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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 25775 | . 2 class iEdg | |
2 | vg | . . 3 setvar 𝑔 | |
3 | cvv 3186 | . . 3 class V | |
4 | 2 | cv 1479 | . . . . 5 class 𝑔 |
5 | 3, 3 | cxp 5072 | . . . . 5 class (V × V) |
6 | 4, 5 | wcel 1987 | . . . 4 wff 𝑔 ∈ (V × V) |
7 | c2nd 7112 | . . . . 5 class 2^{nd} | |
8 | 4, 7 | cfv 5847 | . . . 4 class (2^{nd} ‘𝑔) |
9 | cedgf 25767 | . . . . 5 class .ef | |
10 | 4, 9 | cfv 5847 | . . . 4 class (.ef‘𝑔) |
11 | 6, 8, 10 | cif 4058 | . . 3 class if(𝑔 ∈ (V × V), (2^{nd} ‘𝑔), (.ef‘𝑔)) |
12 | 2, 3, 11 | cmpt 4673 | . 2 class (𝑔 ∈ V ↦ if(𝑔 ∈ (V × V), (2^{nd} ‘𝑔), (.ef‘𝑔))) |
13 | 1, 12 | wceq 1480 | 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 25779 iedgvalOLD 25781 |
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