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 e. Word dom I )
4		wrdf	|- ( F e. 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 )