Theorem ecidg 6589

Description: A set is equal to its converse epsilon coset. (Note: converse epsilon is not an equivalence relation.) (Contributed by Jim Kingdon, 8-Jan-2020.)

Assertion

Ref	Expression
ecidg	⊢ (𝐴 ∈ 𝑉 → [𝐴]◡ E = 𝐴)

Proof of Theorem ecidg

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		vex 2738	. . . 4 ⊢ 𝑦 ∈ V
2		elecg 6563	. . . 4 ⊢ ((𝑦 ∈ V ∧ 𝐴 ∈ 𝑉) → (𝑦 ∈ [𝐴]◡ E ↔ 𝐴◡ E 𝑦))
3	1, 2	mpan 424	. . 3 ⊢ (𝐴 ∈ 𝑉 → (𝑦 ∈ [𝐴]◡ E ↔ 𝐴◡ E 𝑦))
4		brcnvg 4801	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑦 ∈ V) → (𝐴◡ E 𝑦 ↔ 𝑦 E 𝐴))
5	1, 4	mpan2 425	. . 3 ⊢ (𝐴 ∈ 𝑉 → (𝐴◡ E 𝑦 ↔ 𝑦 E 𝐴))
6		epelg 4284	. . 3 ⊢ (𝐴 ∈ 𝑉 → (𝑦 E 𝐴 ↔ 𝑦 ∈ 𝐴))
7	3, 5, 6	3bitrd 214	. 2 ⊢ (𝐴 ∈ 𝑉 → (𝑦 ∈ [𝐴]◡ E ↔ 𝑦 ∈ 𝐴))
8	7	eqrdv 2173	1 ⊢ (𝐴 ∈ 𝑉 → [𝐴]◡ E = 𝐴)

Syntax hints:   → wi 4   ↔ wb 105   = wceq 1353   ∈ wcel 2146  Vcvv 2735   class class class wbr 3998   E cep 4281  ^◡ccnv 4619  [cec 6523

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 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-14 2149  ax-ext 2157  ax-sep 4116  ax-pow 4169  ax-pr 4203

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1459  df-sb 1761  df-eu 2027  df-mo 2028  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ral 2458  df-rex 2459  df-v 2737  df-sbc 2961  df-un 3131  df-in 3133  df-ss 3140  df-pw 3574  df-sn 3595  df-pr 3596  df-op 3598  df-br 3999  df-opab 4060  df-eprel 4283  df-xp 4626  df-cnv 4628  df-dm 4630  df-rn 4631  df-res 4632  df-ima 4633  df-ec 6527

This theorem is referenced by:  addcnsrec  7816  mulcnsrec  7817