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 ∈ ( V × V ) , ( 2nd ‘ 𝐺 ) , ( .ef ‘ 𝐺 ) )
2		fundmge2nop0	⊢ ( ( Fun ( 𝐺 ∖ { ∅ } ) ∧ 2 ≤ ( ♯ ‘ dom 𝐺 ) ) → ¬ 𝐺 ∈ ( V × V ) )
3	2	iffalsed	⊢ ( ( Fun ( 𝐺 ∖ { ∅ } ) ∧ 2 ≤ ( ♯ ‘ dom 𝐺 ) ) → if ( 𝐺 ∈ ( V × V ) , ( 2nd ‘ 𝐺 ) , ( .ef ‘ 𝐺 ) ) = ( .ef ‘ 𝐺 ) )
4	1 3	syl5eq	⊢ ( ( Fun ( 𝐺 ∖ { ∅ } ) ∧ 2 ≤ ( ♯ ‘ dom 𝐺 ) ) → ( iEdg ‘ 𝐺 ) = ( .ef ‘ 𝐺 ) )