Step |
Hyp |
Ref |
Expression |
1 |
|
fvex |
⊢ ( iEdg ‘ 𝐺 ) ∈ V |
2 |
1
|
dmex |
⊢ dom ( iEdg ‘ 𝐺 ) ∈ V |
3 |
|
eqid |
⊢ ( Vtx ‘ 𝐺 ) = ( Vtx ‘ 𝐺 ) |
4 |
|
eqid |
⊢ ( iEdg ‘ 𝐺 ) = ( iEdg ‘ 𝐺 ) |
5 |
3 4
|
usgrf |
⊢ ( 𝐺 ∈ USGraph → ( iEdg ‘ 𝐺 ) : dom ( iEdg ‘ 𝐺 ) –1-1→ { 𝑥 ∈ ( 𝒫 ( Vtx ‘ 𝐺 ) ∖ { ∅ } ) ∣ ( ♯ ‘ 𝑥 ) = 2 } ) |
6 |
|
hashf1rn |
⊢ ( ( dom ( iEdg ‘ 𝐺 ) ∈ V ∧ ( iEdg ‘ 𝐺 ) : dom ( iEdg ‘ 𝐺 ) –1-1→ { 𝑥 ∈ ( 𝒫 ( Vtx ‘ 𝐺 ) ∖ { ∅ } ) ∣ ( ♯ ‘ 𝑥 ) = 2 } ) → ( ♯ ‘ ( iEdg ‘ 𝐺 ) ) = ( ♯ ‘ ran ( iEdg ‘ 𝐺 ) ) ) |
7 |
2 5 6
|
sylancr |
⊢ ( 𝐺 ∈ USGraph → ( ♯ ‘ ( iEdg ‘ 𝐺 ) ) = ( ♯ ‘ ran ( iEdg ‘ 𝐺 ) ) ) |
8 |
|
edgval |
⊢ ( Edg ‘ 𝐺 ) = ran ( iEdg ‘ 𝐺 ) |
9 |
8
|
a1i |
⊢ ( 𝐺 ∈ USGraph → ( Edg ‘ 𝐺 ) = ran ( iEdg ‘ 𝐺 ) ) |
10 |
9
|
fveq2d |
⊢ ( 𝐺 ∈ USGraph → ( ♯ ‘ ( Edg ‘ 𝐺 ) ) = ( ♯ ‘ ran ( iEdg ‘ 𝐺 ) ) ) |
11 |
7 10
|
eqtr4d |
⊢ ( 𝐺 ∈ USGraph → ( ♯ ‘ ( iEdg ‘ 𝐺 ) ) = ( ♯ ‘ ( Edg ‘ 𝐺 ) ) ) |