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 |
|
elex |
⊢ ( 𝐾 ∈ 𝐶 → 𝐾 ∈ V ) |
6 |
|
fveq2 |
⊢ ( 𝑘 = 𝐾 → ( Base ‘ 𝑘 ) = ( Base ‘ 𝐾 ) ) |
7 |
6 1
|
eqtr4di |
⊢ ( 𝑘 = 𝐾 → ( Base ‘ 𝑘 ) = 𝐵 ) |
8 |
|
fveq2 |
⊢ ( 𝑘 = 𝐾 → ( Atoms ‘ 𝑘 ) = ( Atoms ‘ 𝐾 ) ) |
9 |
8 3
|
eqtr4di |
⊢ ( 𝑘 = 𝐾 → ( Atoms ‘ 𝑘 ) = 𝐴 ) |
10 |
|
fveq2 |
⊢ ( 𝑘 = 𝐾 → ( le ‘ 𝑘 ) = ( le ‘ 𝐾 ) ) |
11 |
10 2
|
eqtr4di |
⊢ ( 𝑘 = 𝐾 → ( le ‘ 𝑘 ) = ≤ ) |
12 |
11
|
breqd |
⊢ ( 𝑘 = 𝐾 → ( 𝑎 ( le ‘ 𝑘 ) 𝑥 ↔ 𝑎 ≤ 𝑥 ) ) |
13 |
9 12
|
rabeqbidv |
⊢ ( 𝑘 = 𝐾 → { 𝑎 ∈ ( Atoms ‘ 𝑘 ) ∣ 𝑎 ( le ‘ 𝑘 ) 𝑥 } = { 𝑎 ∈ 𝐴 ∣ 𝑎 ≤ 𝑥 } ) |
14 |
7 13
|
mpteq12dv |
⊢ ( 𝑘 = 𝐾 → ( 𝑥 ∈ ( Base ‘ 𝑘 ) ↦ { 𝑎 ∈ ( Atoms ‘ 𝑘 ) ∣ 𝑎 ( le ‘ 𝑘 ) 𝑥 } ) = ( 𝑥 ∈ 𝐵 ↦ { 𝑎 ∈ 𝐴 ∣ 𝑎 ≤ 𝑥 } ) ) |
15 |
|
df-pmap |
⊢ pmap = ( 𝑘 ∈ V ↦ ( 𝑥 ∈ ( Base ‘ 𝑘 ) ↦ { 𝑎 ∈ ( Atoms ‘ 𝑘 ) ∣ 𝑎 ( le ‘ 𝑘 ) 𝑥 } ) ) |
16 |
14 15 1
|
mptfvmpt |
⊢ ( 𝐾 ∈ V → ( pmap ‘ 𝐾 ) = ( 𝑥 ∈ 𝐵 ↦ { 𝑎 ∈ 𝐴 ∣ 𝑎 ≤ 𝑥 } ) ) |
17 |
4 16
|
syl5eq |
⊢ ( 𝐾 ∈ V → 𝑀 = ( 𝑥 ∈ 𝐵 ↦ { 𝑎 ∈ 𝐴 ∣ 𝑎 ≤ 𝑥 } ) ) |
18 |
5 17
|
syl |
⊢ ( 𝐾 ∈ 𝐶 → 𝑀 = ( 𝑥 ∈ 𝐵 ↦ { 𝑎 ∈ 𝐴 ∣ 𝑎 ≤ 𝑥 } ) ) |