Step |
Hyp |
Ref |
Expression |
1 |
|
edgnbusgreu.e |
⊢ 𝐸 = ( Edg ‘ 𝐺 ) |
2 |
|
edgnbusgreu.n |
⊢ 𝑁 = ( 𝐺 NeighbVtx 𝑀 ) |
3 |
|
simpll |
⊢ ( ( ( 𝐺 ∈ USGraph ∧ 𝑀 ∈ 𝑉 ) ∧ ( 𝐶 ∈ 𝐸 ∧ 𝑀 ∈ 𝐶 ) ) → 𝐺 ∈ USGraph ) |
4 |
1
|
eleq2i |
⊢ ( 𝐶 ∈ 𝐸 ↔ 𝐶 ∈ ( Edg ‘ 𝐺 ) ) |
5 |
4
|
biimpi |
⊢ ( 𝐶 ∈ 𝐸 → 𝐶 ∈ ( Edg ‘ 𝐺 ) ) |
6 |
5
|
ad2antrl |
⊢ ( ( ( 𝐺 ∈ USGraph ∧ 𝑀 ∈ 𝑉 ) ∧ ( 𝐶 ∈ 𝐸 ∧ 𝑀 ∈ 𝐶 ) ) → 𝐶 ∈ ( Edg ‘ 𝐺 ) ) |
7 |
|
simprr |
⊢ ( ( ( 𝐺 ∈ USGraph ∧ 𝑀 ∈ 𝑉 ) ∧ ( 𝐶 ∈ 𝐸 ∧ 𝑀 ∈ 𝐶 ) ) → 𝑀 ∈ 𝐶 ) |
8 |
|
usgredg2vtxeu |
⊢ ( ( 𝐺 ∈ USGraph ∧ 𝐶 ∈ ( Edg ‘ 𝐺 ) ∧ 𝑀 ∈ 𝐶 ) → ∃! 𝑛 ∈ ( Vtx ‘ 𝐺 ) 𝐶 = { 𝑀 , 𝑛 } ) |
9 |
3 6 7 8
|
syl3anc |
⊢ ( ( ( 𝐺 ∈ USGraph ∧ 𝑀 ∈ 𝑉 ) ∧ ( 𝐶 ∈ 𝐸 ∧ 𝑀 ∈ 𝐶 ) ) → ∃! 𝑛 ∈ ( Vtx ‘ 𝐺 ) 𝐶 = { 𝑀 , 𝑛 } ) |
10 |
|
df-reu |
⊢ ( ∃! 𝑛 ∈ ( Vtx ‘ 𝐺 ) 𝐶 = { 𝑀 , 𝑛 } ↔ ∃! 𝑛 ( 𝑛 ∈ ( Vtx ‘ 𝐺 ) ∧ 𝐶 = { 𝑀 , 𝑛 } ) ) |
11 |
|
prcom |
⊢ { 𝑀 , 𝑛 } = { 𝑛 , 𝑀 } |
12 |
11
|
eqeq2i |
⊢ ( 𝐶 = { 𝑀 , 𝑛 } ↔ 𝐶 = { 𝑛 , 𝑀 } ) |
13 |
12
|
biimpi |
⊢ ( 𝐶 = { 𝑀 , 𝑛 } → 𝐶 = { 𝑛 , 𝑀 } ) |
14 |
13
|
eleq1d |
⊢ ( 𝐶 = { 𝑀 , 𝑛 } → ( 𝐶 ∈ 𝐸 ↔ { 𝑛 , 𝑀 } ∈ 𝐸 ) ) |
15 |
14
|
biimpcd |
⊢ ( 𝐶 ∈ 𝐸 → ( 𝐶 = { 𝑀 , 𝑛 } → { 𝑛 , 𝑀 } ∈ 𝐸 ) ) |
16 |
15
|
ad2antrl |
⊢ ( ( ( 𝐺 ∈ USGraph ∧ 𝑀 ∈ 𝑉 ) ∧ ( 𝐶 ∈ 𝐸 ∧ 𝑀 ∈ 𝐶 ) ) → ( 𝐶 = { 𝑀 , 𝑛 } → { 𝑛 , 𝑀 } ∈ 𝐸 ) ) |
17 |
16
|
adantld |
⊢ ( ( ( 𝐺 ∈ USGraph ∧ 𝑀 ∈ 𝑉 ) ∧ ( 𝐶 ∈ 𝐸 ∧ 𝑀 ∈ 𝐶 ) ) → ( ( 𝑛 ∈ ( Vtx ‘ 𝐺 ) ∧ 𝐶 = { 𝑀 , 𝑛 } ) → { 𝑛 , 𝑀 } ∈ 𝐸 ) ) |
18 |
17
|
imp |
