Theorem ecid 6652

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 2763	. . . 4 ⊢ 𝑦 ∈ V
2		ecid.1	. . . 4 ⊢ 𝐴 ∈ V
3	1, 2	elec 6628	. . 3 ⊢ (𝑦 ∈ [𝐴]◡ E ↔ 𝐴◡ E 𝑦)
4	2, 1	brcnv 4845	. . 3 ⊢ (𝐴◡ E 𝑦 ↔ 𝑦 E 𝐴)
5	2	epelc 4322	. . 3 ⊢ (𝑦 E 𝐴 ↔ 𝑦 ∈ 𝐴)
6	3, 4, 5	3bitri 206	. 2 ⊢ (𝑦 ∈ [𝐴]◡ E ↔ 𝑦 ∈ 𝐴)
7	6	eqriv 2190	1 ⊢ [𝐴]◡ E = 𝐴

Syntax hints:   = wceq 1364   ∈ wcel 2164  Vcvv 2760   class class class wbr 4029   E cep 4318  ^◡ccnv 4658  [cec 6585

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-sbc 2986  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-br 4030  df-opab 4091  df-eprel 4320  df-xp 4665  df-cnv 4667  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-ec 6589

This theorem is referenced by:  qsid  6654