Theorem epini 4696

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 2560	. . . . 5 |- x e. _V
3	2	eliniseg 4695	. . . 4 |- ( A e. _V -> ( x e. ( `' _E " { A } ) <-> x _E A ) )
4	1, 3	ax-mp 7	. . 3 |- ( x e. ( `' _E " { A } ) <-> x _E A )
5	1	epelc 4028	. . 3 |- ( x _E A <-> x e. A )
6	4, 5	bitri 173	. 2 |- ( x e. ( `' _E " { A } ) <-> x e. A )
7	6	eqriv 2037	1 |- ( `' _E " { A } ) = A

Syntax hints:   <-> wb 98   = wceq 1243   e. wcel 1393   _Vcvv 2557   {csn 3375   class class class wbr 3764   _E cep 4024   `'ccnv 4344   "cima 4348

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-sbc 2765  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-br 3765  df-opab 3819  df-eprel 4026  df-xp 4351  df-cnv 4353  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358

This theorem is referenced by: (None)