Definition df-vtx 28974

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.)

Assertion

Ref	Expression
df-vtx	⊢ Vtx = (𝑔 ∈ V ↦ if(𝑔 ∈ (V × V), (1st ‘𝑔), (Base‘𝑔)))

Detailed syntax breakdown of Definition df-vtx

Step	Hyp	Ref	Expression
1		cvtx 28972	. 2 class Vtx
2		vg	. . 3 setvar 𝑔
3		cvv 3436	. . 3 class V
4	2	cv 1540	. . . . 5 class 𝑔
5	3, 3	cxp 5614	. . . . 5 class (V × V)
6	4, 5	wcel 2111	. . . 4 wff 𝑔 ∈ (V × V)
7		c1st 7919	. . . . 5 class 1st
8	4, 7	cfv 6481	. . . 4 class (1st ‘𝑔)
9		cbs 17117	. . . . 5 class Base
10	4, 9	cfv 6481	. . . 4 class (Base‘𝑔)
11	6, 8, 10	cif 4475	. . 3 class if(𝑔 ∈ (V × V), (1st ‘𝑔), (Base‘𝑔))
12	2, 3, 11	cmpt 5172	. 2 class (𝑔 ∈ V ↦ if(𝑔 ∈ (V × V), (1st ‘𝑔), (Base‘𝑔)))
13	1, 12	wceq 1541	1 wff Vtx = (𝑔 ∈ V ↦ if(𝑔 ∈ (V × V), (1st ‘𝑔), (Base‘𝑔)))

This definition is referenced by:  vtxval  28976