Step |
Hyp |
Ref |
Expression |
1 |
|
ntrnei.o |
⊢ 𝑂 = ( 𝑖 ∈ V , 𝑗 ∈ V ↦ ( 𝑘 ∈ ( 𝒫 𝑗 ↑m 𝑖 ) ↦ ( 𝑙 ∈ 𝑗 ↦ { 𝑚 ∈ 𝑖 ∣ 𝑙 ∈ ( 𝑘 ‘ 𝑚 ) } ) ) ) |
2 |
|
ntrnei.f |
⊢ 𝐹 = ( 𝒫 𝐵 𝑂 𝐵 ) |
3 |
|
ntrnei.r |
⊢ ( 𝜑 → 𝐼 𝐹 𝑁 ) |
4 |
1 2 3
|
ntrneif1o |
⊢ ( 𝜑 → 𝐹 : ( 𝒫 𝐵 ↑m 𝒫 𝐵 ) –1-1-onto→ ( 𝒫 𝒫 𝐵 ↑m 𝐵 ) ) |
5 |
|
f1orel |
⊢ ( 𝐹 : ( 𝒫 𝐵 ↑m 𝒫 𝐵 ) –1-1-onto→ ( 𝒫 𝒫 𝐵 ↑m 𝐵 ) → Rel 𝐹 ) |
6 |
4 5
|
syl |
⊢ ( 𝜑 → Rel 𝐹 ) |
7 |
|
relelrn |
⊢ ( ( Rel 𝐹 ∧ 𝐼 𝐹 𝑁 ) → 𝑁 ∈ ran 𝐹 ) |
8 |
6 3 7
|
syl2anc |
⊢ ( 𝜑 → 𝑁 ∈ ran 𝐹 ) |
9 |
|
dff1o2 |
⊢ ( 𝐹 : ( 𝒫 𝐵 ↑m 𝒫 𝐵 ) –1-1-onto→ ( 𝒫 𝒫 𝐵 ↑m 𝐵 ) ↔ ( 𝐹 Fn ( 𝒫 𝐵 ↑m 𝒫 𝐵 ) ∧ Fun ◡ 𝐹 ∧ ran 𝐹 = ( 𝒫 𝒫 𝐵 ↑m 𝐵 ) ) ) |
10 |
4 9
|
sylib |
⊢ ( 𝜑 → ( 𝐹 Fn ( 𝒫 𝐵 ↑m 𝒫 𝐵 ) ∧ Fun ◡ 𝐹 ∧ ran 𝐹 = ( 𝒫 𝒫 𝐵 ↑m 𝐵 ) ) ) |
11 |
10
|
simp3d |
⊢ ( 𝜑 → ran 𝐹 = ( 𝒫 𝒫 𝐵 ↑m 𝐵 ) ) |
12 |
8 11
|
eleqtrd |
⊢ ( 𝜑 → 𝑁 ∈ ( 𝒫 𝒫 𝐵 ↑m 𝐵 ) ) |