⊢ ( TopSet ‘ 𝐻 ) = ( TopSet ‘ 𝐻 ) |
20 |
18 19
|
topnval |
⊢ ( ( TopSet ‘ 𝐻 ) ↾t ( Base ‘ 𝐻 ) ) = ( TopOpen ‘ 𝐻 ) |
21 |
14 17 20
|
3eqtr3g |
⊢ ( 𝐴 ∈ V → ( 𝐽 ↾t 𝐴 ) = ( TopOpen ‘ 𝐻 ) ) |
22 |
|
simpr |
⊢ ( ( 𝐽 ∈ V ∧ 𝐴 ∈ V ) → 𝐴 ∈ V ) |
23 |
|
restfn |
⊢ ↾t Fn ( V × V ) |
24 |
23
|
fndmi |
⊢ dom ↾t = ( V × V ) |
25 |
24
|
ndmov |
⊢ ( ¬ ( 𝐽 ∈ V ∧ 𝐴 ∈ V ) → ( 𝐽 ↾t 𝐴 ) = ∅ ) |
26 |
22 25
|
nsyl5 |
⊢ ( ¬ 𝐴 ∈ V → ( 𝐽 ↾t 𝐴 ) = ∅ ) |
27 |
|
reldmress |
⊢ Rel dom ↾s |
28 |
27
|
ovprc2 |
⊢ ( ¬ 𝐴 ∈ V → ( 𝐾 ↾s 𝐴 ) = ∅ ) |
29 |
1 28
|
syl5eq |
⊢ ( ¬ 𝐴 ∈ V → 𝐻 = ∅ ) |
30 |
29
|
fveq2d |
⊢ ( ¬ 𝐴 ∈ V → ( TopSet ‘ 𝐻 ) = ( TopSet ‘ ∅ ) ) |
31 |
|
tsetid |
⊢ TopSet = Slot ( TopSet ‘ ndx ) |
32 |
31
|
str0 |
⊢ ∅ = ( TopSet ‘ ∅ ) |
33 |
30 32
|
eqtr4di |
⊢ ( ¬ 𝐴 ∈ V → ( TopSet ‘ 𝐻 ) = ∅ ) |
34 |
33
|
oveq1d |
⊢ ( ¬ 𝐴 ∈ V → ( ( TopSet ‘ 𝐻 ) ↾t ( Base ‘ 𝐻 ) ) = ( ∅ ↾t ( Base ‘ 𝐻 ) ) ) |
35 |
|
0rest |
⊢ ( ∅ ↾t ( Base ‘ 𝐻 ) ) = ∅ |
36 |
34 20 35
|
3eqtr3g |
⊢ ( ¬ 𝐴 ∈ V → ( TopOpen ‘ 𝐻 ) = ∅ ) |
37 |
26 36
|
eqtr4d |
⊢ ( ¬ 𝐴 ∈ V → ( 𝐽 ↾t 𝐴 ) = ( TopOpen ‘ 𝐻 ) ) |
38 |
21 37
|
pm2.61i |
⊢ ( 𝐽 ↾t 𝐴 ) = ( TopOpen ‘ 𝐻 ) |