Theorem epini 5050

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 2774	. . . . 5 |- x e. _V
3	2	eliniseg 5049	. . . 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 4336	. . 3 |- ( x _E A <-> x e. A )
6	4, 5	bitri 184	. 2 |- ( x e. ( `' _E " { A } ) <-> x e. A )
7	6	eqriv 2201	1 |- ( `' _E " { A } ) = A

Syntax hints:   <-> wb 105   = wceq 1372   e. wcel 2175   _Vcvv 2771   {csn 3632   class class class wbr 4043   _E cep 4332   `'ccnv 4672   "cima 4676

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 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-14 2178  ax-ext 2186  ax-sep 4161  ax-pow 4217  ax-pr 4252

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1375  df-nf 1483  df-sb 1785  df-eu 2056  df-mo 2057  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ral 2488  df-rex 2489  df-v 2773  df-sbc 2998  df-un 3169  df-in 3171  df-ss 3178  df-pw 3617  df-sn 3638  df-pr 3639  df-op 3641  df-br 4044  df-opab 4105  df-eprel 4334  df-xp 4679  df-cnv 4681  df-dm 4683  df-rn 4684  df-res 4685  df-ima 4686

This theorem is referenced by: (None)