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Theorem epini 4623
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 V
Assertion
Ref Expression
epini ( E “ {A}) = A

Proof of Theorem epini
Dummy variable x is distinct from all other variables.
StepHypRef Expression
1 epini.1 . . . 4 A V
2 vex 2538 . . . . 5 x V
32eliniseg 4622 . . . 4 (A V → (x ( E “ {A}) ↔ x E A))
41, 3ax-mp 7 . . 3 (x ( E “ {A}) ↔ x E A)
51epelc 4002 . . 3 (x E Ax A)
64, 5bitri 173 . 2 (x ( E “ {A}) ↔ x A)
76eqriv 2019 1 ( E “ {A}) = A
Colors of variables: wff set class
Syntax hints:  wb 98   = wceq 1228   wcel 1374  Vcvv 2535  {csn 3350   class class class wbr 3738   E cep 3998  ccnv 4271  cima 4275
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 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-sep 3849  ax-pow 3901  ax-pr 3918
This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-nf 1330  df-sb 1628  df-eu 1885  df-mo 1886  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2289  df-rex 2290  df-v 2537  df-sbc 2742  df-un 2899  df-in 2901  df-ss 2908  df-pw 3336  df-sn 3356  df-pr 3357  df-op 3359  df-br 3739  df-opab 3793  df-eprel 4000  df-xp 4278  df-cnv 4280  df-dm 4282  df-rn 4283  df-res 4284  df-ima 4285
This theorem is referenced by: (None)
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