ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  epini GIF version

Theorem epini 4880
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 2663 . . . . 5 𝑥 ∈ V
32eliniseg 4879 . . . 4 (𝐴 ∈ V → (𝑥 ∈ ( E “ {𝐴}) ↔ 𝑥 E 𝐴))
41, 3ax-mp 5 . . 3 (𝑥 ∈ ( E “ {𝐴}) ↔ 𝑥 E 𝐴)
51epelc 4183 . . 3 (𝑥 E 𝐴𝑥𝐴)
64, 5bitri 183 . 2 (𝑥 ∈ ( E “ {𝐴}) ↔ 𝑥𝐴)
76eqriv 2114 1 ( E “ {𝐴}) = 𝐴
Colors of variables: wff set class
Syntax hints:  wb 104   = wceq 1316  wcel 1465  Vcvv 2660  {csn 3497   class class class wbr 3899   E cep 4179  ccnv 4508  cima 4512
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-v 2662  df-sbc 2883  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-br 3900  df-opab 3960  df-eprel 4181  df-xp 4515  df-cnv 4517  df-dm 4519  df-rn 4520  df-res 4521  df-ima 4522
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator