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 |
|
ntrnei.x |
⊢ ( 𝜑 → 𝑋 ∈ 𝐵 ) |
5 |
3
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑠 ∈ 𝒫 𝐵 ) → 𝐼 𝐹 𝑁 ) |
6 |
4
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑠 ∈ 𝒫 𝐵 ) → 𝑋 ∈ 𝐵 ) |
7 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝑠 ∈ 𝒫 𝐵 ) → 𝑠 ∈ 𝒫 𝐵 ) |
8 |
1 2 5 6 7
|
ntrneiel |
⊢ ( ( 𝜑 ∧ 𝑠 ∈ 𝒫 𝐵 ) → ( 𝑋 ∈ ( 𝐼 ‘ 𝑠 ) ↔ 𝑠 ∈ ( 𝑁 ‘ 𝑋 ) ) ) |
9 |
8
|
rexbidva |
⊢ ( 𝜑 → ( ∃ 𝑠 ∈ 𝒫 𝐵 𝑋 ∈ ( 𝐼 ‘ 𝑠 ) ↔ ∃ 𝑠 ∈ 𝒫 𝐵 𝑠 ∈ ( 𝑁 ‘ 𝑋 ) ) ) |
10 |
1 2 3
|
ntrneinex |
⊢ ( 𝜑 → 𝑁 ∈ ( 𝒫 𝒫 𝐵 ↑m 𝐵 ) ) |
11 |
|
elmapi |
⊢ ( 𝑁 ∈ ( 𝒫 𝒫 𝐵 ↑m 𝐵 ) → 𝑁 : 𝐵 ⟶ 𝒫 𝒫 𝐵 ) |
12 |
10 11
|
syl |
⊢ ( 𝜑 → 𝑁 : 𝐵 ⟶ 𝒫 𝒫 𝐵 ) |
13 |
12 4
|
ffvelrnd |
⊢ ( 𝜑 → ( 𝑁 ‘ 𝑋 ) ∈ 𝒫 𝒫 𝐵 ) |
14 |
13
|
elpwid |
⊢ ( 𝜑 → ( 𝑁 ‘ 𝑋 ) ⊆ 𝒫 𝐵 ) |
15 |
14
|
sseld |
⊢ ( 𝜑 → ( 𝑠 ∈ ( 𝑁 ‘ 𝑋 ) → 𝑠 ∈ 𝒫 𝐵 ) ) |
16 |
15
|
pm4.71rd |
⊢ ( 𝜑 → ( 𝑠 ∈ ( 𝑁 ‘ 𝑋 ) ↔ ( 𝑠 ∈ 𝒫 𝐵 ∧ 𝑠 ∈ ( 𝑁 ‘ 𝑋 ) ) ) ) |
17 |
16
|
exbidv |
⊢ ( 𝜑 → ( ∃ 𝑠 𝑠 ∈ ( 𝑁 ‘ 𝑋 ) ↔ ∃ 𝑠 ( 𝑠 ∈ 𝒫 𝐵 ∧ 𝑠 ∈ ( 𝑁 ‘ 𝑋 ) ) ) ) |
18 |
17
|
bicomd |
⊢ ( 𝜑 → ( ∃ 𝑠 ( 𝑠 ∈ 𝒫 𝐵 ∧ 𝑠 ∈ ( 𝑁 ‘ 𝑋 ) ) ↔ ∃ 𝑠 𝑠 ∈ ( 𝑁 ‘ 𝑋 ) ) ) |
19 |
|
df-rex |
⊢ ( ∃ 𝑠 ∈ 𝒫 𝐵 𝑠 ∈ ( 𝑁 ‘ 𝑋 ) ↔ ∃ 𝑠 ( 𝑠 ∈ 𝒫 𝐵 ∧ 𝑠 ∈ ( 𝑁 ‘ 𝑋 ) ) ) |
20 |
|
n0 |
⊢ ( ( 𝑁 ‘ 𝑋 ) ≠ ∅ ↔ ∃ 𝑠 𝑠 ∈ ( 𝑁 ‘ 𝑋 ) ) |
21 |
18 19 20
|
3bitr4g |
⊢ ( 𝜑 → ( ∃ 𝑠 ∈ 𝒫 𝐵 𝑠 ∈ ( 𝑁 ‘ 𝑋 ) ↔ ( 𝑁 ‘ 𝑋 ) ≠ ∅ ) ) |
22 |
9 21
|
bitrd |
⊢ ( 𝜑 → ( ∃ 𝑠 ∈ 𝒫 𝐵 𝑋 ∈ ( 𝐼 ‘ 𝑠 ) ↔ ( 𝑁 ‘ 𝑋 ) ≠ ∅ ) ) |