Theorem ecid 6446

Description: A set is equal to its converse epsilon coset. (Note: converse epsilon is not an equivalence relation.) (Contributed by NM, 13-Aug-1995.) (Revised by Mario Carneiro, 9-Jul-2014.)

Hypothesis

Ref	Expression
ecid.1	|- A e. _V

Assertion

Ref	Expression
ecid	|- [ A ] `' _E = A

Proof of Theorem ecid

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		vex 2660	. . . 4 |- y e. _V
2		ecid.1	. . . 4 |- A e. _V
3	1, 2	elec 6422	. . 3 |- ( y e. [ A ] `' _E <-> A `' _E y )
4	2, 1	brcnv 4682	. . 3 |- ( A `' _E y <-> y _E A )
5	2	epelc 4173	. . 3 |- ( y _E A <-> y e. A )
6	3, 4, 5	3bitri 205	. 2 |- ( y e. [ A ] `' _E <-> y e. A )
7	6	eqriv 2112	1 |- [ A ] `' _E = A

Syntax hints:   = wceq 1314   e. wcel 1463   _Vcvv 2657   class class class wbr 3895   _E cep 4169   `'ccnv 4498   [cec 6381

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4006  ax-pow 4058  ax-pr 4091

This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-ral 2395  df-rex 2396  df-v 2659  df-sbc 2879  df-un 3041  df-in 3043  df-ss 3050  df-pw 3478  df-sn 3499  df-pr 3500  df-op 3502  df-br 3896  df-opab 3950  df-eprel 4171  df-xp 4505  df-cnv 4507  df-dm 4509  df-rn 4510  df-res 4511  df-ima 4512  df-ec 6385

This theorem is referenced by:  qsid  6448