Theorem ecidg 6716

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 2782	. . . 4 ⊢ 𝑦 ∈ V
2		elecg 6690	. . . 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 4880	. . . 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 4358	. . 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 2207	1 ⊢ (𝐴 ∈ 𝑉 → [𝐴]◡ E = 𝐴)

Syntax hints:   → wi 4   ↔ wb 105   = wceq 1375   ∈ wcel 2180  Vcvv 2779   class class class wbr 4062   E cep 4355  ^◡ccnv 4695  [cec 6648

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 713  ax-5 1473  ax-7 1474  ax-gen 1475  ax-ie1 1519  ax-ie2 1520  ax-8 1530  ax-10 1531  ax-11 1532  ax-i12 1533  ax-bndl 1535  ax-4 1536  ax-17 1552  ax-i9 1556  ax-ial 1560  ax-i5r 1561  ax-14 2183  ax-ext 2191  ax-sep 4181  ax-pow 4237  ax-pr 4272

This theorem depends on definitions:  df-bi 117  df-3an 985  df-tru 1378  df-nf 1487  df-sb 1789  df-eu 2060  df-mo 2061  df-clab 2196  df-cleq 2202  df-clel 2205  df-nfc 2341  df-ral 2493  df-rex 2494  df-v 2781  df-sbc 3009  df-un 3181  df-in 3183  df-ss 3190  df-pw 3631  df-sn 3652  df-pr 3653  df-op 3655  df-br 4063  df-opab 4125  df-eprel 4357  df-xp 4702  df-cnv 4704  df-dm 4706  df-rn 4707  df-res 4708  df-ima 4709  df-ec 6652

This theorem is referenced by:  addcnsrec  7997  mulcnsrec  7998