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 |
6
|
cnveqi |
⊢ ◡ 𝐻 = ◡ ( 𝐹 ∘ ( 𝐷 ∘ 𝐺 ) ) |
9 |
|
cnvco |
⊢ ◡ ( 𝐹 ∘ ( 𝐷 ∘ 𝐺 ) ) = ( ◡ ( 𝐷 ∘ 𝐺 ) ∘ ◡ 𝐹 ) |
10 |
|
cnvco |
⊢ ◡ ( 𝐷 ∘ 𝐺 ) = ( ◡ 𝐺 ∘ ◡ 𝐷 ) |
11 |
10
|
coeq1i |
⊢ ( ◡ ( 𝐷 ∘ 𝐺 ) ∘ ◡ 𝐹 ) = ( ( ◡ 𝐺 ∘ ◡ 𝐷 ) ∘ ◡ 𝐹 ) |
12 |
8 9 11
|
3eqtri |
⊢ ◡ 𝐻 = ( ( ◡ 𝐺 ∘ ◡ 𝐷 ) ∘ ◡ 𝐹 ) |
13 |
3 6 7
|
neicvgbex |
⊢ ( 𝜑 → 𝐵 ∈ V ) |
14 |
13
|
pwexd |
⊢ ( 𝜑 → 𝒫 𝐵 ∈ V ) |
15 |
1 13 14 5 4
|
fsovcnvd |
⊢ ( 𝜑 → ◡ 𝐺 = 𝐹 ) |
16 |
2 3 13
|
dssmapnvod |
⊢ ( 𝜑 → ◡ 𝐷 = 𝐷 ) |
17 |
15 16
|
coeq12d |
⊢ ( 𝜑 → ( ◡ 𝐺 ∘ ◡ 𝐷 ) = ( 𝐹 ∘ 𝐷 ) ) |
18 |
1 14 13 4 5
|
fsovcnvd |
⊢ ( 𝜑 → ◡ 𝐹 = 𝐺 ) |
19 |
17 18
|
coeq12d |
⊢ ( 𝜑 → ( ( ◡ 𝐺 ∘ ◡ 𝐷 ) ∘ ◡ 𝐹 ) = ( ( 𝐹 ∘ 𝐷 ) ∘ 𝐺 ) ) |
20 |
12 19
|
syl5eq |
⊢ ( 𝜑 → ◡ 𝐻 = ( ( 𝐹 ∘ 𝐷 ) ∘ 𝐺 ) ) |
21 |
|
coass |
⊢ ( ( 𝐹 ∘ 𝐷 ) ∘ 𝐺 ) = ( 𝐹 ∘ ( 𝐷 ∘ 𝐺 ) ) |
22 |
21 6
|
eqtr4i |
⊢ ( ( 𝐹 ∘ 𝐷 ) ∘ 𝐺 ) = 𝐻 |
23 |
20 22
|
eqtrdi |
⊢ ( 𝜑 → ◡ 𝐻 = 𝐻 ) |