Step |
Hyp |
Ref |
Expression |
1 |
|
pmapfval.b |
⊢ 𝐵 = ( Base ‘ 𝐾 ) |
2 |
|
pmapfval.l |
⊢ ≤ = ( le ‘ 𝐾 ) |
3 |
|
pmapfval.a |
⊢ 𝐴 = ( Atoms ‘ 𝐾 ) |
4 |
|
pmapfval.m |
⊢ 𝑀 = ( pmap ‘ 𝐾 ) |
5 |
1 2 3 4
|
pmapfval |
⊢ ( 𝐾 ∈ 𝐶 → 𝑀 = ( 𝑥 ∈ 𝐵 ↦ { 𝑎 ∈ 𝐴 ∣ 𝑎 ≤ 𝑥 } ) ) |
6 |
5
|
fveq1d |
⊢ ( 𝐾 ∈ 𝐶 → ( 𝑀 ‘ 𝑋 ) = ( ( 𝑥 ∈ 𝐵 ↦ { 𝑎 ∈ 𝐴 ∣ 𝑎 ≤ 𝑥 } ) ‘ 𝑋 ) ) |
7 |
|
breq2 |
⊢ ( 𝑥 = 𝑋 → ( 𝑎 ≤ 𝑥 ↔ 𝑎 ≤ 𝑋 ) ) |
8 |
7
|
rabbidv |
⊢ ( 𝑥 = 𝑋 → { 𝑎 ∈ 𝐴 ∣ 𝑎 ≤ 𝑥 } = { 𝑎 ∈ 𝐴 ∣ 𝑎 ≤ 𝑋 } ) |
9 |
|
eqid |
⊢ ( 𝑥 ∈ 𝐵 ↦ { 𝑎 ∈ 𝐴 ∣ 𝑎 ≤ 𝑥 } ) = ( 𝑥 ∈ 𝐵 ↦ { 𝑎 ∈ 𝐴 ∣ 𝑎 ≤ 𝑥 } ) |
10 |
3
|
fvexi |
⊢ 𝐴 ∈ V |
11 |
10
|
rabex |
⊢ { 𝑎 ∈ 𝐴 ∣ 𝑎 ≤ 𝑋 } ∈ V |
12 |
8 9 11
|
fvmpt |
⊢ ( 𝑋 ∈ 𝐵 → ( ( 𝑥 ∈ 𝐵 ↦ { 𝑎 ∈ 𝐴 ∣ 𝑎 ≤ 𝑥 } ) ‘ 𝑋 ) = { 𝑎 ∈ 𝐴 ∣ 𝑎 ≤ 𝑋 } ) |
13 |
6 12
|
sylan9eq |
⊢ ( ( 𝐾 ∈ 𝐶 ∧ 𝑋 ∈ 𝐵 ) → ( 𝑀 ‘ 𝑋 ) = { 𝑎 ∈ 𝐴 ∣ 𝑎 ≤ 𝑋 } ) |