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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 25673 | . 2 class Vtx | |
2 | vg | . . 3 setvar 𝑔 | |
3 | cvv 3173 | . . 3 class V | |
4 | 2 | cv 1474 | . . . . 5 class 𝑔 |
5 | 3, 3 | cxp 5036 | . . . . 5 class (V × V) |
6 | 4, 5 | wcel 1977 | . . . 4 wff 𝑔 ∈ (V × V) |
7 | c1st 7057 | . . . . 5 class 1^{st} | |
8 | 4, 7 | cfv 5804 | . . . 4 class (1^{st} ‘𝑔) |
9 | cbs 15695 | . . . . 5 class Base | |
10 | 4, 9 | cfv 5804 | . . . 4 class (Base‘𝑔) |
11 | 6, 8, 10 | cif 4036 | . . 3 class if(𝑔 ∈ (V × V), (1^{st} ‘𝑔), (Base‘𝑔)) |
12 | 2, 3, 11 | cmpt 4643 | . 2 class (𝑔 ∈ V ↦ if(𝑔 ∈ (V × V), (1^{st} ‘𝑔), (Base‘𝑔))) |
13 | 1, 12 | wceq 1475 | 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 25677 |
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