Step |
Hyp |
Ref |
Expression |
1 |
|
psrbas.s |
⊢ 𝑆 = ( 𝐼 mPwSer 𝑅 ) |
2 |
|
psrbas.k |
⊢ 𝐾 = ( Base ‘ 𝑅 ) |
3 |
|
psrbas.d |
⊢ 𝐷 = { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin } |
4 |
|
psrbas.b |
⊢ 𝐵 = ( Base ‘ 𝑆 ) |
5 |
|
psrelbas.x |
⊢ ( 𝜑 → 𝑋 ∈ 𝐵 ) |
6 |
|
reldmpsr |
⊢ Rel dom mPwSer |
7 |
6 1 4
|
elbasov |
⊢ ( 𝑋 ∈ 𝐵 → ( 𝐼 ∈ V ∧ 𝑅 ∈ V ) ) |
8 |
5 7
|
syl |
⊢ ( 𝜑 → ( 𝐼 ∈ V ∧ 𝑅 ∈ V ) ) |
9 |
8
|
simpld |
⊢ ( 𝜑 → 𝐼 ∈ V ) |
10 |
1 2 3 4 9
|
psrbas |
⊢ ( 𝜑 → 𝐵 = ( 𝐾 ↑m 𝐷 ) ) |
11 |
5 10
|
eleqtrd |
⊢ ( 𝜑 → 𝑋 ∈ ( 𝐾 ↑m 𝐷 ) ) |
12 |
2
|
fvexi |
⊢ 𝐾 ∈ V |
13 |
|
ovex |
⊢ ( ℕ0 ↑m 𝐼 ) ∈ V |
14 |
3 13
|
rabex2 |
⊢ 𝐷 ∈ V |
15 |
12 14
|
elmap |
⊢ ( 𝑋 ∈ ( 𝐾 ↑m 𝐷 ) ↔ 𝑋 : 𝐷 ⟶ 𝐾 ) |
16 |
11 15
|
sylib |
⊢ ( 𝜑 → 𝑋 : 𝐷 ⟶ 𝐾 ) |