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Theorem epini 4639
 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 2554 . . . . 5 x V
32eliniseg 4638 . . . 4 (A V → (x ( E “ {A}) ↔ x E A))
41, 3ax-mp 7 . . 3 (x ( E “ {A}) ↔ x E A)
51epelc 4019 . . 3 (x E Ax A)
64, 5bitri 173 . 2 (x ( E “ {A}) ↔ x A)
76eqriv 2034 1 ( E “ {A}) = A
 Colors of variables: wff set class Syntax hints:   ↔ wb 98   = wceq 1242   ∈ wcel 1390  Vcvv 2551  {csn 3367   class class class wbr 3755   E cep 4015  ◡ccnv 4287   “ cima 4291 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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935 This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-sbc 2759  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-opab 3810  df-eprel 4017  df-xp 4294  df-cnv 4296  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301 This theorem is referenced by: (None)
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