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Theorem epini 5401
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 3176 . . . . 5 𝑥 ∈ V
32eliniseg 5400 . . . 4 (𝐴 ∈ V → (𝑥 ∈ ( E “ {𝐴}) ↔ 𝑥 E 𝐴))
41, 3ax-mp 5 . . 3 (𝑥 ∈ ( E “ {𝐴}) ↔ 𝑥 E 𝐴)
51epelc 4941 . . 3 (𝑥 E 𝐴𝑥𝐴)
64, 5bitri 263 . 2 (𝑥 ∈ ( E “ {𝐴}) ↔ 𝑥𝐴)
76eqriv 2607 1 ( E “ {𝐴}) = 𝐴
Colors of variables: wff setvar class
Syntax hints:  wb 195   = wceq 1475  wcel 1977  Vcvv 3173  {csn 4125   class class class wbr 4578   E cep 4937  ccnv 5027  cima 5031
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4704  ax-nul 4712  ax-pr 4828
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4579  df-opab 4639  df-eprel 4939  df-xp 5034  df-cnv 5036  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041
This theorem is referenced by:  infxpenlem  8697  fz1isolem  13057
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