Theorem pthsfval

Description: The set of paths (in an undirected graph). (Contributed by Alexander van der Vekens, 20-Oct-2017) (Revised by AV, 9-Jan-2021) (Revised by AV, 29-Oct-2021)

	Ref	Expression
Assertion	pthsfval	⊢ ( Paths ‘ 𝐺 ) = { ⟨ 𝑓 , 𝑝 ⟩ ∣ ( 𝑓 ( Trails ‘ 𝐺 ) 𝑝 ∧ Fun ◡ ( 𝑝 ↾ ( 1 ..^ ( ♯ ‘ 𝑓 ) ) ) ∧ ( ( 𝑝 “ { 0 , ( ♯ ‘ 𝑓 ) } ) ∩ ( 𝑝 “ ( 1 ..^ ( ♯ ‘ 𝑓 ) ) ) ) = ∅ ) }

Proof

Step	Hyp	Ref	Expression
1		biidd	⊢ ( ( ⊤ ∧ 𝑔 = 𝐺 ) → ( ( Fun ◡ ( 𝑝 ↾ ( 1 ..^ ( ♯ ‘ 𝑓 ) ) ) ∧ ( ( 𝑝 “ { 0 , ( ♯ ‘ 𝑓 ) } ) ∩ ( 𝑝 “ ( 1 ..^ ( ♯ ‘ 𝑓 ) ) ) ) = ∅ ) ↔ ( Fun ◡ ( 𝑝 ↾ ( 1 ..^ ( ♯ ‘ 𝑓 ) ) ) ∧ ( ( 𝑝 “ { 0 , ( ♯ ‘ 𝑓 ) } ) ∩ ( 𝑝 “ ( 1 ..^ ( ♯ ‘ 𝑓 ) ) ) ) = ∅ ) ) )
2		wksv	⊢ { ⟨ 𝑓 , 𝑝 ⟩ ∣ 𝑓 ( Walks ‘ 𝐺 ) 𝑝 } ∈ V
3		trliswlk	⊢ ( 𝑓 ( Trails ‘ 𝐺 ) 𝑝 → 𝑓 ( Walks ‘ 𝐺 ) 𝑝 )
4	3	ssopab2i	⊢ { ⟨ 𝑓 , 𝑝 ⟩ ∣ 𝑓 ( Trails ‘ 𝐺 ) 𝑝 } ⊆ { ⟨ 𝑓 , 𝑝 ⟩ ∣ 𝑓 ( Walks ‘ 𝐺 ) 𝑝 }
5	2 4	ssexi	⊢ { ⟨ 𝑓 , 𝑝 ⟩ ∣ 𝑓 ( Trails ‘ 𝐺 ) 𝑝 } ∈ V
6	5	a1i	⊢ ( ⊤ → { ⟨ 𝑓 , 𝑝 ⟩ ∣ 𝑓 ( Trails ‘ 𝐺 ) 𝑝 } ∈ V )
7		df-pths	⊢ Paths = ( 𝑔 ∈ V ↦ { ⟨ 𝑓 , 𝑝 ⟩ ∣ ( 𝑓 ( Trails ‘ 𝑔 ) 𝑝 ∧ Fun ◡ ( 𝑝 ↾ ( 1 ..^ ( ♯ ‘ 𝑓 ) ) ) ∧ ( ( 𝑝 “ { 0 , ( ♯ ‘ 𝑓 ) } ) ∩ ( 𝑝 “ ( 1 ..^ ( ♯ ‘ 𝑓 ) ) ) ) = ∅ ) } )
