Step |
Hyp |
Ref |
Expression |
1 |
|
fsovd.fs |
⊢ 𝑂 = ( 𝑎 ∈ V , 𝑏 ∈ V ↦ ( 𝑓 ∈ ( 𝒫 𝑏 ↑m 𝑎 ) ↦ ( 𝑦 ∈ 𝑏 ↦ { 𝑥 ∈ 𝑎 ∣ 𝑦 ∈ ( 𝑓 ‘ 𝑥 ) } ) ) ) |
2 |
|
fsovd.a |
⊢ ( 𝜑 → 𝐴 ∈ 𝑉 ) |
3 |
|
fsovd.b |
⊢ ( 𝜑 → 𝐵 ∈ 𝑊 ) |
4 |
|
fsovfvd.g |
⊢ 𝐺 = ( 𝐴 𝑂 𝐵 ) |
5 |
|
fsovfvd.f |
⊢ ( 𝜑 → 𝐹 ∈ ( 𝒫 𝐵 ↑m 𝐴 ) ) |
6 |
|
fsovfvfvd.h |
⊢ 𝐻 = ( 𝐺 ‘ 𝐹 ) |
7 |
|
fsovfvfvd.y |
⊢ ( 𝜑 → 𝑌 ∈ 𝐵 ) |
8 |
1 2 3 4 5
|
fsovfvd |
⊢ ( 𝜑 → ( 𝐺 ‘ 𝐹 ) = ( 𝑦 ∈ 𝐵 ↦ { 𝑥 ∈ 𝐴 ∣ 𝑦 ∈ ( 𝐹 ‘ 𝑥 ) } ) ) |
9 |
6 8
|
syl5eq |
⊢ ( 𝜑 → 𝐻 = ( 𝑦 ∈ 𝐵 ↦ { 𝑥 ∈ 𝐴 ∣ 𝑦 ∈ ( 𝐹 ‘ 𝑥 ) } ) ) |
10 |
|
eleq1 |
⊢ ( 𝑦 = 𝑌 → ( 𝑦 ∈ ( 𝐹 ‘ 𝑥 ) ↔ 𝑌 ∈ ( 𝐹 ‘ 𝑥 ) ) ) |
11 |
10
|
rabbidv |
⊢ ( 𝑦 = 𝑌 → { 𝑥 ∈ 𝐴 ∣ 𝑦 ∈ ( 𝐹 ‘ 𝑥 ) } = { 𝑥 ∈ 𝐴 ∣ 𝑌 ∈ ( 𝐹 ‘ 𝑥 ) } ) |
12 |
11
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑦 = 𝑌 ) → { 𝑥 ∈ 𝐴 ∣ 𝑦 ∈ ( 𝐹 ‘ 𝑥 ) } = { 𝑥 ∈ 𝐴 ∣ 𝑌 ∈ ( 𝐹 ‘ 𝑥 ) } ) |
13 |
|
rabexg |
⊢ ( 𝐴 ∈ 𝑉 → { 𝑥 ∈ 𝐴 ∣ 𝑌 ∈ ( 𝐹 ‘ 𝑥 ) } ∈ V ) |
14 |
2 13
|
syl |
⊢ ( 𝜑 → { 𝑥 ∈ 𝐴 ∣ 𝑌 ∈ ( 𝐹 ‘ 𝑥 ) } ∈ V ) |
15 |
9 12 7 14
|
fvmptd |
⊢ ( 𝜑 → ( 𝐻 ‘ 𝑌 ) = { 𝑥 ∈ 𝐴 ∣ 𝑌 ∈ ( 𝐹 ‘ 𝑥 ) } ) |