Step |
Hyp |
Ref |
Expression |
1 |
|
dfnbgr3.v |
⊢ 𝑉 = ( Vtx ‘ 𝐺 ) |
2 |
|
dfnbgr3.i |
⊢ 𝐼 = ( iEdg ‘ 𝐺 ) |
3 |
|
eqid |
⊢ ( Edg ‘ 𝐺 ) = ( Edg ‘ 𝐺 ) |
4 |
1 3
|
nbgrval |
⊢ ( 𝑁 ∈ 𝑉 → ( 𝐺 NeighbVtx 𝑁 ) = { 𝑛 ∈ ( 𝑉 ∖ { 𝑁 } ) ∣ ∃ 𝑒 ∈ ( Edg ‘ 𝐺 ) { 𝑁 , 𝑛 } ⊆ 𝑒 } ) |
5 |
4
|
adantr |
⊢ ( ( 𝑁 ∈ 𝑉 ∧ Fun 𝐼 ) → ( 𝐺 NeighbVtx 𝑁 ) = { 𝑛 ∈ ( 𝑉 ∖ { 𝑁 } ) ∣ ∃ 𝑒 ∈ ( Edg ‘ 𝐺 ) { 𝑁 , 𝑛 } ⊆ 𝑒 } ) |
6 |
|
edgval |
⊢ ( Edg ‘ 𝐺 ) = ran ( iEdg ‘ 𝐺 ) |
7 |
2
|
eqcomi |
⊢ ( iEdg ‘ 𝐺 ) = 𝐼 |
8 |
7
|
rneqi |
⊢ ran ( iEdg ‘ 𝐺 ) = ran 𝐼 |
9 |
6 8
|
eqtri |
⊢ ( Edg ‘ 𝐺 ) = ran 𝐼 |
10 |
9
|
rexeqi |
⊢ ( ∃ 𝑒 ∈ ( Edg ‘ 𝐺 ) { 𝑁 , 𝑛 } ⊆ 𝑒 ↔ ∃ 𝑒 ∈ ran 𝐼 { 𝑁 , 𝑛 } ⊆ 𝑒 ) |
11 |
|
funfn |
⊢ ( Fun 𝐼 ↔ 𝐼 Fn dom 𝐼 ) |
12 |
11
|
biimpi |
⊢ ( Fun 𝐼 → 𝐼 Fn dom 𝐼 ) |
13 |
12
|
adantl |
⊢ ( ( 𝑁 ∈ 𝑉 ∧ Fun 𝐼 ) → 𝐼 Fn dom 𝐼 ) |
14 |
|
sseq2 |
⊢ ( 𝑒 = ( 𝐼 ‘ 𝑖 ) → ( { 𝑁 , 𝑛 } ⊆ 𝑒 ↔ { 𝑁 , 𝑛 } ⊆ ( 𝐼 ‘ 𝑖 ) ) ) |
15 |
14
|
rexrn |
⊢ ( 𝐼 Fn dom 𝐼 → ( ∃ 𝑒 ∈ ran 𝐼 { 𝑁 , 𝑛 } ⊆ 𝑒 ↔ ∃ 𝑖 ∈ dom 𝐼 { 𝑁 , 𝑛 } ⊆ ( 𝐼 ‘ 𝑖 ) ) ) |
16 |
13 15
|
syl |
⊢ ( ( 𝑁 ∈ 𝑉 ∧ Fun 𝐼 ) → ( ∃ 𝑒 ∈ ran 𝐼 { 𝑁 , 𝑛 } ⊆ 𝑒 ↔ ∃ 𝑖 ∈ dom 𝐼 { 𝑁 , 𝑛 } ⊆ ( 𝐼 ‘ 𝑖 ) ) ) |
17 |
10 16
|
syl5bb |
⊢ ( ( 𝑁 ∈ 𝑉 ∧ Fun 𝐼 ) → ( ∃ 𝑒 ∈ ( Edg ‘ 𝐺 ) { 𝑁 , 𝑛 } ⊆ 𝑒 ↔ ∃ 𝑖 ∈ dom 𝐼 { 𝑁 , 𝑛 } ⊆ ( 𝐼 ‘ 𝑖 ) ) ) |
18 |
17
|
rabbidv |
⊢ ( ( 𝑁 ∈ 𝑉 ∧ Fun 𝐼 ) → { 𝑛 ∈ ( 𝑉 ∖ { 𝑁 } ) ∣ ∃ 𝑒 ∈ ( Edg ‘ 𝐺 ) { 𝑁 , 𝑛 } ⊆ 𝑒 } = { 𝑛 ∈ ( 𝑉 ∖ { 𝑁 } ) ∣ ∃ 𝑖 ∈ dom 𝐼 { 𝑁 , 𝑛 } ⊆ ( 𝐼 ‘ 𝑖 ) } ) |
19 |
5 18
|
eqtrd |
⊢ ( ( 𝑁 ∈ 𝑉 ∧ Fun 𝐼 ) → ( 𝐺 NeighbVtx 𝑁 ) = { 𝑛 ∈ ( 𝑉 ∖ { 𝑁 } ) ∣ ∃ 𝑖 ∈ dom 𝐼 { 𝑁 , 𝑛 } ⊆ ( 𝐼 ‘ 𝑖 ) } ) |