Theorem epini 5099

Description: Any set is equal to its preimage under the converse epsilon relation. (Contributed by Mario Carneiro, 9-Mar-2013.)

Hypothesis

Ref	Expression
epini.1	|- A e. _V

Assertion

Ref	Expression
epini	|- ( `' _E " { A } ) = A

Proof of Theorem epini

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		epini.1	. . . 4 |- A e. _V
2		vex 2802	. . . . 5 |- x e. _V
3	2	eliniseg 5098	. . . 4 |- ( A e. _V -> ( x e. ( `' _E " { A } ) <-> x _E A ) )
4	1, 3	ax-mp 5	. . 3 |- ( x e. ( `' _E " { A } ) <-> x _E A )
5	1	epelc 4382	. . 3 |- ( x _E A <-> x e. A )
6	4, 5	bitri 184	. 2 |- ( x e. ( `' _E " { A } ) <-> x e. A )
7	6	eqriv 2226	1 |- ( `' _E " { A } ) = A

Syntax hints:   <-> wb 105   = wceq 1395   e. wcel 2200   _Vcvv 2799   {csn 3666   class class class wbr 4083   _E cep 4378   `'ccnv 4718   "cima 4722

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 4202  ax-pow 4258  ax-pr 4293

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 2801  df-sbc 3029  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-br 4084  df-opab 4146  df-eprel 4380  df-xp 4725  df-cnv 4727  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732

This theorem is referenced by: (None)