Definition df-vtx 16169

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 16167	. 2 class Vtx
2		vg	. . 3 setvar 𝑔
3		cvv 2821	. . 3 class V
4	2	cv 1401	. . . . 5 class 𝑔
5	3, 3	cxp 4767	. . . . 5 class (V × V)
6	4, 5	wcel 2209	. . . 4 wff 𝑔 ∈ (V × V)
7		c1st 6362	. . . . 5 class 1st
8	4, 7	cfv 5372	. . . 4 class (1st ‘𝑔)
9		cbs 13330	. . . . 5 class Base
10	4, 9	cfv 5372	. . . 4 class (Base‘𝑔)
11	6, 8, 10	cif 3635	. . 3 class if(𝑔 ∈ (V × V), (1st ‘𝑔), (Base‘𝑔))
12	2, 3, 11	cmpt 4187	. 2 class (𝑔 ∈ V ↦ if(𝑔 ∈ (V × V), (1st ‘𝑔), (Base‘𝑔)))
13	1, 12	wceq 1402	1 wff Vtx = (𝑔 ∈ V ↦ if(𝑔 ∈ (V × V), (1st ‘𝑔), (Base‘𝑔)))

This definition is referenced by:  vtxvalg  16171  1vgrex  16175  wlkreslem  16533  clwwlknonmpo  16583  trlsegvdegfi  16622  eupth2lem3lem1fi  16623  eupth2lem3lem2fi  16624  eupth2lem3lem6fi  16626  eupth2lem3lem4fi  16628