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Theorem epini 5926
Description: Any set is equal to its preimage under the converse membership 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 3444 . . . . 5 𝑥 ∈ V
32eliniseg 5925 . . . 4 (𝐴 ∈ V → (𝑥 ∈ ( E “ {𝐴}) ↔ 𝑥 E 𝐴))
41, 3ax-mp 5 . . 3 (𝑥 ∈ ( E “ {𝐴}) ↔ 𝑥 E 𝐴)
51epeli 5432 . . 3 (𝑥 E 𝐴𝑥𝐴)
64, 5bitri 278 . 2 (𝑥 ∈ ( E “ {𝐴}) ↔ 𝑥𝐴)
76eqriv 2795 1 ( E “ {𝐴}) = 𝐴
Colors of variables: wff setvar class
Syntax hints:  wb 209   = wceq 1538  wcel 2111  Vcvv 3441  {csn 4525   class class class wbr 5030   E cep 5429  ccnv 5518  cima 5522
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pr 5295
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-br 5031  df-opab 5093  df-eprel 5430  df-xp 5525  df-cnv 5527  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532
This theorem is referenced by:  infxpenlem  9424  fz1isolem  13815
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