| Step |
Hyp |
Ref |
Expression |
| 1 |
|
eqid |
⊢ ( Vtx ‘ 𝐺 ) = ( Vtx ‘ 𝐺 ) |
| 2 |
|
eqid |
⊢ ( Edg ‘ 𝐺 ) = ( Edg ‘ 𝐺 ) |
| 3 |
1 2
|
iscplgredg |
⊢ ( 𝐺 ∈ ComplGraph → ( 𝐺 ∈ ComplGraph ↔ ∀ 𝑣 ∈ ( Vtx ‘ 𝐺 ) ∀ 𝑛 ∈ ( ( Vtx ‘ 𝐺 ) ∖ { 𝑣 } ) ∃ 𝑒 ∈ ( Edg ‘ 𝐺 ) { 𝑣 , 𝑛 } ⊆ 𝑒 ) ) |
| 4 |
|
edgval |
⊢ ( Edg ‘ 𝐺 ) = ran ( iEdg ‘ 𝐺 ) |
| 5 |
4
|
a1i |
⊢ ( 𝐺 ∈ ComplGraph → ( Edg ‘ 𝐺 ) = ran ( iEdg ‘ 𝐺 ) ) |
| 6 |
|
simpl |
⊢ ( ( ( Vtx ‘ 𝑔 ) = ( Vtx ‘ 𝐺 ) ∧ ( iEdg ‘ 𝑔 ) = ( iEdg ‘ 𝐺 ) ) → ( Vtx ‘ 𝑔 ) = ( Vtx ‘ 𝐺 ) ) |
| 7 |
6
|
adantl |
⊢ ( ( ( Edg ‘ 𝐺 ) = ran ( iEdg ‘ 𝐺 ) ∧ ( ( Vtx ‘ 𝑔 ) = ( Vtx ‘ 𝐺 ) ∧ ( iEdg ‘ 𝑔 ) = ( iEdg ‘ 𝐺 ) ) ) → ( Vtx ‘ 𝑔 ) = ( Vtx ‘ 𝐺 ) ) |
| 8 |
6
|
difeq1d |
⊢ ( ( ( Vtx ‘ 𝑔 ) = ( Vtx ‘ 𝐺 ) ∧ ( iEdg ‘ 𝑔 ) = ( iEdg ‘ 𝐺 ) ) → ( ( Vtx ‘ 𝑔 ) ∖ { 𝑣 } ) = ( ( Vtx ‘ 𝐺 ) ∖ { 𝑣 } ) ) |
| 9 |
8
|
adantl |
⊢ ( ( ( Edg ‘ 𝐺 ) = ran ( iEdg ‘ 𝐺 ) ∧ ( ( Vtx ‘ 𝑔 ) = ( Vtx ‘ 𝐺 ) ∧ ( iEdg ‘ 𝑔 ) = ( iEdg ‘ 𝐺 ) ) ) → ( ( Vtx ‘ 𝑔 ) ∖ { 𝑣 } ) = ( ( Vtx ‘ 𝐺 ) ∖ { 𝑣 } ) ) |
| 10 |
|
edgval |
⊢ ( Edg ‘ 𝑔 ) = ran ( iEdg ‘ 𝑔 ) |
| 11 |
|
simpr |
⊢ ( ( ( Vtx ‘ 𝑔 ) = ( Vtx ‘ 𝐺 ) ∧ ( iEdg ‘ 𝑔 ) = ( iEdg ‘ 𝐺 ) ) → ( iEdg ‘ 𝑔 ) = ( iEdg ‘ 𝐺 ) ) |
| 12 |
11
|
rneqd |
