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 |
1 2 3
|
fsovd |
⊢ ( 𝜑 → ( 𝐴 𝑂 𝐵 ) = ( 𝑓 ∈ ( 𝒫 𝐵 ↑m 𝐴 ) ↦ ( 𝑦 ∈ 𝐵 ↦ { 𝑥 ∈ 𝐴 ∣ 𝑦 ∈ ( 𝑓 ‘ 𝑥 ) } ) ) ) |
7 |
4 6
|
syl5eq |
⊢ ( 𝜑 → 𝐺 = ( 𝑓 ∈ ( 𝒫 𝐵 ↑m 𝐴 ) ↦ ( 𝑦 ∈ 𝐵 ↦ { 𝑥 ∈ 𝐴 ∣ 𝑦 ∈ ( 𝑓 ‘ 𝑥 ) } ) ) ) |
8 |
|
fveq1 |
⊢ ( 𝑓 = 𝐹 → ( 𝑓 ‘ 𝑥 ) = ( 𝐹 ‘ 𝑥 ) ) |
9 |
8
|
eleq2d |
⊢ ( 𝑓 = 𝐹 → ( 𝑦 ∈ ( 𝑓 ‘ 𝑥 ) ↔ 𝑦 ∈ ( 𝐹 ‘ 𝑥 ) ) ) |
10 |
9
|
rabbidv |
⊢ ( 𝑓 = 𝐹 → { 𝑥 ∈ 𝐴 ∣ 𝑦 ∈ ( 𝑓 ‘ 𝑥 ) } = { 𝑥 ∈ 𝐴 ∣ 𝑦 ∈ ( 𝐹 ‘ 𝑥 ) } ) |
11 |
10
|
mpteq2dv |
⊢ ( 𝑓 = 𝐹 → ( 𝑦 ∈ 𝐵 ↦ { 𝑥 ∈ 𝐴 ∣ 𝑦 ∈ ( 𝑓 ‘ 𝑥 ) } ) = ( 𝑦 ∈ 𝐵 ↦ { 𝑥 ∈ 𝐴 ∣ 𝑦 ∈ ( 𝐹 ‘ 𝑥 ) } ) ) |
12 |
11
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑓 = 𝐹 ) → ( 𝑦 ∈ 𝐵 ↦ { 𝑥 ∈ 𝐴 ∣ 𝑦 ∈ ( 𝑓 ‘ 𝑥 ) } ) = ( 𝑦 ∈ 𝐵 ↦ { 𝑥 ∈ 𝐴 ∣ 𝑦 ∈ ( 𝐹 ‘ 𝑥 ) } ) ) |
13 |
3
|
mptexd |
⊢ ( 𝜑 → ( 𝑦 ∈ 𝐵 ↦ { 𝑥 ∈ 𝐴 ∣ 𝑦 ∈ ( 𝐹 ‘ 𝑥 ) } ) ∈ V ) |
14 |
7 12 5 13
|
fvmptd |
⊢ ( 𝜑 → ( 𝐺 ‘ 𝐹 ) = ( 𝑦 ∈ 𝐵 ↦ { 𝑥 ∈ 𝐴 ∣ 𝑦 ∈ ( 𝐹 ‘ 𝑥 ) } ) ) |