Theorem ecidg 6565

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 2729	. . . 4 |- y e. _V
2		elecg 6539	. . . 4 |- ( ( y e. _V /\ A e. V ) -> ( y e. [ A ] `' _E <-> A `' _E y ) )
3	1, 2	mpan 421	. . 3 |- ( A e. V -> ( y e. [ A ] `' _E <-> A `' _E y ) )
4		brcnvg 4785	. . . 4 |- ( ( A e. V /\ y e. _V ) -> ( A `' _E y <-> y _E A ) )
5	1, 4	mpan2 422	. . 3 |- ( A e. V -> ( A `' _E y <-> y _E A ) )
6		epelg 4268	. . 3 |- ( A e. V -> ( y _E A <-> y e. A ) )
7	3, 5, 6	3bitrd 213	. 2 |- ( A e. V -> ( y e. [ A ] `' _E <-> y e. A ) )
8	7	eqrdv 2163	1 |- ( A e. V -> [ A ] `' _E = A )

Syntax hints:   -> wi 4   <-> wb 104   = wceq 1343   e. wcel 2136   _Vcvv 2726   class class class wbr 3982   _E cep 4265   `'ccnv 4603   [cec 6499

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-sbc 2952  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-br 3983  df-opab 4044  df-eprel 4267  df-xp 4610  df-cnv 4612  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-ec 6503

This theorem is referenced by:  addcnsrec  7783  mulcnsrec  7784