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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), (1^{st} ‘𝑔), (Base‘𝑔))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cvtx 26484 | . 2 class Vtx | |
2 | vg | . . 3 setvar 𝑔 | |
3 | cvv 3415 | . . 3 class V | |
4 | 2 | cv 1506 | . . . . 5 class 𝑔 |
5 | 3, 3 | cxp 5405 | . . . . 5 class (V × V) |
6 | 4, 5 | wcel 2050 | . . . 4 wff 𝑔 ∈ (V × V) |
7 | c1st 7499 | . . . . 5 class 1^{st} | |
8 | 4, 7 | cfv 6188 | . . . 4 class (1^{st} ‘𝑔) |
9 | cbs 16339 | . . . . 5 class Base | |
10 | 4, 9 | cfv 6188 | . . . 4 class (Base‘𝑔) |
11 | 6, 8, 10 | cif 4350 | . . 3 class if(𝑔 ∈ (V × V), (1^{st} ‘𝑔), (Base‘𝑔)) |
12 | 2, 3, 11 | cmpt 5008 | . 2 class (𝑔 ∈ V ↦ if(𝑔 ∈ (V × V), (1^{st} ‘𝑔), (Base‘𝑔))) |
13 | 1, 12 | wceq 1507 | 1 wff Vtx = (𝑔 ∈ V ↦ if(𝑔 ∈ (V × V), (1^{st} ‘𝑔), (Base‘𝑔))) |
Colors of variables: wff setvar class |
This definition is referenced by: vtxval 26488 |
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