8		3anass	⊢ ( ( 𝑓 ( Trails ‘ 𝑔 ) 𝑝 ∧ Fun ◡ ( 𝑝 ↾ ( 1 ..^ ( ♯ ‘ 𝑓 ) ) ) ∧ ( ( 𝑝 “ { 0 , ( ♯ ‘ 𝑓 ) } ) ∩ ( 𝑝 “ ( 1 ..^ ( ♯ ‘ 𝑓 ) ) ) ) = ∅ ) ↔ ( 𝑓 ( Trails ‘ 𝑔 ) 𝑝 ∧ ( Fun ◡ ( 𝑝 ↾ ( 1 ..^ ( ♯ ‘ 𝑓 ) ) ) ∧ ( ( 𝑝 “ { 0 , ( ♯ ‘ 𝑓 ) } ) ∩ ( 𝑝 “ ( 1 ..^ ( ♯ ‘ 𝑓 ) ) ) ) = ∅ ) ) )
9	8	opabbii	⊢ { ⟨ 𝑓 , 𝑝 ⟩ ∣ ( 𝑓 ( Trails ‘ 𝑔 ) 𝑝 ∧ Fun ◡ ( 𝑝 ↾ ( 1 ..^ ( ♯ ‘ 𝑓 ) ) ) ∧ ( ( 𝑝 “ { 0 , ( ♯ ‘ 𝑓 ) } ) ∩ ( 𝑝 “ ( 1 ..^ ( ♯ ‘ 𝑓 ) ) ) ) = ∅ ) } = { ⟨ 𝑓 , 𝑝 ⟩ ∣ ( 𝑓 ( Trails ‘ 𝑔 ) 𝑝 ∧ ( Fun ◡ ( 𝑝 ↾ ( 1 ..^ ( ♯ ‘ 𝑓 ) ) ) ∧ ( ( 𝑝 “ { 0 , ( ♯ ‘ 𝑓 ) } ) ∩ ( 𝑝 “ ( 1 ..^ ( ♯ ‘ 𝑓 ) ) ) ) = ∅ ) ) }
10	9	mpteq2i	⊢ ( 𝑔 ∈ V ↦ { ⟨ 𝑓 , 𝑝 ⟩ ∣ ( 𝑓 ( Trails ‘ 𝑔 ) 𝑝 ∧ Fun ◡ ( 𝑝 ↾ ( 1 ..^ ( ♯ ‘ 𝑓 ) ) ) ∧ ( ( 𝑝 “ { 0 , ( ♯ ‘ 𝑓 ) } ) ∩ ( 𝑝 “ ( 1 ..^ ( ♯ ‘ 𝑓 ) ) ) ) = ∅ ) } ) = ( 𝑔 ∈ V ↦ { ⟨ 𝑓 , 𝑝 ⟩ ∣ ( 𝑓 ( Trails ‘ 𝑔 ) 𝑝 ∧ ( Fun ◡ ( 𝑝 ↾ ( 1 ..^ ( ♯ ‘ 𝑓 ) ) ) ∧ ( ( 𝑝 “ { 0 , ( ♯ ‘ 𝑓 ) } ) ∩ ( 𝑝 “ ( 1 ..^ ( ♯ ‘ 𝑓 ) ) ) ) = ∅ ) ) } )
11	7 10	eqtri	⊢ Paths = ( 𝑔 ∈ V ↦ { ⟨ 𝑓 , 𝑝 ⟩ ∣ ( 𝑓 ( Trails ‘ 𝑔 ) 𝑝 ∧ ( Fun ◡ ( 𝑝 ↾ ( 1 ..^ ( ♯ ‘ 𝑓 ) ) ) ∧ ( ( 𝑝 “ { 0 , ( ♯ ‘ 𝑓 ) } ) ∩ ( 𝑝 “ ( 1 ..^ ( ♯ ‘ 𝑓 ) ) ) ) = ∅ ) ) } )
12	1 6 11	fvmptopab	⊢ ( ⊤ → ( Paths ‘ 𝐺 ) = { ⟨ 𝑓 , 𝑝 ⟩ ∣ ( 𝑓 ( Trails ‘ 𝐺 ) 𝑝 ∧ ( Fun ◡ ( 𝑝 ↾ ( 1 ..^ ( ♯ ‘ 𝑓 ) ) ) ∧ ( ( 𝑝 “ { 0 , ( ♯ ‘ 𝑓 ) } ) ∩ ( 𝑝 “ ( 1 ..^ ( ♯ ‘ 𝑓 ) ) ) ) = ∅ ) ) } )
