Theorem ecidg 6763

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 2803	. . . 4 |- y e. _V
2		elecg 6737	. . . 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 4909	. . . 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 4385	. . 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 2227	1 |- ( A e. V -> [ A ] `' _E = A )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1395   e. wcel 2200   _Vcvv 2800   class class class wbr 4086   _E cep 4382   `'ccnv 4722   [cec 6695

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4205  ax-pow 4262  ax-pr 4297

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2802  df-sbc 3030  df-un 3202  df-in 3204  df-ss 3211  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-br 4087  df-opab 4149  df-eprel 4384  df-xp 4729  df-cnv 4731  df-dm 4733  df-rn 4734  df-res 4735  df-ima 4736  df-ec 6699

This theorem is referenced by:  addcnsrec  8052  mulcnsrec  8053