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Theorem epin 6104
Description: Any set is equal to its preimage under the converse membership relation. (Contributed by Mario Carneiro, 9-Mar-2013.) Put in closed form. (Revised by BJ, 16-Oct-2024.)
Assertion
Ref Expression
epin (𝐴𝑉 → ( E “ {𝐴}) = 𝐴)

Proof of Theorem epin
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 vex 3477 . . . 4 𝑥 ∈ V
21eliniseg 6103 . . 3 (𝐴𝑉 → (𝑥 ∈ ( E “ {𝐴}) ↔ 𝑥 E 𝐴))
3 epelg 5587 . . 3 (𝐴𝑉 → (𝑥 E 𝐴𝑥𝐴))
42, 3bitrd 278 . 2 (𝐴𝑉 → (𝑥 ∈ ( E “ {𝐴}) ↔ 𝑥𝐴))
54eqrdv 2726 1 (𝐴𝑉 → ( E “ {𝐴}) = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1533  wcel 2098  {csn 4632   class class class wbr 5152   E cep 5585  ccnv 5681  cima 5685
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pr 5433
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2706  df-cleq 2720  df-clel 2806  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-br 5153  df-opab 5215  df-eprel 5586  df-xp 5688  df-cnv 5690  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695
This theorem is referenced by:  epini  6105
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