⊢ ( ( ( ( 𝐺 ∈ USGraph ∧ 𝑀 ∈ 𝑉 ) ∧ ( 𝐶 ∈ 𝐸 ∧ 𝑀 ∈ 𝐶 ) ) ∧ ( 𝑛 ∈ ( Vtx ‘ 𝐺 ) ∧ 𝐶 = { 𝑀 , 𝑛 } ) ) → { 𝑛 , 𝑀 } ∈ 𝐸 ) |
19 |
|
simprr |
⊢ ( ( ( ( 𝐺 ∈ USGraph ∧ 𝑀 ∈ 𝑉 ) ∧ ( 𝐶 ∈ 𝐸 ∧ 𝑀 ∈ 𝐶 ) ) ∧ ( 𝑛 ∈ ( Vtx ‘ 𝐺 ) ∧ 𝐶 = { 𝑀 , 𝑛 } ) ) → 𝐶 = { 𝑀 , 𝑛 } ) |
20 |
18 19
|
jca |
⊢ ( ( ( ( 𝐺 ∈ USGraph ∧ 𝑀 ∈ 𝑉 ) ∧ ( 𝐶 ∈ 𝐸 ∧ 𝑀 ∈ 𝐶 ) ) ∧ ( 𝑛 ∈ ( Vtx ‘ 𝐺 ) ∧ 𝐶 = { 𝑀 , 𝑛 } ) ) → ( { 𝑛 , 𝑀 } ∈ 𝐸 ∧ 𝐶 = { 𝑀 , 𝑛 } ) ) |
21 |
|
simpl |
⊢ ( ( { 𝑛 , 𝑀 } ∈ 𝐸 ∧ 𝐶 = { 𝑀 , 𝑛 } ) → { 𝑛 , 𝑀 } ∈ 𝐸 ) |
22 |
|
eqid |
⊢ ( Vtx ‘ 𝐺 ) = ( Vtx ‘ 𝐺 ) |
23 |
1 22
|
usgrpredgv |
⊢ ( ( 𝐺 ∈ USGraph ∧ { 𝑛 , 𝑀 } ∈ 𝐸 ) → ( 𝑛 ∈ ( Vtx ‘ 𝐺 ) ∧ 𝑀 ∈ ( Vtx ‘ 𝐺 ) ) ) |
24 |
23
|
simpld |
⊢ ( ( 𝐺 ∈ USGraph ∧ { 𝑛 , 𝑀 } ∈ 𝐸 ) → 𝑛 ∈ ( Vtx ‘ 𝐺 ) ) |
25 |
3 21 24
|
syl2an |
⊢ ( ( ( ( 𝐺 ∈ USGraph ∧ 𝑀 ∈ 𝑉 ) ∧ ( 𝐶 ∈ 𝐸 ∧ 𝑀 ∈ 𝐶 ) ) ∧ ( { 𝑛 , 𝑀 } ∈ 𝐸 ∧ 𝐶 = { 𝑀 , 𝑛 } ) ) → 𝑛 ∈ ( Vtx ‘ 𝐺 ) ) |
26 |
|
simprr |
⊢ ( ( ( ( 𝐺 ∈ USGraph ∧ 𝑀 ∈ 𝑉 ) ∧ ( 𝐶 ∈ 𝐸 ∧ 𝑀 ∈ 𝐶 ) ) ∧ ( { 𝑛 , 𝑀 } ∈ 𝐸 ∧ 𝐶 = { 𝑀 , 𝑛 } ) ) → 𝐶 = { 𝑀 , 𝑛 } ) |
27 |
25 26
|
jca |
⊢ ( ( ( ( 𝐺 ∈ USGraph ∧ 𝑀 ∈ 𝑉 ) ∧ ( 𝐶 ∈ 𝐸 ∧ 𝑀 ∈ 𝐶 ) ) ∧ ( { 𝑛 , 𝑀 } ∈ 𝐸 ∧ 𝐶 = { 𝑀 , 𝑛 } ) ) → ( 𝑛 ∈ ( Vtx ‘ 𝐺 ) ∧ 𝐶 = { 𝑀 , 𝑛 } ) ) |
28 |
20 27
|
impbida |
⊢ ( ( ( 𝐺 ∈ USGraph ∧ 𝑀 ∈ 𝑉 ) ∧ ( 𝐶 ∈ 𝐸 ∧ 𝑀 ∈ 𝐶 ) ) → ( ( 𝑛 ∈ ( Vtx ‘ 𝐺 ) ∧ 𝐶 = { 𝑀 , 𝑛 } ) ↔ ( { 𝑛 , 𝑀 } ∈ 𝐸 ∧ 𝐶 = { 𝑀 , 𝑛 } ) ) ) |
29 |
28
|
eubidv |
⊢ ( ( ( 𝐺 ∈ USGraph ∧ 𝑀 ∈ 𝑉 ) ∧ ( 𝐶 ∈ 𝐸 ∧ 𝑀 ∈ 𝐶 ) ) → ( ∃! 𝑛 ( 𝑛 ∈ ( Vtx ‘ 𝐺 ) ∧ 𝐶 = { 𝑀 , 𝑛 } ) ↔ ∃! 𝑛 ( { 𝑛 , 𝑀 } ∈ 𝐸 ∧ 𝐶 = { 𝑀 , 𝑛 } ) ) ) |
