Step |
Hyp |
Ref |
Expression |
1 |
|
eupths.i |
⊢ 𝐼 = ( iEdg ‘ 𝐺 ) |
2 |
|
fveq2 |
⊢ ( 𝑔 = 𝐺 → ( iEdg ‘ 𝑔 ) = ( iEdg ‘ 𝐺 ) ) |
3 |
2 1
|
eqtr4di |
⊢ ( 𝑔 = 𝐺 → ( iEdg ‘ 𝑔 ) = 𝐼 ) |
4 |
3
|
dmeqd |
⊢ ( 𝑔 = 𝐺 → dom ( iEdg ‘ 𝑔 ) = dom 𝐼 ) |
5 |
|
foeq3 |
⊢ ( dom ( iEdg ‘ 𝑔 ) = dom 𝐼 → ( 𝑓 : ( 0 ..^ ( ♯ ‘ 𝑓 ) ) –onto→ dom ( iEdg ‘ 𝑔 ) ↔ 𝑓 : ( 0 ..^ ( ♯ ‘ 𝑓 ) ) –onto→ dom 𝐼 ) ) |
6 |
4 5
|
syl |
⊢ ( 𝑔 = 𝐺 → ( 𝑓 : ( 0 ..^ ( ♯ ‘ 𝑓 ) ) –onto→ dom ( iEdg ‘ 𝑔 ) ↔ 𝑓 : ( 0 ..^ ( ♯ ‘ 𝑓 ) ) –onto→ dom 𝐼 ) ) |
7 |
|
df-eupth |
⊢ EulerPaths = ( 𝑔 ∈ V ↦ { 〈 𝑓 , 𝑝 〉 ∣ ( 𝑓 ( Trails ‘ 𝑔 ) 𝑝 ∧ 𝑓 : ( 0 ..^ ( ♯ ‘ 𝑓 ) ) –onto→ dom ( iEdg ‘ 𝑔 ) ) } ) |
8 |
6 7
|
fvmptopab |
⊢ ( EulerPaths ‘ 𝐺 ) = { 〈 𝑓 , 𝑝 〉 ∣ ( 𝑓 ( Trails ‘ 𝐺 ) 𝑝 ∧ 𝑓 : ( 0 ..^ ( ♯ ‘ 𝑓 ) ) –onto→ dom 𝐼 ) } |