Definition df-vtx 16477

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 16475	. 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 4774	. . . . 5 class (V × V)
6	4, 5	wcel 2209	. . . 4 wff 𝑔 ∈ (V × V)
7		c1st 6376	. . . . 5 class 1st
8	4, 7	cfv 5380	. . . 4 class (1st ‘𝑔)
9		cbs 13409	. . . . 5 class Base
10	4, 9	cfv 5380	. . . 4 class (Base‘𝑔)
11	6, 8, 10	cif 3638	. . 3 class if(𝑔 ∈ (V × V), (1st ‘𝑔), (Base‘𝑔))
12	2, 3, 11	cmpt 4193	. 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 used by:  vtxvalg  16479  1vgrex  16483  wlkreslem  16841  clwwlknonmpo  16891  trlsegvdegfi  16930  eupth2lem3lem1fi  16931  eupth2lem3lem2fi  16932  eupth2lem3lem6fi  16934  eupth2lem3lem4fi  16936