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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 26708 | . 2 class Vtx | |
2 | vg | . . 3 setvar 𝑔 | |
3 | cvv 3492 | . . 3 class V | |
4 | 2 | cv 1527 | . . . . 5 class 𝑔 |
5 | 3, 3 | cxp 5546 | . . . . 5 class (V × V) |
6 | 4, 5 | wcel 2105 | . . . 4 wff 𝑔 ∈ (V × V) |
7 | c1st 7676 | . . . . 5 class 1st | |
8 | 4, 7 | cfv 6348 | . . . 4 class (1st ‘𝑔) |
9 | cbs 16471 | . . . . 5 class Base | |
10 | 4, 9 | cfv 6348 | . . . 4 class (Base‘𝑔) |
11 | 6, 8, 10 | cif 4463 | . . 3 class if(𝑔 ∈ (V × V), (1st ‘𝑔), (Base‘𝑔)) |
12 | 2, 3, 11 | cmpt 5137 | . 2 class (𝑔 ∈ V ↦ if(𝑔 ∈ (V × V), (1st ‘𝑔), (Base‘𝑔))) |
13 | 1, 12 | wceq 1528 | 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 26712 |
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