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		df-pths	⊢ Paths = ( 𝑔 ∈ V ↦ { ⟨ 𝑓 , 𝑝 ⟩ ∣ ( 𝑓 ( Trails ‘ 𝑔 ) 𝑝 ∧ Fun ◡ ( 𝑝 ↾ ( 1 ..^ ( ♯ ‘ 𝑓 ) ) ) ∧ ( ( 𝑝 “ { 0 , ( ♯ ‘ 𝑓 ) } ) ∩ ( 𝑝 “ ( 1 ..^ ( ♯ ‘ 𝑓 ) ) ) ) = ∅ ) } )
3		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 ) ) ) ) = ∅ ) ) )
4	3	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 ) ) ) ) = ∅ ) ) }
5	4	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 ) ) ) ) = ∅ ) ) } )
6	2 5	eqtri	⊢ Paths = ( 𝑔 ∈ V ↦ { ⟨ 𝑓 , 𝑝 ⟩ ∣ ( 𝑓 ( Trails ‘ 𝑔 ) 𝑝 ∧ ( Fun ◡ ( 𝑝 ↾ ( 1 ..^ ( ♯ ‘ 𝑓 ) ) ) ∧ ( ( 𝑝 “ { 0 , ( ♯ ‘ 𝑓 ) } ) ∩ ( 𝑝 “ ( 1 ..^ ( ♯ ‘ 𝑓 ) ) ) ) = ∅ ) ) } )
7	1 6	fvmptopab	⊢ ( Paths ‘ 𝐺 ) = { ⟨ 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	7 9	eqtr4i	⊢ ( Paths ‘ 𝐺 ) = { ⟨ 𝑓 , 𝑝 ⟩ ∣ ( 𝑓 ( Trails ‘ 𝐺 ) 𝑝 ∧ Fun ◡ ( 𝑝 ↾ ( 1 ..^ ( ♯ ‘ 𝑓 ) ) ) ∧ ( ( 𝑝 “ { 0 , ( ♯ ‘ 𝑓 ) } ) ∩ ( 𝑝 “ ( 1 ..^ ( ♯ ‘ 𝑓 ) ) ) ) = ∅ ) }