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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 ∈ 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 2560 . . . . 5 𝑥 ∈ V
32eliniseg 4695 . . . 4 (𝐴 ∈ V → (𝑥 ∈ ( E “ {𝐴}) ↔ 𝑥 E 𝐴))
41, 3ax-mp 7 . . 3 (𝑥 ∈ ( E “ {𝐴}) ↔ 𝑥 E 𝐴)
51epelc 4028 . . 3 (𝑥 E 𝐴𝑥𝐴)
64, 5bitri 173 . 2 (𝑥 ∈ ( E “ {𝐴}) ↔ 𝑥𝐴)
76eqriv 2037 1 ( E “ {𝐴}) = 𝐴
 Colors of variables: wff set class Syntax hints:   ↔ wb 98   = wceq 1243   ∈ 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)
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