Definition df-iedg

Description: Define the function mapping a graph to its indexed edges. This definition is very general: It defines the indexed edges for any ordered pair as its second component, and for any other class as its "edge function". It is meaningful, however, only if the ordered pair represents a graph resp. the class is an extensible structure (containing a slot for "edge functions") representing a graph. (Contributed by AV, 20-Sep-2020)

	Ref	Expression
Assertion	df-iedg	⊢ iEdg = ( 𝑔 ∈ V ↦ if ( 𝑔 ∈ ( V × V ) , ( 2nd ‘ 𝑔 ) , ( .ef ‘ 𝑔 ) ) )

Detailed syntax breakdown

Step	Hyp	Ref	Expression
0		ciedg	⊢ iEdg
1		vg	⊢ 𝑔
2		cvv	⊢ V
3	1	cv	⊢ 𝑔
4	2 2	cxp	⊢ ( V × V )
5	3 4	wcel	⊢ 𝑔 ∈ ( V × V )
6		c2nd	⊢ 2nd
7	3 6	cfv	⊢ ( 2nd ‘ 𝑔 )
8		cedgf	⊢ .ef
9	3 8	cfv	⊢ ( .ef ‘ 𝑔 )
10	5 7 9	cif	⊢ if ( 𝑔 ∈ ( V × V ) , ( 2nd ‘ 𝑔 ) , ( .ef ‘ 𝑔 ) )
11	1 2 10	cmpt	⊢ ( 𝑔 ∈ V ↦ if ( 𝑔 ∈ ( V × V ) , ( 2nd ‘ 𝑔 ) , ( .ef ‘ 𝑔 ) ) )
12	0 11	wceq	⊢ iEdg = ( 𝑔 ∈ V ↦ if ( 𝑔 ∈ ( V × V ) , ( 2nd ‘ 𝑔 ) , ( .ef ‘ 𝑔 ) ) )