Theorem ecidg 6863

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 e. 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 e. _V
2		elecg 6837	. . . 4 |- ( ( y e. _V /\ A e. V ) -> ( y e. [ A ] `' _E <-> A `' _E y ) )
3	1, 2	mpan 428	. . 3 |- ( A e. V -> ( y e. [ A ] `' _E <-> A `' _E y ) )
4		brcnvg 4956	. . . 4 |- ( ( A e. V /\ y e. _V ) -> ( A `' _E y <-> y _E A ) )
5	1, 4	mpan2 429	. . 3 |- ( A e. V -> ( A `' _E y <-> y _E A ) )
6		epelg 4430	. . 3 |- ( A e. V -> ( y _E A <-> y e. A ) )
7	3, 5, 6	3bitrd 214	. 2 |- ( A e. V -> ( y e. [ A ] `' _E <-> y e. A ) )
8	7	eqrdv 2236	1 |- ( A e. V -> [ A ] `' _E = A )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1402   e. wcel 2209   _Vcvv 2821   class class class wbr 4125   _E cep 4427   `'ccnv 4768   [cec 6795

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 4244  ax-pow 4306  ax-pr 4341

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 3687  df-sn 3711  df-pr 3712  df-op 3714  df-br 4126  df-opab 4188  df-eprel 4429  df-xp 4775  df-cnv 4777  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-ec 6799

This theorem is referenced by:  addcnsrec  8199  mulcnsrec  8200