Definition df-member 36705

Description: Define the membership equivalence relation on the class 𝐴 (or, the restricted elementhood equivalence relation on its domain quotient 𝐴.) Alternate definitions are dfmember2 36712 and dfmember3 36713.

Later on, in an application of set theory I make a distinction between the default elementhood concept and a special membership concept: membership equivalence relation will be an integral part of that membership concept. (Contributed by Peter Mazsa, 26-Jun-2021.) (Revised by Peter Mazsa, 28-Nov-2022.)

Assertion

Ref	Expression
df-member	⊢ ( MembEr 𝐴 ↔ ≀ (◡ E ↾ 𝐴) ErALTV 𝐴)

Detailed syntax breakdown of Definition df-member

Step	Hyp	Ref	Expression
1		cA	. . 3 class 𝐴
2	1	wmember 36288	. 2 wff MembEr 𝐴
3		cep 5485	. . . . . 6 class E
4	3	ccnv 5579	. . . . 5 class ◡ E
5	4, 1	cres 5582	. . . 4 class (◡ E ↾ 𝐴)
6	5	ccoss 36260	. . 3 class ≀ (◡ E ↾ 𝐴)
7	1, 6	werALTV 36286	. 2 wff ≀ (◡ E ↾ 𝐴) ErALTV 𝐴
8	2, 7	wb 205	1 wff ( MembEr 𝐴 ↔ ≀ (◡ E ↾ 𝐴) ErALTV 𝐴)

This definition is referenced by:  dfmember  36711