⊢ ( ( ( Vtx ‘ 𝑔 ) = ( Vtx ‘ 𝐺 ) ∧ ( iEdg ‘ 𝑔 ) = ( iEdg ‘ 𝐺 ) ) → ran ( iEdg ‘ 𝑔 ) = ran ( iEdg ‘ 𝐺 ) ) |
| 13 |
10 12
|
syl5eq |
⊢ ( ( ( Vtx ‘ 𝑔 ) = ( Vtx ‘ 𝐺 ) ∧ ( iEdg ‘ 𝑔 ) = ( iEdg ‘ 𝐺 ) ) → ( Edg ‘ 𝑔 ) = ran ( iEdg ‘ 𝐺 ) ) |
| 14 |
13
|
adantl |
⊢ ( ( ( Edg ‘ 𝐺 ) = ran ( iEdg ‘ 𝐺 ) ∧ ( ( Vtx ‘ 𝑔 ) = ( Vtx ‘ 𝐺 ) ∧ ( iEdg ‘ 𝑔 ) = ( iEdg ‘ 𝐺 ) ) ) → ( Edg ‘ 𝑔 ) = ran ( iEdg ‘ 𝐺 ) ) |
| 15 |
|
simpl |
⊢ ( ( ( Edg ‘ 𝐺 ) = ran ( iEdg ‘ 𝐺 ) ∧ ( ( Vtx ‘ 𝑔 ) = ( Vtx ‘ 𝐺 ) ∧ ( iEdg ‘ 𝑔 ) = ( iEdg ‘ 𝐺 ) ) ) → ( Edg ‘ 𝐺 ) = ran ( iEdg ‘ 𝐺 ) ) |
| 16 |
14 15
|
eqtr4d |
⊢ ( ( ( Edg ‘ 𝐺 ) = ran ( iEdg ‘ 𝐺 ) ∧ ( ( Vtx ‘ 𝑔 ) = ( Vtx ‘ 𝐺 ) ∧ ( iEdg ‘ 𝑔 ) = ( iEdg ‘ 𝐺 ) ) ) → ( Edg ‘ 𝑔 ) = ( Edg ‘ 𝐺 ) ) |
| 17 |
16
|
rexeqdv |
⊢ ( ( ( Edg ‘ 𝐺 ) = ran ( iEdg ‘ 𝐺 ) ∧ ( ( Vtx ‘ 𝑔 ) = ( Vtx ‘ 𝐺 ) ∧ ( iEdg ‘ 𝑔 ) = ( iEdg ‘ 𝐺 ) ) ) → ( ∃ 𝑒 ∈ ( Edg ‘ 𝑔 ) { 𝑣 , 𝑛 } ⊆ 𝑒 ↔ ∃ 𝑒 ∈ ( Edg ‘ 𝐺 ) { 𝑣 , 𝑛 } ⊆ 𝑒 ) ) |
| 18 |
9 17
|
raleqbidv |
⊢ ( ( ( Edg ‘ 𝐺 ) = ran ( iEdg ‘ 𝐺 ) ∧ ( ( Vtx ‘ 𝑔 ) = ( Vtx ‘ 𝐺 ) ∧ ( iEdg ‘ 𝑔 ) = ( iEdg ‘ 𝐺 ) ) ) → ( ∀ 𝑛 ∈ ( ( Vtx ‘ 𝑔 ) ∖ { 𝑣 } ) ∃ 𝑒 ∈ ( Edg ‘ 𝑔 ) { 𝑣 , 𝑛 } ⊆ 𝑒 ↔ ∀ 𝑛 ∈ ( ( Vtx ‘ 𝐺 ) ∖ { 𝑣 } ) ∃ 𝑒 ∈ ( Edg ‘ 𝐺 ) { 𝑣 , 𝑛 } ⊆ 𝑒 ) ) |
| 19 |
7 18
|
raleqbidv |
