Theorem ecidg 6767

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 2805	. . . 4 |- y e. _V
2		elecg 6741	. . . 4 |- ( ( y e. _V /\ A e. V ) -> ( y e. [ A ] `' _E <-> A `' _E y ) )
3	1, 2	mpan 424	. . 3 |- ( A e. V -> ( y e. [ A ] `' _E <-> A `' _E y ) )
4		brcnvg 4911	. . . 4 |- ( ( A e. V /\ y e. _V ) -> ( A `' _E y <-> y _E A ) )
5	1, 4	mpan2 425	. . 3 |- ( A e. V -> ( A `' _E y <-> y _E A ) )
6		epelg 4387	. . 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 2229	1 |- ( A e. V -> [ A ] `' _E = A )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1397   e. wcel 2202   _Vcvv 2802   class class class wbr 4088   _E cep 4384   `'ccnv 4724   [cec 6699

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-v 2804  df-sbc 3032  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-br 4089  df-opab 4151  df-eprel 4386  df-xp 4731  df-cnv 4733  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-ec 6703

This theorem is referenced by:  addcnsrec  8061  mulcnsrec  8062