Theorem ecid 6422

Description: A set is equal to its converse epsilon coset. (Note: converse epsilon is not an equivalence relation.) (Contributed by NM, 13-Aug-1995.) (Revised by Mario Carneiro, 9-Jul-2014.)

Hypothesis

Ref	Expression
ecid.1	⊢ 𝐴 ∈ V

Assertion

Ref	Expression
ecid	⊢ [𝐴]◡ E = 𝐴

Proof of Theorem ecid

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		vex 2644	. . . 4 ⊢ 𝑦 ∈ V
2		ecid.1	. . . 4 ⊢ 𝐴 ∈ V
3	1, 2	elec 6398	. . 3 ⊢ (𝑦 ∈ [𝐴]◡ E ↔ 𝐴◡ E 𝑦)
4	2, 1	brcnv 4660	. . 3 ⊢ (𝐴◡ E 𝑦 ↔ 𝑦 E 𝐴)
5	2	epelc 4151	. . 3 ⊢ (𝑦 E 𝐴 ↔ 𝑦 ∈ 𝐴)
6	3, 4, 5	3bitri 205	. 2 ⊢ (𝑦 ∈ [𝐴]◡ E ↔ 𝑦 ∈ 𝐴)
7	6	eqriv 2097	1 ⊢ [𝐴]◡ E = 𝐴

Syntax hints:   = wceq 1299   ∈ wcel 1448  Vcvv 2641   class class class wbr 3875   E cep 4147  ^◡ccnv 4476  [cec 6357

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-14 1460  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-sep 3986  ax-pow 4038  ax-pr 4069

This theorem depends on definitions:  df-bi 116  df-3an 932  df-tru 1302  df-nf 1405  df-sb 1704  df-eu 1963  df-mo 1964  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ral 2380  df-rex 2381  df-v 2643  df-sbc 2863  df-un 3025  df-in 3027  df-ss 3034  df-pw 3459  df-sn 3480  df-pr 3481  df-op 3483  df-br 3876  df-opab 3930  df-eprel 4149  df-xp 4483  df-cnv 4485  df-dm 4487  df-rn 4488  df-res 4489  df-ima 4490  df-ec 6361

This theorem is referenced by:  qsid  6424