Step |
Hyp |
Ref |
Expression |
1 |
|
clsnei.o |
⊢ 𝑂 = ( 𝑖 ∈ V , 𝑗 ∈ V ↦ ( 𝑘 ∈ ( 𝒫 𝑗 ↑m 𝑖 ) ↦ ( 𝑙 ∈ 𝑗 ↦ { 𝑚 ∈ 𝑖 ∣ 𝑙 ∈ ( 𝑘 ‘ 𝑚 ) } ) ) ) |
2 |
|
clsnei.p |
⊢ 𝑃 = ( 𝑛 ∈ V ↦ ( 𝑝 ∈ ( 𝒫 𝑛 ↑m 𝒫 𝑛 ) ↦ ( 𝑜 ∈ 𝒫 𝑛 ↦ ( 𝑛 ∖ ( 𝑝 ‘ ( 𝑛 ∖ 𝑜 ) ) ) ) ) ) |
3 |
|
clsnei.d |
⊢ 𝐷 = ( 𝑃 ‘ 𝐵 ) |
4 |
|
clsnei.f |
⊢ 𝐹 = ( 𝒫 𝐵 𝑂 𝐵 ) |
5 |
|
clsnei.h |
⊢ 𝐻 = ( 𝐹 ∘ 𝐷 ) |
6 |
|
clsnei.r |
⊢ ( 𝜑 → 𝐾 𝐻 𝑁 ) |
7 |
3 5 6
|
clsneibex |
⊢ ( 𝜑 → 𝐵 ∈ V ) |
8 |
|
pwexg |
⊢ ( 𝐵 ∈ V → 𝒫 𝐵 ∈ V ) |
9 |
8
|
adantl |
⊢ ( ( 𝜑 ∧ 𝐵 ∈ V ) → 𝒫 𝐵 ∈ V ) |
10 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝐵 ∈ V ) → 𝐵 ∈ V ) |
11 |
1 9 10 4
|
fsovf1od |
⊢ ( ( 𝜑 ∧ 𝐵 ∈ V ) → 𝐹 : ( 𝒫 𝐵 ↑m 𝒫 𝐵 ) –1-1-onto→ ( 𝒫 𝒫 𝐵 ↑m 𝐵 ) ) |
12 |
|
f1ofn |
⊢ ( 𝐹 : ( 𝒫 𝐵 ↑m 𝒫 𝐵 ) –1-1-onto→ ( 𝒫 𝒫 𝐵 ↑m 𝐵 ) → 𝐹 Fn ( 𝒫 𝐵 ↑m 𝒫 𝐵 ) ) |
13 |
11 12
|
syl |
⊢ ( ( 𝜑 ∧ 𝐵 ∈ V ) → 𝐹 Fn ( 𝒫 𝐵 ↑m 𝒫 𝐵 ) ) |
14 |
2 3 10
|
dssmapf1od |
⊢ ( ( 𝜑 ∧ 𝐵 ∈ V ) → 𝐷 : ( 𝒫 𝐵 ↑m 𝒫 𝐵 ) –1-1-onto→ ( 𝒫 𝐵 ↑m 𝒫 𝐵 ) ) |
15 |
|
f1of |
⊢ ( 𝐷 : ( 𝒫 𝐵 ↑m 𝒫 𝐵 ) –1-1-onto→ ( 𝒫 𝐵 ↑m 𝒫 𝐵 ) → 𝐷 : ( 𝒫 𝐵 ↑m 𝒫 𝐵 ) ⟶ ( 𝒫 𝐵 ↑m 𝒫 𝐵 ) ) |
16 |
14 15
|
syl |
⊢ ( ( 𝜑 ∧ 𝐵 ∈ V ) → 𝐷 : ( 𝒫 𝐵 ↑m 𝒫 𝐵 ) ⟶ ( 𝒫 𝐵 ↑m 𝒫 𝐵 ) ) |
17 |
6
|
adantr |
⊢ ( ( 𝜑 ∧ 𝐵 ∈ V ) → 𝐾 𝐻 𝑁 ) |
18 |
5
|
breqi |
⊢ ( 𝐾 𝐻 𝑁 ↔ 𝐾 ( 𝐹 ∘ 𝐷 ) 𝑁 ) |
19 |
17 18
|
sylib |
⊢ ( ( 𝜑 ∧ 𝐵 ∈ V ) → 𝐾 ( 𝐹 ∘ 𝐷 ) 𝑁 ) |
20 |
13 16 19
|
brcoffn |
⊢ ( ( 𝜑 ∧ 𝐵 ∈ V ) → ( 𝐾 𝐷 ( 𝐷 ‘ 𝐾 ) ∧ ( 𝐷 ‘ 𝐾 ) 𝐹 𝑁 ) ) |
21 |
7 20
|
mpdan |
⊢ ( 𝜑 → ( 𝐾 𝐷 ( 𝐷 ‘ 𝐾 ) ∧ ( 𝐷 ‘ 𝐾 ) 𝐹 𝑁 ) ) |
22 |
21
|
simprd |
⊢ ( 𝜑 → ( 𝐷 ‘ 𝐾 ) 𝐹 𝑁 ) |
23 |
1 4 22
|
ntrneinex |
⊢ ( 𝜑 → 𝑁 ∈ ( 𝒫 𝒫 𝐵 ↑m 𝐵 ) ) |