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Metamath Proof Explorer

Theorem eupths

Description: The Eulerian paths on the graph G . (Contributed by AV, 18-Feb-2021) (Revised by AV, 29-Oct-2021)

Ref Expression
Hypothesis eupths.i ⊢ 𝐼 = ( iEdg ‘ 𝐺 )
Assertion eupths ( EulerPaths ‘ 𝐺 ) = { ⟨ 𝑓 , 𝑝 ⟩ ∣ ( 𝑓 ( Trails ‘ 𝐺 ) 𝑝 ∧ 𝑓 : ( 0 ..^ ( ♯ ‘ 𝑓 ) ) –onto→ dom 𝐼 ) }

Proof

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 ) }