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Mirrors > Home > MPE Home > Th. List > df-vtx | Structured version Visualization version GIF version |
Description: Define the function mapping a graph to the set of its vertices. This definition is very general: It defines the set of vertices for any ordered pair as its first component, and for any other class as its "base set". It is meaningful, however, only if the ordered pair represents a graph resp. the class is an extensible structure representing a graph. (Contributed by AV, 9-Jan-2020.) (Revised by AV, 20-Sep-2020.) |
Ref | Expression |
---|---|
df-vtx | ⊢ Vtx = (𝑔 ∈ V ↦ if(𝑔 ∈ (V × V), (1st ‘𝑔), (Base‘𝑔))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cvtx 28881 | . 2 class Vtx | |
2 | vg | . . 3 setvar 𝑔 | |
3 | cvv 3461 | . . 3 class V | |
4 | 2 | cv 1532 | . . . . 5 class 𝑔 |
5 | 3, 3 | cxp 5676 | . . . . 5 class (V × V) |
6 | 4, 5 | wcel 2098 | . . . 4 wff 𝑔 ∈ (V × V) |
7 | c1st 7992 | . . . . 5 class 1st | |
8 | 4, 7 | cfv 6549 | . . . 4 class (1st ‘𝑔) |
9 | cbs 17183 | . . . . 5 class Base | |
10 | 4, 9 | cfv 6549 | . . . 4 class (Base‘𝑔) |
11 | 6, 8, 10 | cif 4530 | . . 3 class if(𝑔 ∈ (V × V), (1st ‘𝑔), (Base‘𝑔)) |
12 | 2, 3, 11 | cmpt 5232 | . 2 class (𝑔 ∈ V ↦ if(𝑔 ∈ (V × V), (1st ‘𝑔), (Base‘𝑔))) |
13 | 1, 12 | wceq 1533 | 1 wff Vtx = (𝑔 ∈ V ↦ if(𝑔 ∈ (V × V), (1st ‘𝑔), (Base‘𝑔))) |
Colors of variables: wff setvar class |
This definition is referenced by: vtxval 28885 |
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