Theorem ecid 6685

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 2775	. . . 4 |- y e. _V
2		ecid.1	. . . 4 |- A e. _V
3	1, 2	elec 6661	. . 3 |- ( y e. [ A ] `' _E <-> A `' _E y )
4	2, 1	brcnv 4861	. . 3 |- ( A `' _E y <-> y _E A )
5	2	epelc 4338	. . 3 |- ( y _E A <-> y e. A )
6	3, 4, 5	3bitri 206	. 2 |- ( y e. [ A ] `' _E <-> y e. A )
7	6	eqriv 2202	1 |- [ A ] `' _E = A

Syntax hints:   = wceq 1373   e. wcel 2176   _Vcvv 2772   class class class wbr 4044   _E cep 4334   `'ccnv 4674   [cec 6618

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-14 2179  ax-ext 2187  ax-sep 4162  ax-pow 4218  ax-pr 4253

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-rex 2490  df-v 2774  df-sbc 2999  df-un 3170  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-br 4045  df-opab 4106  df-eprel 4336  df-xp 4681  df-cnv 4683  df-dm 4685  df-rn 4686  df-res 4687  df-ima 4688  df-ec 6622

This theorem is referenced by:  qsid  6687