Theorem ecidg 6577

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 2733	. . . 4 |- y e. _V
2		elecg 6551	. . . 4 |- ( ( y e. _V /\ A e. V ) -> ( y e. [ A ] `' _E <-> A `' _E y ) )
3	1, 2	mpan 422	. . 3 |- ( A e. V -> ( y e. [ A ] `' _E <-> A `' _E y ) )
4		brcnvg 4792	. . . 4 |- ( ( A e. V /\ y e. _V ) -> ( A `' _E y <-> y _E A ) )
5	1, 4	mpan2 423	. . 3 |- ( A e. V -> ( A `' _E y <-> y _E A ) )
6		epelg 4275	. . 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 2168	1 |- ( A e. V -> [ A ] `' _E = A )

Syntax hints:   -> wi 4   <-> wb 104   = wceq 1348   e. wcel 2141   _Vcvv 2730   class class class wbr 3989   _E cep 4272   `'ccnv 4610   [cec 6511

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-sbc 2956  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-br 3990  df-opab 4051  df-eprel 4274  df-xp 4617  df-cnv 4619  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-ec 6515

This theorem is referenced by:  addcnsrec  7804  mulcnsrec  7805