Step |
Hyp |
Ref |
Expression |
1 |
|
diaval.b |
⊢ 𝐵 = ( Base ‘ 𝐾 ) |
2 |
|
diaval.l |
⊢ ≤ = ( le ‘ 𝐾 ) |
3 |
|
diaval.h |
⊢ 𝐻 = ( LHyp ‘ 𝐾 ) |
4 |
|
diaval.t |
⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) |
5 |
|
diaval.r |
⊢ 𝑅 = ( ( trL ‘ 𝐾 ) ‘ 𝑊 ) |
6 |
|
diaval.i |
⊢ 𝐼 = ( ( DIsoA ‘ 𝐾 ) ‘ 𝑊 ) |
7 |
1 2 3 4 5 6
|
diaval |
⊢ ( ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) → ( 𝐼 ‘ 𝑋 ) = { 𝑓 ∈ 𝑇 ∣ ( 𝑅 ‘ 𝑓 ) ≤ 𝑋 } ) |
8 |
7
|
eleq2d |
⊢ ( ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) → ( 𝐹 ∈ ( 𝐼 ‘ 𝑋 ) ↔ 𝐹 ∈ { 𝑓 ∈ 𝑇 ∣ ( 𝑅 ‘ 𝑓 ) ≤ 𝑋 } ) ) |
9 |
|
fveq2 |
⊢ ( 𝑓 = 𝐹 → ( 𝑅 ‘ 𝑓 ) = ( 𝑅 ‘ 𝐹 ) ) |
10 |
9
|
breq1d |
⊢ ( 𝑓 = 𝐹 → ( ( 𝑅 ‘ 𝑓 ) ≤ 𝑋 ↔ ( 𝑅 ‘ 𝐹 ) ≤ 𝑋 ) ) |
11 |
10
|
elrab |
⊢ ( 𝐹 ∈ { 𝑓 ∈ 𝑇 ∣ ( 𝑅 ‘ 𝑓 ) ≤ 𝑋 } ↔ ( 𝐹 ∈ 𝑇 ∧ ( 𝑅 ‘ 𝐹 ) ≤ 𝑋 ) ) |
12 |
8 11
|
syl6bb |
⊢ ( ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) → ( 𝐹 ∈ ( 𝐼 ‘ 𝑋 ) ↔ ( 𝐹 ∈ 𝑇 ∧ ( 𝑅 ‘ 𝐹 ) ≤ 𝑋 ) ) ) |