Step |
Hyp |
Ref |
Expression |
1 |
|
dibval.b |
⊢ 𝐵 = ( Base ‘ 𝐾 ) |
2 |
|
dibval.h |
⊢ 𝐻 = ( LHyp ‘ 𝐾 ) |
3 |
|
elex |
⊢ ( 𝐾 ∈ 𝑉 → 𝐾 ∈ V ) |
4 |
|
fveq2 |
⊢ ( 𝑘 = 𝐾 → ( LHyp ‘ 𝑘 ) = ( LHyp ‘ 𝐾 ) ) |
5 |
4 2
|
eqtr4di |
⊢ ( 𝑘 = 𝐾 → ( LHyp ‘ 𝑘 ) = 𝐻 ) |
6 |
|
fveq2 |
⊢ ( 𝑘 = 𝐾 → ( DIsoA ‘ 𝑘 ) = ( DIsoA ‘ 𝐾 ) ) |
7 |
6
|
fveq1d |
⊢ ( 𝑘 = 𝐾 → ( ( DIsoA ‘ 𝑘 ) ‘ 𝑤 ) = ( ( DIsoA ‘ 𝐾 ) ‘ 𝑤 ) ) |
8 |
7
|
dmeqd |
⊢ ( 𝑘 = 𝐾 → dom ( ( DIsoA ‘ 𝑘 ) ‘ 𝑤 ) = dom ( ( DIsoA ‘ 𝐾 ) ‘ 𝑤 ) ) |
9 |
7
|
fveq1d |
⊢ ( 𝑘 = 𝐾 → ( ( ( DIsoA ‘ 𝑘 ) ‘ 𝑤 ) ‘ 𝑥 ) = ( ( ( DIsoA ‘ 𝐾 ) ‘ 𝑤 ) ‘ 𝑥 ) ) |
10 |
|
fveq2 |
⊢ ( 𝑘 = 𝐾 → ( LTrn ‘ 𝑘 ) = ( LTrn ‘ 𝐾 ) ) |
11 |
10
|
fveq1d |
⊢ ( 𝑘 = 𝐾 → ( ( LTrn ‘ 𝑘 ) ‘ 𝑤 ) = ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) ) |
12 |
|
fveq2 |
⊢ ( 𝑘 = 𝐾 → ( Base ‘ 𝑘 ) = ( Base ‘ 𝐾 ) ) |
13 |
12 1
|
eqtr4di |
⊢ ( 𝑘 = 𝐾 → ( Base ‘ 𝑘 ) = 𝐵 ) |
14 |
13
|
reseq2d |
⊢ ( 𝑘 = 𝐾 → ( I ↾ ( Base ‘ 𝑘 ) ) = ( I ↾ 𝐵 ) ) |
15 |
11 14
|
mpteq12dv |
⊢ ( 𝑘 = 𝐾 → ( 𝑓 ∈ ( ( LTrn ‘ 𝑘 ) ‘ 𝑤 ) ↦ ( I ↾ ( Base ‘ 𝑘 ) ) ) = ( 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) ↦ ( I ↾ 𝐵 ) ) ) |
16 |
15
|
sneqd |
⊢ ( 𝑘 = 𝐾 → { ( 𝑓 ∈ ( ( LTrn ‘ 𝑘 ) ‘ 𝑤 ) ↦ ( I ↾ ( Base ‘ 𝑘 ) ) ) } = { ( 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) ↦ ( I ↾ 𝐵 ) ) } ) |
17 |
9 16
|
xpeq12d |
⊢ ( 𝑘 = 𝐾 → ( ( ( ( DIsoA ‘ 𝑘 ) ‘ 𝑤 ) ‘ 𝑥 ) × { ( 𝑓 ∈ ( ( LTrn ‘ 𝑘 ) ‘ 𝑤 ) ↦ ( I ↾ ( Base ‘ 𝑘 ) ) ) } ) = ( ( ( ( DIsoA ‘ 𝐾 ) ‘ 𝑤 ) ‘ 𝑥 ) × { ( 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) ↦ ( I ↾ 𝐵 ) ) } ) ) |