30 |
29
|
biimpd |
⊢ ( ( ( 𝐺 ∈ USGraph ∧ 𝑀 ∈ 𝑉 ) ∧ ( 𝐶 ∈ 𝐸 ∧ 𝑀 ∈ 𝐶 ) ) → ( ∃! 𝑛 ( 𝑛 ∈ ( Vtx ‘ 𝐺 ) ∧ 𝐶 = { 𝑀 , 𝑛 } ) → ∃! 𝑛 ( { 𝑛 , 𝑀 } ∈ 𝐸 ∧ 𝐶 = { 𝑀 , 𝑛 } ) ) ) |
31 |
10 30
|
syl5bi |
⊢ ( ( ( 𝐺 ∈ USGraph ∧ 𝑀 ∈ 𝑉 ) ∧ ( 𝐶 ∈ 𝐸 ∧ 𝑀 ∈ 𝐶 ) ) → ( ∃! 𝑛 ∈ ( Vtx ‘ 𝐺 ) 𝐶 = { 𝑀 , 𝑛 } → ∃! 𝑛 ( { 𝑛 , 𝑀 } ∈ 𝐸 ∧ 𝐶 = { 𝑀 , 𝑛 } ) ) ) |
32 |
9 31
|
mpd |
⊢ ( ( ( 𝐺 ∈ USGraph ∧ 𝑀 ∈ 𝑉 ) ∧ ( 𝐶 ∈ 𝐸 ∧ 𝑀 ∈ 𝐶 ) ) → ∃! 𝑛 ( { 𝑛 , 𝑀 } ∈ 𝐸 ∧ 𝐶 = { 𝑀 , 𝑛 } ) ) |
33 |
2
|
eleq2i |
⊢ ( 𝑛 ∈ 𝑁 ↔ 𝑛 ∈ ( 𝐺 NeighbVtx 𝑀 ) ) |
34 |
1
|
nbusgreledg |
⊢ ( 𝐺 ∈ USGraph → ( 𝑛 ∈ ( 𝐺 NeighbVtx 𝑀 ) ↔ { 𝑛 , 𝑀 } ∈ 𝐸 ) ) |
35 |
33 34
|
syl5bb |
⊢ ( 𝐺 ∈ USGraph → ( 𝑛 ∈ 𝑁 ↔ { 𝑛 , 𝑀 } ∈ 𝐸 ) ) |
36 |
35
|
anbi1d |
⊢ ( 𝐺 ∈ USGraph → ( ( 𝑛 ∈ 𝑁 ∧ 𝐶 = { 𝑀 , 𝑛 } ) ↔ ( { 𝑛 , 𝑀 } ∈ 𝐸 ∧ 𝐶 = { 𝑀 , 𝑛 } ) ) ) |
37 |
36
|
ad2antrr |
⊢ ( ( ( 𝐺 ∈ USGraph ∧ 𝑀 ∈ 𝑉 ) ∧ ( 𝐶 ∈ 𝐸 ∧ 𝑀 ∈ 𝐶 ) ) → ( ( 𝑛 ∈ 𝑁 ∧ 𝐶 = { 𝑀 , 𝑛 } ) ↔ ( { 𝑛 , 𝑀 } ∈ 𝐸 ∧ 𝐶 = { 𝑀 , 𝑛 } ) ) ) |
38 |
37
|
eubidv |
⊢ ( ( ( 𝐺 ∈ USGraph ∧ 𝑀 ∈ 𝑉 ) ∧ ( 𝐶 ∈ 𝐸 ∧ 𝑀 ∈ 𝐶 ) ) → ( ∃! 𝑛 ( 𝑛 ∈ 𝑁 ∧ 𝐶 = { 𝑀 , 𝑛 } ) ↔ ∃! 𝑛 ( { 𝑛 , 𝑀 } ∈ 𝐸 ∧ 𝐶 = { 𝑀 , 𝑛 } ) ) ) |
39 |
32 38
|
mpbird |
⊢ ( ( ( 𝐺 ∈ USGraph ∧ 𝑀 ∈ 𝑉 ) ∧ ( 𝐶 ∈ 𝐸 ∧ 𝑀 ∈ 𝐶 ) ) → ∃! 𝑛 ( 𝑛 ∈ 𝑁 ∧ 𝐶 = { 𝑀 , 𝑛 } ) ) |
40 |
|
df-reu |
⊢ ( ∃! 𝑛 ∈ 𝑁 𝐶 = { 𝑀 , 𝑛 } ↔ ∃! 𝑛 ( 𝑛 ∈ 𝑁 ∧ 𝐶 = { 𝑀 , 𝑛 } ) ) |
41 |
39 40
|
sylibr |
⊢ ( ( ( 𝐺 ∈ USGraph ∧ 𝑀 ∈ 𝑉 ) ∧ ( 𝐶 ∈ 𝐸 ∧ 𝑀 ∈ 𝐶 ) ) → ∃! 𝑛 ∈ 𝑁 𝐶 = { 𝑀 , 𝑛 } ) |