Theorem funiedgdm2val

Description: The set of indexed edges of an extensible structure with (at least) two slots. (Contributed by AV, 22-Sep-2020) (Revised by AV, 7-Jun-2021) (Revised by AV, 12-Nov-2021)

	Ref	Expression
Hypotheses	funvtxdm2val.a	|- A e. _V
	funvtxdm2val.b	|- B e. _V
Assertion	funiedgdm2val	|- ( ( Fun ( G \ { (/) } ) /\ A =/= B /\ { A , B } C_ dom G ) -> ( iEdg ` G ) = ( .ef ` G ) )

Proof

Step	Hyp	Ref	Expression
1		funvtxdm2val.a	|- A e. _V
2		funvtxdm2val.b	|- B e. _V
3		iedgval	|- ( iEdg ` G ) = if ( G e. ( _V X. _V ) , ( 2nd ` G ) , ( .ef ` G ) )
4	1 2	fun2dmnop0	|- ( ( Fun ( G \ { (/) } ) /\ A =/= B /\ { A , B } C_ dom G ) -> -. G e. ( _V X. _V ) )
5	4	iffalsed	|- ( ( Fun ( G \ { (/) } ) /\ A =/= B /\ { A , B } C_ dom G ) -> if ( G e. ( _V X. _V ) , ( 2nd ` G ) , ( .ef ` G ) ) = ( .ef ` G ) )
6	3 5	syl5eq	|- ( ( Fun ( G \ { (/) } ) /\ A =/= B /\ { A , B } C_ dom G ) -> ( iEdg ` G ) = ( .ef ` G ) )