18 |
8 17
|
mpteq12dv |
⊢ ( 𝑘 = 𝐾 → ( 𝑥 ∈ dom ( ( DIsoA ‘ 𝑘 ) ‘ 𝑤 ) ↦ ( ( ( ( DIsoA ‘ 𝑘 ) ‘ 𝑤 ) ‘ 𝑥 ) × { ( 𝑓 ∈ ( ( LTrn ‘ 𝑘 ) ‘ 𝑤 ) ↦ ( I ↾ ( Base ‘ 𝑘 ) ) ) } ) ) = ( 𝑥 ∈ dom ( ( DIsoA ‘ 𝐾 ) ‘ 𝑤 ) ↦ ( ( ( ( DIsoA ‘ 𝐾 ) ‘ 𝑤 ) ‘ 𝑥 ) × { ( 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) ↦ ( I ↾ 𝐵 ) ) } ) ) ) |
19 |
5 18
|
mpteq12dv |
⊢ ( 𝑘 = 𝐾 → ( 𝑤 ∈ ( LHyp ‘ 𝑘 ) ↦ ( 𝑥 ∈ dom ( ( DIsoA ‘ 𝑘 ) ‘ 𝑤 ) ↦ ( ( ( ( DIsoA ‘ 𝑘 ) ‘ 𝑤 ) ‘ 𝑥 ) × { ( 𝑓 ∈ ( ( LTrn ‘ 𝑘 ) ‘ 𝑤 ) ↦ ( I ↾ ( Base ‘ 𝑘 ) ) ) } ) ) ) = ( 𝑤 ∈ 𝐻 ↦ ( 𝑥 ∈ dom ( ( DIsoA ‘ 𝐾 ) ‘ 𝑤 ) ↦ ( ( ( ( DIsoA ‘ 𝐾 ) ‘ 𝑤 ) ‘ 𝑥 ) × { ( 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) ↦ ( I ↾ 𝐵 ) ) } ) ) ) ) |
20 |
|
df-dib |
⊢ DIsoB = ( 𝑘 ∈ V ↦ ( 𝑤 ∈ ( LHyp ‘ 𝑘 ) ↦ ( 𝑥 ∈ dom ( ( DIsoA ‘ 𝑘 ) ‘ 𝑤 ) ↦ ( ( ( ( DIsoA ‘ 𝑘 ) ‘ 𝑤 ) ‘ 𝑥 ) × { ( 𝑓 ∈ ( ( LTrn ‘ 𝑘 ) ‘ 𝑤 ) ↦ ( I ↾ ( Base ‘ 𝑘 ) ) ) } ) ) ) ) |
21 |
19 20 2
|
mptfvmpt |
⊢ ( 𝐾 ∈ V → ( DIsoB ‘ 𝐾 ) = ( 𝑤 ∈ 𝐻 ↦ ( 𝑥 ∈ dom ( ( DIsoA ‘ 𝐾 ) ‘ 𝑤 ) ↦ ( ( ( ( DIsoA ‘ 𝐾 ) ‘ 𝑤 ) ‘ 𝑥 ) × { ( 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) ↦ ( I ↾ 𝐵 ) ) } ) ) ) ) |
22 |
3 21
|
syl |
⊢ ( 𝐾 ∈ 𝑉 → ( DIsoB ‘ 𝐾 ) = ( 𝑤 ∈ 𝐻 ↦ ( 𝑥 ∈ dom ( ( DIsoA ‘ 𝐾 ) ‘ 𝑤 ) ↦ ( ( ( ( DIsoA ‘ 𝐾 ) ‘ 𝑤 ) ‘ 𝑥 ) × { ( 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) ↦ ( I ↾ 𝐵 ) ) } ) ) ) ) |