⊢ ( ( ( Edg ‘ 𝐺 ) = ran ( iEdg ‘ 𝐺 ) ∧ ( ( Vtx ‘ 𝑔 ) = ( Vtx ‘ 𝐺 ) ∧ ( iEdg ‘ 𝑔 ) = ( iEdg ‘ 𝐺 ) ) ) → ( ∀ 𝑣 ∈ ( Vtx ‘ 𝑔 ) ∀ 𝑛 ∈ ( ( Vtx ‘ 𝑔 ) ∖ { 𝑣 } ) ∃ 𝑒 ∈ ( Edg ‘ 𝑔 ) { 𝑣 , 𝑛 } ⊆ 𝑒 ↔ ∀ 𝑣 ∈ ( Vtx ‘ 𝐺 ) ∀ 𝑛 ∈ ( ( Vtx ‘ 𝐺 ) ∖ { 𝑣 } ) ∃ 𝑒 ∈ ( Edg ‘ 𝐺 ) { 𝑣 , 𝑛 } ⊆ 𝑒 ) ) |
| 20 |
19
|
biimpar |
⊢ ( ( ( ( Edg ‘ 𝐺 ) = ran ( iEdg ‘ 𝐺 ) ∧ ( ( Vtx ‘ 𝑔 ) = ( Vtx ‘ 𝐺 ) ∧ ( iEdg ‘ 𝑔 ) = ( iEdg ‘ 𝐺 ) ) ) ∧ ∀ 𝑣 ∈ ( Vtx ‘ 𝐺 ) ∀ 𝑛 ∈ ( ( Vtx ‘ 𝐺 ) ∖ { 𝑣 } ) ∃ 𝑒 ∈ ( Edg ‘ 𝐺 ) { 𝑣 , 𝑛 } ⊆ 𝑒 ) → ∀ 𝑣 ∈ ( Vtx ‘ 𝑔 ) ∀ 𝑛 ∈ ( ( Vtx ‘ 𝑔 ) ∖ { 𝑣 } ) ∃ 𝑒 ∈ ( Edg ‘ 𝑔 ) { 𝑣 , 𝑛 } ⊆ 𝑒 ) |
| 21 |
|
eqid |
⊢ ( Vtx ‘ 𝑔 ) = ( Vtx ‘ 𝑔 ) |
| 22 |
|
eqid |
⊢ ( Edg ‘ 𝑔 ) = ( Edg ‘ 𝑔 ) |
| 23 |
21 22
|
iscplgredg |
⊢ ( 𝑔 ∈ V → ( 𝑔 ∈ ComplGraph ↔ ∀ 𝑣 ∈ ( Vtx ‘ 𝑔 ) ∀ 𝑛 ∈ ( ( Vtx ‘ 𝑔 ) ∖ { 𝑣 } ) ∃ 𝑒 ∈ ( Edg ‘ 𝑔 ) { 𝑣 , 𝑛 } ⊆ 𝑒 ) ) |
| 24 |
23
|
elv |
⊢ ( 𝑔 ∈ ComplGraph ↔ ∀ 𝑣 ∈ ( Vtx ‘ 𝑔 ) ∀ 𝑛 ∈ ( ( Vtx ‘ 𝑔 ) ∖ { 𝑣 } ) ∃ 𝑒 ∈ ( Edg ‘ 𝑔 ) { 𝑣 , 𝑛 } ⊆ 𝑒 ) |
| 25 |
20 24
|
sylibr |
⊢ ( ( ( ( Edg ‘ 𝐺 ) = ran ( iEdg ‘ 𝐺 ) ∧ ( ( Vtx ‘ 𝑔 ) = ( Vtx ‘ 𝐺 ) ∧ ( iEdg ‘ 𝑔 ) = ( iEdg ‘ 𝐺 ) ) ) ∧ ∀ 𝑣 ∈ ( Vtx ‘ 𝐺 ) ∀ 𝑛 ∈ ( ( Vtx ‘ 𝐺 ) ∖ { 𝑣 } ) ∃ 𝑒 ∈ ( Edg ‘ 𝐺 ) { 𝑣 , 𝑛 } ⊆ 𝑒 ) → 𝑔 ∈ ComplGraph ) |
| 26 |
25
|
expcom |
