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 |
|
f1ofn |
⊢ ( 𝐹 : ( 𝒫 𝐵 ↑m 𝒫 𝐵 ) –1-1-onto→ ( 𝒫 𝒫 𝐵 ↑m 𝐵 ) → 𝐹 Fn ( 𝒫 𝐵 ↑m 𝒫 𝐵 ) ) |
6 |
4 5
|
syl |
⊢ ( 𝜑 → 𝐹 Fn ( 𝒫 𝐵 ↑m 𝒫 𝐵 ) ) |
7 |
1 2 3
|
ntrneiiex |
⊢ ( 𝜑 → 𝐼 ∈ ( 𝒫 𝐵 ↑m 𝒫 𝐵 ) ) |
8 |
6 7
|
jca |
⊢ ( 𝜑 → ( 𝐹 Fn ( 𝒫 𝐵 ↑m 𝒫 𝐵 ) ∧ 𝐼 ∈ ( 𝒫 𝐵 ↑m 𝒫 𝐵 ) ) ) |
9 |
|
fnfun |
⊢ ( 𝐹 Fn ( 𝒫 𝐵 ↑m 𝒫 𝐵 ) → Fun 𝐹 ) |
10 |
9
|
adantr |
⊢ ( ( 𝐹 Fn ( 𝒫 𝐵 ↑m 𝒫 𝐵 ) ∧ 𝐼 ∈ ( 𝒫 𝐵 ↑m 𝒫 𝐵 ) ) → Fun 𝐹 ) |
11 |
|
fndm |
⊢ ( 𝐹 Fn ( 𝒫 𝐵 ↑m 𝒫 𝐵 ) → dom 𝐹 = ( 𝒫 𝐵 ↑m 𝒫 𝐵 ) ) |
12 |
11
|
eleq2d |
⊢ ( 𝐹 Fn ( 𝒫 𝐵 ↑m 𝒫 𝐵 ) → ( 𝐼 ∈ dom 𝐹 ↔ 𝐼 ∈ ( 𝒫 𝐵 ↑m 𝒫 𝐵 ) ) ) |
13 |
12
|
biimpar |
⊢ ( ( 𝐹 Fn ( 𝒫 𝐵 ↑m 𝒫 𝐵 ) ∧ 𝐼 ∈ ( 𝒫 𝐵 ↑m 𝒫 𝐵 ) ) → 𝐼 ∈ dom 𝐹 ) |
14 |
10 13
|
jca |
⊢ ( ( 𝐹 Fn ( 𝒫 𝐵 ↑m 𝒫 𝐵 ) ∧ 𝐼 ∈ ( 𝒫 𝐵 ↑m 𝒫 𝐵 ) ) → ( Fun 𝐹 ∧ 𝐼 ∈ dom 𝐹 ) ) |
15 |
|
funbrfvb |
⊢ ( ( Fun 𝐹 ∧ 𝐼 ∈ dom 𝐹 ) → ( ( 𝐹 ‘ 𝐼 ) = 𝑁 ↔ 𝐼 𝐹 𝑁 ) ) |
16 |
8 14 15
|
3syl |
⊢ ( 𝜑 → ( ( 𝐹 ‘ 𝐼 ) = 𝑁 ↔ 𝐼 𝐹 𝑁 ) ) |
17 |
3 16
|
mpbird |
⊢ ( 𝜑 → ( 𝐹 ‘ 𝐼 ) = 𝑁 ) |