Theorem trlf1

Description: The enumeration F of a trail <. F , P >. is injective. (Contributed by AV, 20-Feb-2021) (Proof shortened by AV, 29-Oct-2021)

	Ref	Expression
Hypothesis	trlf1.i	⊢ 𝐼 = ( iEdg ‘ 𝐺 )
Assertion	trlf1	⊢ ( 𝐹 ( Trails ‘ 𝐺 ) 𝑃 → 𝐹 : ( 0 ..^ ( ♯ ‘ 𝐹 ) ) –1-1→ dom 𝐼 )

Proof

Step	Hyp	Ref	Expression
1		trlf1.i	⊢ 𝐼 = ( iEdg ‘ 𝐺 )
2		istrl	⊢ ( 𝐹 ( Trails ‘ 𝐺 ) 𝑃 ↔ ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 ∧ Fun ◡ 𝐹 ) )
3	1	wlkf	⊢ ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 → 𝐹 ∈ Word dom 𝐼 )
4		wrdf	⊢ ( 𝐹 ∈ Word dom 𝐼 → 𝐹 : ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ⟶ dom 𝐼 )
5		df-f1	⊢ ( 𝐹 : ( 0 ..^ ( ♯ ‘ 𝐹 ) ) –1-1→ dom 𝐼 ↔ ( 𝐹 : ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ⟶ dom 𝐼 ∧ Fun ◡ 𝐹 ) )
6	5	simplbi2	⊢ ( 𝐹 : ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ⟶ dom 𝐼 → ( Fun ◡ 𝐹 → 𝐹 : ( 0 ..^ ( ♯ ‘ 𝐹 ) ) –1-1→ dom 𝐼 ) )
7	3 4 6	3syl	⊢ ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 → ( Fun ◡ 𝐹 → 𝐹 : ( 0 ..^ ( ♯ ‘ 𝐹 ) ) –1-1→ dom 𝐼 ) )
8	7	imp	⊢ ( ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 ∧ Fun ◡ 𝐹 ) → 𝐹 : ( 0 ..^ ( ♯ ‘ 𝐹 ) ) –1-1→ dom 𝐼 )
9	2 8	sylbi	⊢ ( 𝐹 ( Trails ‘ 𝐺 ) 𝑃 → 𝐹 : ( 0 ..^ ( ♯ ‘ 𝐹 ) ) –1-1→ dom 𝐼 )