⊢ ( ∀ 𝑣 ∈ ( Vtx ‘ 𝐺 ) ∀ 𝑛 ∈ ( ( Vtx ‘ 𝐺 ) ∖ { 𝑣 } ) ∃ 𝑒 ∈ ( Edg ‘ 𝐺 ) { 𝑣 , 𝑛 } ⊆ 𝑒 → ( ( ( Edg ‘ 𝐺 ) = ran ( iEdg ‘ 𝐺 ) ∧ ( ( Vtx ‘ 𝑔 ) = ( Vtx ‘ 𝐺 ) ∧ ( iEdg ‘ 𝑔 ) = ( iEdg ‘ 𝐺 ) ) ) → 𝑔 ∈ ComplGraph ) ) |
| 27 |
26
|
expd |
⊢ ( ∀ 𝑣 ∈ ( Vtx ‘ 𝐺 ) ∀ 𝑛 ∈ ( ( Vtx ‘ 𝐺 ) ∖ { 𝑣 } ) ∃ 𝑒 ∈ ( Edg ‘ 𝐺 ) { 𝑣 , 𝑛 } ⊆ 𝑒 → ( ( Edg ‘ 𝐺 ) = ran ( iEdg ‘ 𝐺 ) → ( ( ( Vtx ‘ 𝑔 ) = ( Vtx ‘ 𝐺 ) ∧ ( iEdg ‘ 𝑔 ) = ( iEdg ‘ 𝐺 ) ) → 𝑔 ∈ ComplGraph ) ) ) |
| 28 |
5 27
|
syl5com |
⊢ ( 𝐺 ∈ ComplGraph → ( ∀ 𝑣 ∈ ( Vtx ‘ 𝐺 ) ∀ 𝑛 ∈ ( ( Vtx ‘ 𝐺 ) ∖ { 𝑣 } ) ∃ 𝑒 ∈ ( Edg ‘ 𝐺 ) { 𝑣 , 𝑛 } ⊆ 𝑒 → ( ( ( Vtx ‘ 𝑔 ) = ( Vtx ‘ 𝐺 ) ∧ ( iEdg ‘ 𝑔 ) = ( iEdg ‘ 𝐺 ) ) → 𝑔 ∈ ComplGraph ) ) ) |
| 29 |
3 28
|
sylbid |
⊢ ( 𝐺 ∈ ComplGraph → ( 𝐺 ∈ ComplGraph → ( ( ( Vtx ‘ 𝑔 ) = ( Vtx ‘ 𝐺 ) ∧ ( iEdg ‘ 𝑔 ) = ( iEdg ‘ 𝐺 ) ) → 𝑔 ∈ ComplGraph ) ) ) |
| 30 |
29
|
pm2.43i |
⊢ ( 𝐺 ∈ ComplGraph → ( ( ( Vtx ‘ 𝑔 ) = ( Vtx ‘ 𝐺 ) ∧ ( iEdg ‘ 𝑔 ) = ( iEdg ‘ 𝐺 ) ) → 𝑔 ∈ ComplGraph ) ) |
| 31 |
30
|
alrimiv |
⊢ ( 𝐺 ∈ ComplGraph → ∀ 𝑔 ( ( ( Vtx ‘ 𝑔 ) = ( Vtx ‘ 𝐺 ) ∧ ( iEdg ‘ 𝑔 ) = ( iEdg ‘ 𝐺 ) ) → 𝑔 ∈ ComplGraph ) ) |
| 32 |
|
fvexd |
⊢ ( 𝐺 ∈ ComplGraph → ( Vtx ‘ 𝐺 ) ∈ V ) |
| 33 |
|
fvexd |
⊢ ( 𝐺 ∈ ComplGraph → ( iEdg ‘ 𝐺 ) ∈ V ) |
| 34 |
31 32 33
|
gropeld |
⊢ ( 𝐺 ∈ ComplGraph → ⟨ ( Vtx ‘ 𝐺 ) , ( iEdg ‘ 𝐺 ) ⟩ ∈ ComplGraph ) |