Theorem funiedgdmge2val

Description: The set of indexed edges of an extensible structure with (at least) two slots. (Contributed by AV, 12-Oct-2020) (Revised by AV, 7-Jun-2021) (Revised by AV, 12-Nov-2021)

	Ref	Expression
Assertion	funiedgdmge2val	|- ( ( Fun ( G \ { (/) } ) /\ 2 <_ ( # ` dom G ) ) -> ( iEdg ` G ) = ( .ef ` G ) )

Proof

Step	Hyp	Ref	Expression
1		iedgval	|- ( iEdg ` G ) = if ( G e. ( _V X. _V ) , ( 2nd ` G ) , ( .ef ` G ) )
2		fundmge2nop0	|- ( ( Fun ( G \ { (/) } ) /\ 2 <_ ( # ` dom G ) ) -> -. G e. ( _V X. _V ) )
3	2	iffalsed	|- ( ( Fun ( G \ { (/) } ) /\ 2 <_ ( # ` dom G ) ) -> if ( G e. ( _V X. _V ) , ( 2nd ` G ) , ( .ef ` G ) ) = ( .ef ` G ) )
4	1 3	eqtrid	|- ( ( Fun ( G \ { (/) } ) /\ 2 <_ ( # ` dom G ) ) -> ( iEdg ` G ) = ( .ef ` G ) )