Metamath Proof Explorer


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 2 nd g ef g

Detailed syntax breakdown

Step Hyp Ref Expression
0 ciedg class iEdg
1 vg setvar g
2 cvv class V
3 1 cv setvar g
4 2 2 cxp class V × V
5 3 4 wcel wff g V × V
6 c2nd class 2 nd
7 3 6 cfv class 2 nd g
8 cedgf class ef
9 3 8 cfv class ef g
10 5 7 9 cif class if g V × V 2 nd g ef g
11 1 2 10 cmpt class g V if g V × V 2 nd g ef g
12 0 11 wceq wff iEdg = g V if g V × V 2 nd g ef g