Theorem upgrle

Description: An edge of an undirected pseudograph has at most two ends. (Contributed by Mario Carneiro, 11-Mar-2015) (Revised by AV, 10-Oct-2020)

	Ref	Expression
Hypotheses	isupgr.v	|- V = ( Vtx ` G )
	isupgr.e	|- E = ( iEdg ` G )
Assertion	upgrle	|- ( ( G e. UPGraph /\ E Fn A /\ F e. A ) -> ( # ` ( E ` F ) ) <_ 2 )

Proof

Step	Hyp	Ref	Expression
1		isupgr.v	|- V = ( Vtx ` G )
2		isupgr.e	|- E = ( iEdg ` G )
3	1 2	upgrfn	|- ( ( G e. UPGraph /\ E Fn A ) -> E : A --> { x e. ( ~P V \ { (/) } ) | ( # ` x ) <_ 2 } )
4	3	ffvelcdmda	|- ( ( ( G e. UPGraph /\ E Fn A ) /\ F e. A ) -> ( E ` F ) e. { x e. ( ~P V \ { (/) } ) | ( # ` x ) <_ 2 } )
5	4	3impa	|- ( ( G e. UPGraph /\ E Fn A /\ F e. A ) -> ( E ` F ) e. { x e. ( ~P V \ { (/) } ) | ( # ` x ) <_ 2 } )
6		fveq2	|- ( x = ( E ` F ) -> ( # ` x ) = ( # ` ( E ` F ) ) )
7	6	breq1d	|- ( x = ( E ` F ) -> ( ( # ` x ) <_ 2 <-> ( # ` ( E ` F ) ) <_ 2 ) )
8	7	elrab	|- ( ( E ` F ) e. { x e. ( ~P V \ { (/) } ) | ( # ` x ) <_ 2 } <-> ( ( E ` F ) e. ( ~P V \ { (/) } ) /\ ( # ` ( E ` F ) ) <_ 2 ) )
9	8	simprbi	|- ( ( E ` F ) e. { x e. ( ~P V \ { (/) } ) | ( # ` x ) <_ 2 } -> ( # ` ( E ` F ) ) <_ 2 )
10	5 9	syl	|- ( ( G e. UPGraph /\ E Fn A /\ F e. A ) -> ( # ` ( E ` F ) ) <_ 2 )