Theorem epini 4997

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 2740	. . . . 5 |- x e. _V
3	2	eliniseg 4996	. . . 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 4290	. . 3 |- ( x _E A <-> x e. A )
6	4, 5	bitri 184	. 2 |- ( x e. ( `' _E " { A } ) <-> x e. A )
7	6	eqriv 2174	1 |- ( `' _E " { A } ) = A

Syntax hints:   <-> wb 105   = wceq 1353   e. wcel 2148   _Vcvv 2737   {csn 3592   class class class wbr 4002   _E cep 4286   `'ccnv 4624   "cima 4628

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4120  ax-pow 4173  ax-pr 4208

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-sbc 2963  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-br 4003  df-opab 4064  df-eprel 4288  df-xp 4631  df-cnv 4633  df-dm 4635  df-rn 4636  df-res 4637  df-ima 4638

This theorem is referenced by: (None)