Theorem ecidg 6867

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 2824	. . . 4 ⊢ 𝑦 ∈ V
2		elecg 6841	. . . 4 ⊢ ((𝑦 ∈ V ∧ 𝐴 ∈ 𝑉) → (𝑦 ∈ [𝐴]◡ E ↔ 𝐴◡ E 𝑦))
3	1, 2	mpan 428	. . 3 ⊢ (𝐴 ∈ 𝑉 → (𝑦 ∈ [𝐴]◡ E ↔ 𝐴◡ E 𝑦))
4		brcnvg 4959	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑦 ∈ V) → (𝐴◡ E 𝑦 ↔ 𝑦 E 𝐴))
5	1, 4	mpan2 429	. . 3 ⊢ (𝐴 ∈ 𝑉 → (𝐴◡ E 𝑦 ↔ 𝑦 E 𝐴))
6		epelg 4433	. . 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 2236	1 ⊢ (𝐴 ∈ 𝑉 → [𝐴]◡ E = 𝐴)

Syntax hints:   → wi 4   ↔ wb 105   = wceq 1402   ∈ wcel 2209  Vcvv 2821   class class class wbr 4128   E cep 4430  ^◡ccnv 4771  [cec 6799

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4247  ax-pow 4309  ax-pr 4344

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-sbc 3052  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3714  df-pr 3715  df-op 3717  df-br 4129  df-opab 4191  df-eprel 4432  df-xp 4778  df-cnv 4780  df-dm 4782  df-rn 4783  df-res 4784  df-ima 4785  df-ec 6803

This theorem is referenced by:  addcnsrec  8203  mulcnsrec  8204