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 |
|
dssmapfv3d.s |
⊢ ( 𝜑 → 𝑆 ∈ 𝒫 𝐵 ) |
7 |
|
dssmapfv3d.t |
⊢ 𝑇 = ( 𝐺 ‘ 𝑆 ) |
8 |
1 2 3 4 5
|
dssmapfv2d |
⊢ ( 𝜑 → 𝐺 = ( 𝑠 ∈ 𝒫 𝐵 ↦ ( 𝐵 ∖ ( 𝐹 ‘ ( 𝐵 ∖ 𝑠 ) ) ) ) ) |
9 |
|
difeq2 |
⊢ ( 𝑠 = 𝑆 → ( 𝐵 ∖ 𝑠 ) = ( 𝐵 ∖ 𝑆 ) ) |
10 |
9
|
fveq2d |
⊢ ( 𝑠 = 𝑆 → ( 𝐹 ‘ ( 𝐵 ∖ 𝑠 ) ) = ( 𝐹 ‘ ( 𝐵 ∖ 𝑆 ) ) ) |
11 |
10
|
difeq2d |
⊢ ( 𝑠 = 𝑆 → ( 𝐵 ∖ ( 𝐹 ‘ ( 𝐵 ∖ 𝑠 ) ) ) = ( 𝐵 ∖ ( 𝐹 ‘ ( 𝐵 ∖ 𝑆 ) ) ) ) |
12 |
11
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑠 = 𝑆 ) → ( 𝐵 ∖ ( 𝐹 ‘ ( 𝐵 ∖ 𝑠 ) ) ) = ( 𝐵 ∖ ( 𝐹 ‘ ( 𝐵 ∖ 𝑆 ) ) ) ) |
13 |
|
difexg |
⊢ ( 𝐵 ∈ 𝑉 → ( 𝐵 ∖ ( 𝐹 ‘ ( 𝐵 ∖ 𝑆 ) ) ) ∈ V ) |
14 |
3 13
|
syl |
⊢ ( 𝜑 → ( 𝐵 ∖ ( 𝐹 ‘ ( 𝐵 ∖ 𝑆 ) ) ) ∈ V ) |
15 |
8 12 6 14
|
fvmptd |
⊢ ( 𝜑 → ( 𝐺 ‘ 𝑆 ) = ( 𝐵 ∖ ( 𝐹 ‘ ( 𝐵 ∖ 𝑆 ) ) ) ) |
16 |
7 15
|
syl5eq |
⊢ ( 𝜑 → 𝑇 = ( 𝐵 ∖ ( 𝐹 ‘ ( 𝐵 ∖ 𝑆 ) ) ) ) |