13	12	mptru	⊢ ( Paths ‘ 𝐺 ) = { ⟨ 𝑓 , 𝑝 ⟩ ∣ ( 𝑓 ( Trails ‘ 𝐺 ) 𝑝 ∧ ( Fun ◡ ( 𝑝 ↾ ( 1 ..^ ( ♯ ‘ 𝑓 ) ) ) ∧ ( ( 𝑝 “ { 0 , ( ♯ ‘ 𝑓 ) } ) ∩ ( 𝑝 “ ( 1 ..^ ( ♯ ‘ 𝑓 ) ) ) ) = ∅ ) ) }
14		3anass	⊢ ( ( 𝑓 ( Trails ‘ 𝐺 ) 𝑝 ∧ Fun ◡ ( 𝑝 ↾ ( 1 ..^ ( ♯ ‘ 𝑓 ) ) ) ∧ ( ( 𝑝 “ { 0 , ( ♯ ‘ 𝑓 ) } ) ∩ ( 𝑝 “ ( 1 ..^ ( ♯ ‘ 𝑓 ) ) ) ) = ∅ ) ↔ ( 𝑓 ( Trails ‘ 𝐺 ) 𝑝 ∧ ( Fun ◡ ( 𝑝 ↾ ( 1 ..^ ( ♯ ‘ 𝑓 ) ) ) ∧ ( ( 𝑝 “ { 0 , ( ♯ ‘ 𝑓 ) } ) ∩ ( 𝑝 “ ( 1 ..^ ( ♯ ‘ 𝑓 ) ) ) ) = ∅ ) ) )
15	14	bicomi	⊢ ( ( 𝑓 ( Trails ‘ 𝐺 ) 𝑝 ∧ ( Fun ◡ ( 𝑝 ↾ ( 1 ..^ ( ♯ ‘ 𝑓 ) ) ) ∧ ( ( 𝑝 “ { 0 , ( ♯ ‘ 𝑓 ) } ) ∩ ( 𝑝 “ ( 1 ..^ ( ♯ ‘ 𝑓 ) ) ) ) = ∅ ) ) ↔ ( 𝑓 ( Trails ‘ 𝐺 ) 𝑝 ∧ Fun ◡ ( 𝑝 ↾ ( 1 ..^ ( ♯ ‘ 𝑓 ) ) ) ∧ ( ( 𝑝 “ { 0 , ( ♯ ‘ 𝑓 ) } ) ∩ ( 𝑝 “ ( 1 ..^ ( ♯ ‘ 𝑓 ) ) ) ) = ∅ ) )
16	15	opabbii	⊢ { ⟨ 𝑓 , 𝑝 ⟩ ∣ ( 𝑓 ( Trails ‘ 𝐺 ) 𝑝 ∧ ( Fun ◡ ( 𝑝 ↾ ( 1 ..^ ( ♯ ‘ 𝑓 ) ) ) ∧ ( ( 𝑝 “ { 0 , ( ♯ ‘ 𝑓 ) } ) ∩ ( 𝑝 “ ( 1 ..^ ( ♯ ‘ 𝑓 ) ) ) ) = ∅ ) ) } = { ⟨ 𝑓 , 𝑝 ⟩ ∣ ( 𝑓 ( Trails ‘ 𝐺 ) 𝑝 ∧ Fun ◡ ( 𝑝 ↾ ( 1 ..^ ( ♯ ‘ 𝑓 ) ) ) ∧ ( ( 𝑝 “ { 0 , ( ♯ ‘ 𝑓 ) } ) ∩ ( 𝑝 “ ( 1 ..^ ( ♯ ‘ 𝑓 ) ) ) ) = ∅ ) }
17	13 16	eqtri	⊢ ( Paths ‘ 𝐺 ) = { ⟨ 𝑓 , 𝑝 ⟩ ∣ ( 𝑓 ( Trails ‘ 𝐺 ) 𝑝 ∧ Fun ◡ ( 𝑝 ↾ ( 1 ..^ ( ♯ ‘ 𝑓 ) ) ) ∧ ( ( 𝑝 “ { 0 , ( ♯ ‘ 𝑓 ) } ) ∩ ( 𝑝 “ ( 1 ..^ ( ♯ ‘ 𝑓 ) ) ) ) = ∅ ) }