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Mirrors > Home > MPE Home > Th. List > df-iedg | Structured version Visualization version GIF version |
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 |
---|---|
df-iedg | ⊢ iEdg = (𝑔 ∈ V ↦ if(𝑔 ∈ (V × V), (2^{nd} ‘𝑔), (.ef‘𝑔))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ciedg 25674 | . 2 class iEdg | |
2 | vg | . . 3 setvar 𝑔 | |
3 | cvv 3173 | . . 3 class V | |
4 | 2 | cv 1474 | . . . . 5 class 𝑔 |
5 | 3, 3 | cxp 5036 | . . . . 5 class (V × V) |
6 | 4, 5 | wcel 1977 | . . . 4 wff 𝑔 ∈ (V × V) |
7 | c2nd 7058 | . . . . 5 class 2^{nd} | |
8 | 4, 7 | cfv 5804 | . . . 4 class (2^{nd} ‘𝑔) |
9 | cedgf 25667 | . . . . 5 class .ef | |
10 | 4, 9 | cfv 5804 | . . . 4 class (.ef‘𝑔) |
11 | 6, 8, 10 | cif 4036 | . . 3 class if(𝑔 ∈ (V × V), (2^{nd} ‘𝑔), (.ef‘𝑔)) |
12 | 2, 3, 11 | cmpt 4643 | . 2 class (𝑔 ∈ V ↦ if(𝑔 ∈ (V × V), (2^{nd} ‘𝑔), (.ef‘𝑔))) |
13 | 1, 12 | wceq 1475 | 1 wff iEdg = (𝑔 ∈ V ↦ if(𝑔 ∈ (V × V), (2^{nd} ‘𝑔), (.ef‘𝑔))) |
Colors of variables: wff setvar class |
This definition is referenced by: iedgval 25678 |
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