Step |
Hyp |
Ref |
Expression |
1 |
|
ntrcls.o |
⊢ 𝑂 = ( 𝑖 ∈ V ↦ ( 𝑘 ∈ ( 𝒫 𝑖 ↑m 𝒫 𝑖 ) ↦ ( 𝑗 ∈ 𝒫 𝑖 ↦ ( 𝑖 ∖ ( 𝑘 ‘ ( 𝑖 ∖ 𝑗 ) ) ) ) ) ) |
2 |
|
ntrcls.d |
⊢ 𝐷 = ( 𝑂 ‘ 𝐵 ) |
3 |
|
ntrcls.r |
⊢ ( 𝜑 → 𝐼 𝐷 𝐾 ) |
4 |
1 2 3
|
ntrclsf1o |
⊢ ( 𝜑 → 𝐷 : ( 𝒫 𝐵 ↑m 𝒫 𝐵 ) –1-1-onto→ ( 𝒫 𝐵 ↑m 𝒫 𝐵 ) ) |
5 |
|
f1orel |
⊢ ( 𝐷 : ( 𝒫 𝐵 ↑m 𝒫 𝐵 ) –1-1-onto→ ( 𝒫 𝐵 ↑m 𝒫 𝐵 ) → Rel 𝐷 ) |
6 |
4 5
|
syl |
⊢ ( 𝜑 → Rel 𝐷 ) |
7 |
|
releldm |
⊢ ( ( Rel 𝐷 ∧ 𝐼 𝐷 𝐾 ) → 𝐼 ∈ dom 𝐷 ) |
8 |
6 3 7
|
syl2anc |
⊢ ( 𝜑 → 𝐼 ∈ dom 𝐷 ) |
9 |
|
f1odm |
⊢ ( 𝐷 : ( 𝒫 𝐵 ↑m 𝒫 𝐵 ) –1-1-onto→ ( 𝒫 𝐵 ↑m 𝒫 𝐵 ) → dom 𝐷 = ( 𝒫 𝐵 ↑m 𝒫 𝐵 ) ) |
10 |
4 9
|
syl |
⊢ ( 𝜑 → dom 𝐷 = ( 𝒫 𝐵 ↑m 𝒫 𝐵 ) ) |
11 |
8 10
|
eleqtrd |
⊢ ( 𝜑 → 𝐼 ∈ ( 𝒫 𝐵 ↑m 𝒫 𝐵 ) ) |