Step |
Hyp |
Ref |
Expression |
1 |
|
neicvg.o |
⊢ 𝑂 = ( 𝑖 ∈ V , 𝑗 ∈ V ↦ ( 𝑘 ∈ ( 𝒫 𝑗 ↑m 𝑖 ) ↦ ( 𝑙 ∈ 𝑗 ↦ { 𝑚 ∈ 𝑖 ∣ 𝑙 ∈ ( 𝑘 ‘ 𝑚 ) } ) ) ) |
2 |
|
neicvg.p |
⊢ 𝑃 = ( 𝑛 ∈ V ↦ ( 𝑝 ∈ ( 𝒫 𝑛 ↑m 𝒫 𝑛 ) ↦ ( 𝑜 ∈ 𝒫 𝑛 ↦ ( 𝑛 ∖ ( 𝑝 ‘ ( 𝑛 ∖ 𝑜 ) ) ) ) ) ) |
3 |
|
neicvg.d |
⊢ 𝐷 = ( 𝑃 ‘ 𝐵 ) |
4 |
|
neicvg.f |
⊢ 𝐹 = ( 𝒫 𝐵 𝑂 𝐵 ) |
5 |
|
neicvg.g |
⊢ 𝐺 = ( 𝐵 𝑂 𝒫 𝐵 ) |
6 |
|
neicvg.h |
⊢ 𝐻 = ( 𝐹 ∘ ( 𝐷 ∘ 𝐺 ) ) |
7 |
|
neicvg.r |
⊢ ( 𝜑 → 𝑁 𝐻 𝑀 ) |
8 |
3 6 7
|
neicvgbex |
⊢ ( 𝜑 → 𝐵 ∈ V ) |
9 |
8
|
pwexd |
⊢ ( 𝜑 → 𝒫 𝐵 ∈ V ) |
10 |
1 9 8 4
|
fsovf1od |
⊢ ( 𝜑 → 𝐹 : ( 𝒫 𝐵 ↑m 𝒫 𝐵 ) –1-1-onto→ ( 𝒫 𝒫 𝐵 ↑m 𝐵 ) ) |
11 |
2 3 8
|
dssmapf1od |
⊢ ( 𝜑 → 𝐷 : ( 𝒫 𝐵 ↑m 𝒫 𝐵 ) –1-1-onto→ ( 𝒫 𝐵 ↑m 𝒫 𝐵 ) ) |
12 |
1 8 9 5
|
fsovf1od |
⊢ ( 𝜑 → 𝐺 : ( 𝒫 𝒫 𝐵 ↑m 𝐵 ) –1-1-onto→ ( 𝒫 𝐵 ↑m 𝒫 𝐵 ) ) |
13 |
|
f1oco |
⊢ ( ( 𝐷 : ( 𝒫 𝐵 ↑m 𝒫 𝐵 ) –1-1-onto→ ( 𝒫 𝐵 ↑m 𝒫 𝐵 ) ∧ 𝐺 : ( 𝒫 𝒫 𝐵 ↑m 𝐵 ) –1-1-onto→ ( 𝒫 𝐵 ↑m 𝒫 𝐵 ) ) → ( 𝐷 ∘ 𝐺 ) : ( 𝒫 𝒫 𝐵 ↑m 𝐵 ) –1-1-onto→ ( 𝒫 𝐵 ↑m 𝒫 𝐵 ) ) |
14 |
11 12 13
|
syl2anc |
⊢ ( 𝜑 → ( 𝐷 ∘ 𝐺 ) : ( 𝒫 𝒫 𝐵 ↑m 𝐵 ) –1-1-onto→ ( 𝒫 𝐵 ↑m 𝒫 𝐵 ) ) |
15 |
|
f1oco |
⊢ ( ( 𝐹 : ( 𝒫 𝐵 ↑m 𝒫 𝐵 ) –1-1-onto→ ( 𝒫 𝒫 𝐵 ↑m 𝐵 ) ∧ ( 𝐷 ∘ 𝐺 ) : ( 𝒫 𝒫 𝐵 ↑m 𝐵 ) –1-1-onto→ ( 𝒫 𝐵 ↑m 𝒫 𝐵 ) ) → ( 𝐹 ∘ ( 𝐷 ∘ 𝐺 ) ) : ( 𝒫 𝒫 𝐵 ↑m 𝐵 ) –1-1-onto→ ( 𝒫 𝒫 𝐵 ↑m 𝐵 ) ) |
16 |
10 14 15
|
syl2anc |
⊢ ( 𝜑 → ( 𝐹 ∘ ( 𝐷 ∘ 𝐺 ) ) : ( 𝒫 𝒫 𝐵 ↑m 𝐵 ) –1-1-onto→ ( 𝒫 𝒫 𝐵 ↑m 𝐵 ) ) |
17 |
|
f1oeq1 |
⊢ ( 𝐻 = ( 𝐹 ∘ ( 𝐷 ∘ 𝐺 ) ) → ( 𝐻 : ( 𝒫 𝒫 𝐵 ↑m 𝐵 ) –1-1-onto→ ( 𝒫 𝒫 𝐵 ↑m 𝐵 ) ↔ ( 𝐹 ∘ ( 𝐷 ∘ 𝐺 ) ) : ( 𝒫 𝒫 𝐵 ↑m 𝐵 ) –1-1-onto→ ( 𝒫 𝒫 𝐵 ↑m 𝐵 ) ) ) |
18 |
6 17
|
ax-mp |
⊢ ( 𝐻 : ( 𝒫 𝒫 𝐵 ↑m 𝐵 ) –1-1-onto→ ( 𝒫 𝒫 𝐵 ↑m 𝐵 ) ↔ ( 𝐹 ∘ ( 𝐷 ∘ 𝐺 ) ) : ( 𝒫 𝒫 𝐵 ↑m 𝐵 ) –1-1-onto→ ( 𝒫 𝒫 𝐵 ↑m 𝐵 ) ) |
19 |
16 18
|
sylibr |
⊢ ( 𝜑 → 𝐻 : ( 𝒫 𝒫 𝐵 ↑m 𝐵 ) –1-1-onto→ ( 𝒫 𝒫 𝐵 ↑m 𝐵 ) ) |