Theorem pthdifv

Description: The vertices of a path are distinct (except the first and last vertex), so the restricted vertex function is one-to-one. (Contributed by AV, 2-Oct-2025)

	Ref	Expression
Assertion	pthdifv	⊢ ( 𝐹 ( Paths ‘ 𝐺 ) 𝑃 → ( 𝑃 ↾ ( 1 ... ( ♯ ‘ 𝐹 ) ) ) : ( 1 ... ( ♯ ‘ 𝐹 ) ) –1-1→ ( Vtx ‘ 𝐺 ) )

Proof

Step	Hyp	Ref	Expression
1		trliswlk	⊢ ( 𝐹 ( Trails ‘ 𝐺 ) 𝑃 → 𝐹 ( Walks ‘ 𝐺 ) 𝑃 )
2		eqid	⊢ ( Vtx ‘ 𝐺 ) = ( Vtx ‘ 𝐺 )
3	2	wlkp	⊢ ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 → 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ ( Vtx ‘ 𝐺 ) )
4		fz1ssfz0	⊢ ( 1 ... ( ♯ ‘ 𝐹 ) ) ⊆ ( 0 ... ( ♯ ‘ 𝐹 ) )
5	4	a1i	⊢ ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 → ( 1 ... ( ♯ ‘ 𝐹 ) ) ⊆ ( 0 ... ( ♯ ‘ 𝐹 ) ) )
6	3 5	fssresd	⊢ ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 → ( 𝑃 ↾ ( 1 ... ( ♯ ‘ 𝐹 ) ) ) : ( 1 ... ( ♯ ‘ 𝐹 ) ) ⟶ ( Vtx ‘ 𝐺 ) )
7	1 6	syl	⊢ ( 𝐹 ( Trails ‘ 𝐺 ) 𝑃 → ( 𝑃 ↾ ( 1 ... ( ♯ ‘ 𝐹 ) ) ) : ( 1 ... ( ♯ ‘ 𝐹 ) ) ⟶ ( Vtx ‘ 𝐺 ) )
8	7	anim1i	⊢ ( ( 𝐹 ( Trails ‘ 𝐺 ) 𝑃 ∧ Fun ◡ ( 𝑃 ↾ ( 1 ... ( ♯ ‘ 𝐹 ) ) ) ) → ( ( 𝑃 ↾ ( 1 ... ( ♯ ‘ 𝐹 ) ) ) : ( 1 ... ( ♯ ‘ 𝐹 ) ) ⟶ ( Vtx ‘ 𝐺 ) ∧ Fun ◡ ( 𝑃 ↾ ( 1 ... ( ♯ ‘ 𝐹 ) ) ) ) )
9	8	3adant3	⊢ ( ( 𝐹 ( Trails ‘ 𝐺 ) 𝑃 ∧ Fun ◡ ( 𝑃 ↾ ( 1 ... ( ♯ ‘ 𝐹 ) ) ) ∧ ( 𝑃 ‘ 0 ) ∉ ( 𝑃 “ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ) ) → ( ( 𝑃 ↾ ( 1 ... ( ♯ ‘ 𝐹 ) ) ) : ( 1 ... ( ♯ ‘ 𝐹 ) ) ⟶ ( Vtx ‘ 𝐺 ) ∧ Fun ◡ ( 𝑃 ↾ ( 1 ... ( ♯ ‘ 𝐹 ) ) ) ) )
10		dfpth2	⊢ ( 𝐹 ( Paths ‘ 𝐺 ) 𝑃 ↔ ( 𝐹 ( Trails ‘ 𝐺 ) 𝑃 ∧ Fun ◡ ( 𝑃 ↾ ( 1 ... ( ♯ ‘ 𝐹 ) ) ) ∧ ( 𝑃 ‘ 0 ) ∉ ( 𝑃 “ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ) ) )
11		df-f1	⊢ ( ( 𝑃 ↾ ( 1 ... ( ♯ ‘ 𝐹 ) ) ) : ( 1 ... ( ♯ ‘ 𝐹 ) ) –1-1→ ( Vtx ‘ 𝐺 ) ↔ ( ( 𝑃 ↾ ( 1 ... ( ♯ ‘ 𝐹 ) ) ) : ( 1 ... ( ♯ ‘ 𝐹 ) ) ⟶ ( Vtx ‘ 𝐺 ) ∧ Fun ◡ ( 𝑃 ↾ ( 1 ... ( ♯ ‘ 𝐹 ) ) ) ) )
12	9 10 11	3imtr4i	⊢ ( 𝐹 ( Paths ‘ 𝐺 ) 𝑃 → ( 𝑃 ↾ ( 1 ... ( ♯ ‘ 𝐹 ) ) ) : ( 1 ... ( ♯ ‘ 𝐹 ) ) –1-1→ ( Vtx ‘ 𝐺 ) )