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	6	adantl	⊢ ( ( ⊤ ∧ 𝑔 = 𝐺 ) → ( 𝑓 : ( 0 ..^ ( ♯ ‘ 𝑓 ) ) –onto→ dom ( iEdg ‘ 𝑔 ) ↔ 𝑓 : ( 0 ..^ ( ♯ ‘ 𝑓 ) ) –onto→ dom 𝐼 ) )
8		wksv	⊢ { ⟨ 𝑓 , 𝑝 ⟩ ∣ 𝑓 ( Walks ‘ 𝐺 ) 𝑝 } ∈ V
9		trliswlk	⊢ ( 𝑓 ( Trails ‘ 𝐺 ) 𝑝 → 𝑓 ( Walks ‘ 𝐺 ) 𝑝 )
10	9	ssopab2i	⊢ { ⟨ 𝑓 , 𝑝 ⟩ ∣ 𝑓 ( Trails ‘ 𝐺 ) 𝑝 } ⊆ { ⟨ 𝑓 , 𝑝 ⟩ ∣ 𝑓 ( Walks ‘ 𝐺 ) 𝑝 }
11	8 10	ssexi	⊢ { ⟨ 𝑓 , 𝑝 ⟩ ∣ 𝑓 ( Trails ‘ 𝐺 ) 𝑝 } ∈ V
12	11	a1i	⊢ ( ⊤ → { ⟨ 𝑓 , 𝑝 ⟩ ∣ 𝑓 ( Trails ‘ 𝐺 ) 𝑝 } ∈ V )
13		df-eupth	⊢ EulerPaths = ( 𝑔 ∈ V ↦ { ⟨ 𝑓 , 𝑝 ⟩ ∣ ( 𝑓 ( Trails ‘ 𝑔 ) 𝑝 ∧ 𝑓 : ( 0 ..^ ( ♯ ‘ 𝑓 ) ) –onto→ dom ( iEdg ‘ 𝑔 ) ) } )
14	7 12 13	fvmptopab	⊢ ( ⊤ → ( EulerPaths ‘ 𝐺 ) = { ⟨ 𝑓 , 𝑝 ⟩ ∣ ( 𝑓 ( Trails ‘ 𝐺 ) 𝑝 ∧ 𝑓 : ( 0 ..^ ( ♯ ‘ 𝑓 ) ) –onto→ dom 𝐼 ) } )
15	14	mptru	⊢ ( EulerPaths ‘ 𝐺 ) = { ⟨ 𝑓 , 𝑝 ⟩ ∣ ( 𝑓 ( Trails ‘ 𝐺 ) 𝑝 ∧ 𝑓 : ( 0 ..^ ( ♯ ‘ 𝑓 ) ) –onto→ dom 𝐼 ) }