Theorem epini 5072

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 2779	. . . . 5 |- x e. _V
3	2	eliniseg 5071	. . . 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 4356	. . 3 |- ( x _E A <-> x e. A )
6	4, 5	bitri 184	. 2 |- ( x e. ( `' _E " { A } ) <-> x e. A )
7	6	eqriv 2204	1 |- ( `' _E " { A } ) = A

Syntax hints:   <-> wb 105   = wceq 1373   e. wcel 2178   _Vcvv 2776   {csn 3643   class class class wbr 4059   _E cep 4352   `'ccnv 4692   "cima 4696

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 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-14 2181  ax-ext 2189  ax-sep 4178  ax-pow 4234  ax-pr 4269

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ral 2491  df-rex 2492  df-v 2778  df-sbc 3006  df-un 3178  df-in 3180  df-ss 3187  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-br 4060  df-opab 4122  df-eprel 4354  df-xp 4699  df-cnv 4701  df-dm 4703  df-rn 4704  df-res 4705  df-ima 4706

This theorem is referenced by: (None)