Step |
Hyp |
Ref |
Expression |
1 |
|
dssmapfvd.o |
⊢ 𝑂 = ( 𝑏 ∈ V ↦ ( 𝑓 ∈ ( 𝒫 𝑏 ↑m 𝒫 𝑏 ) ↦ ( 𝑠 ∈ 𝒫 𝑏 ↦ ( 𝑏 ∖ ( 𝑓 ‘ ( 𝑏 ∖ 𝑠 ) ) ) ) ) ) |
2 |
|
dssmapfvd.d |
⊢ 𝐷 = ( 𝑂 ‘ 𝐵 ) |
3 |
|
dssmapfvd.b |
⊢ ( 𝜑 → 𝐵 ∈ 𝑉 ) |
4 |
|
dssmapfv2d.f |
⊢ ( 𝜑 → 𝐹 ∈ ( 𝒫 𝐵 ↑m 𝒫 𝐵 ) ) |
5 |
|
dssmapfv2d.g |
⊢ 𝐺 = ( 𝐷 ‘ 𝐹 ) |
6 |
1 2 3
|
dssmapfvd |
⊢ ( 𝜑 → 𝐷 = ( 𝑓 ∈ ( 𝒫 𝐵 ↑m 𝒫 𝐵 ) ↦ ( 𝑠 ∈ 𝒫 𝐵 ↦ ( 𝐵 ∖ ( 𝑓 ‘ ( 𝐵 ∖ 𝑠 ) ) ) ) ) ) |
7 |
|
fveq1 |
⊢ ( 𝑓 = 𝐹 → ( 𝑓 ‘ ( 𝐵 ∖ 𝑠 ) ) = ( 𝐹 ‘ ( 𝐵 ∖ 𝑠 ) ) ) |
8 |
7
|
difeq2d |
⊢ ( 𝑓 = 𝐹 → ( 𝐵 ∖ ( 𝑓 ‘ ( 𝐵 ∖ 𝑠 ) ) ) = ( 𝐵 ∖ ( 𝐹 ‘ ( 𝐵 ∖ 𝑠 ) ) ) ) |
9 |
8
|
mpteq2dv |
⊢ ( 𝑓 = 𝐹 → ( 𝑠 ∈ 𝒫 𝐵 ↦ ( 𝐵 ∖ ( 𝑓 ‘ ( 𝐵 ∖ 𝑠 ) ) ) ) = ( 𝑠 ∈ 𝒫 𝐵 ↦ ( 𝐵 ∖ ( 𝐹 ‘ ( 𝐵 ∖ 𝑠 ) ) ) ) ) |
10 |
9
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑓 = 𝐹 ) → ( 𝑠 ∈ 𝒫 𝐵 ↦ ( 𝐵 ∖ ( 𝑓 ‘ ( 𝐵 ∖ 𝑠 ) ) ) ) = ( 𝑠 ∈ 𝒫 𝐵 ↦ ( 𝐵 ∖ ( 𝐹 ‘ ( 𝐵 ∖ 𝑠 ) ) ) ) ) |
11 |
|
pwexg |
⊢ ( 𝐵 ∈ 𝑉 → 𝒫 𝐵 ∈ V ) |
12 |
|
mptexg |
⊢ ( 𝒫 𝐵 ∈ V → ( 𝑠 ∈ 𝒫 𝐵 ↦ ( 𝐵 ∖ ( 𝐹 ‘ ( 𝐵 ∖ 𝑠 ) ) ) ) ∈ V ) |
13 |
3 11 12
|
3syl |
⊢ ( 𝜑 → ( 𝑠 ∈ 𝒫 𝐵 ↦ ( 𝐵 ∖ ( 𝐹 ‘ ( 𝐵 ∖ 𝑠 ) ) ) ) ∈ V ) |
14 |
6 10 4 13
|
fvmptd |
⊢ ( 𝜑 → ( 𝐷 ‘ 𝐹 ) = ( 𝑠 ∈ 𝒫 𝐵 ↦ ( 𝐵 ∖ ( 𝐹 ‘ ( 𝐵 ∖ 𝑠 ) ) ) ) ) |
15 |
5 14
|
syl5eq |
⊢ ( 𝜑 → 𝐺 = ( 𝑠 ∈ 𝒫 𝐵 ↦ ( 𝐵 ∖ ( 𝐹 ‘ ( 𝐵 ∖ 𝑠 ) ) ) ) ) |