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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 26798 | . 2 class Vtx | |
2 | vg | . . 3 setvar 𝑔 | |
3 | cvv 3480 | . . 3 class V | |
4 | 2 | cv 1537 | . . . . 5 class 𝑔 |
5 | 3, 3 | cxp 5541 | . . . . 5 class (V × V) |
6 | 4, 5 | wcel 2115 | . . . 4 wff 𝑔 ∈ (V × V) |
7 | c1st 7684 | . . . . 5 class 1^{st} | |
8 | 4, 7 | cfv 6345 | . . . 4 class (1^{st} ‘𝑔) |
9 | cbs 16485 | . . . . 5 class Base | |
10 | 4, 9 | cfv 6345 | . . . 4 class (Base‘𝑔) |
11 | 6, 8, 10 | cif 4450 | . . 3 class if(𝑔 ∈ (V × V), (1^{st} ‘𝑔), (Base‘𝑔)) |
12 | 2, 3, 11 | cmpt 5133 | . 2 class (𝑔 ∈ V ↦ if(𝑔 ∈ (V × V), (1^{st} ‘𝑔), (Base‘𝑔))) |
13 | 1, 12 | wceq 1538 | 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 26802 |
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