Step |
Hyp |
Ref |
Expression |
1 |
|
rfovd.rf |
⊢ 𝑂 = ( 𝑎 ∈ V , 𝑏 ∈ V ↦ ( 𝑟 ∈ 𝒫 ( 𝑎 × 𝑏 ) ↦ ( 𝑥 ∈ 𝑎 ↦ { 𝑦 ∈ 𝑏 ∣ 𝑥 𝑟 𝑦 } ) ) ) |
2 |
|
rfovd.a |
⊢ ( 𝜑 → 𝐴 ∈ 𝑉 ) |
3 |
|
rfovd.b |
⊢ ( 𝜑 → 𝐵 ∈ 𝑊 ) |
4 |
|
rfovfvd.r |
⊢ ( 𝜑 → 𝑅 ∈ 𝒫 ( 𝐴 × 𝐵 ) ) |
5 |
|
rfovfvd.f |
⊢ 𝐹 = ( 𝐴 𝑂 𝐵 ) |
6 |
|
rfovfvfvd.x |
⊢ ( 𝜑 → 𝑋 ∈ 𝐴 ) |
7 |
|
rfovfvfvd.g |
⊢ 𝐺 = ( 𝐹 ‘ 𝑅 ) |
8 |
1 2 3 4 5
|
rfovfvd |
⊢ ( 𝜑 → ( 𝐹 ‘ 𝑅 ) = ( 𝑥 ∈ 𝐴 ↦ { 𝑦 ∈ 𝐵 ∣ 𝑥 𝑅 𝑦 } ) ) |
9 |
7 8
|
syl5eq |
⊢ ( 𝜑 → 𝐺 = ( 𝑥 ∈ 𝐴 ↦ { 𝑦 ∈ 𝐵 ∣ 𝑥 𝑅 𝑦 } ) ) |
10 |
|
breq1 |
⊢ ( 𝑥 = 𝑋 → ( 𝑥 𝑅 𝑦 ↔ 𝑋 𝑅 𝑦 ) ) |
11 |
10
|
rabbidv |
⊢ ( 𝑥 = 𝑋 → { 𝑦 ∈ 𝐵 ∣ 𝑥 𝑅 𝑦 } = { 𝑦 ∈ 𝐵 ∣ 𝑋 𝑅 𝑦 } ) |
12 |
11
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑥 = 𝑋 ) → { 𝑦 ∈ 𝐵 ∣ 𝑥 𝑅 𝑦 } = { 𝑦 ∈ 𝐵 ∣ 𝑋 𝑅 𝑦 } ) |
13 |
|
rabexg |
⊢ ( 𝐵 ∈ 𝑊 → { 𝑦 ∈ 𝐵 ∣ 𝑋 𝑅 𝑦 } ∈ V ) |
14 |
3 13
|
syl |
⊢ ( 𝜑 → { 𝑦 ∈ 𝐵 ∣ 𝑋 𝑅 𝑦 } ∈ V ) |
15 |
9 12 6 14
|
fvmptd |
⊢ ( 𝜑 → ( 𝐺 ‘ 𝑋 ) = { 𝑦 ∈ 𝐵 ∣ 𝑋 𝑅 𝑦 } ) |