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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 26349 | . 2 class iEdg | |
2 | vg | . . 3 setvar 𝑔 | |
3 | cvv 3398 | . . 3 class V | |
4 | 2 | cv 1600 | . . . . 5 class 𝑔 |
5 | 3, 3 | cxp 5355 | . . . . 5 class (V × V) |
6 | 4, 5 | wcel 2107 | . . . 4 wff 𝑔 ∈ (V × V) |
7 | c2nd 7446 | . . . . 5 class 2^{nd} | |
8 | 4, 7 | cfv 6137 | . . . 4 class (2^{nd} ‘𝑔) |
9 | cedgf 26341 | . . . . 5 class .ef | |
10 | 4, 9 | cfv 6137 | . . . 4 class (.ef‘𝑔) |
11 | 6, 8, 10 | cif 4307 | . . 3 class if(𝑔 ∈ (V × V), (2^{nd} ‘𝑔), (.ef‘𝑔)) |
12 | 2, 3, 11 | cmpt 4967 | . 2 class (𝑔 ∈ V ↦ if(𝑔 ∈ (V × V), (2^{nd} ‘𝑔), (.ef‘𝑔))) |
13 | 1, 12 | wceq 1601 | 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 26353 |
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