MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  epin Structured version   Visualization version   GIF version

Theorem epin 6051
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 3451 . . . 4 𝑥 ∈ V
21eliniseg 6050 . . 3 (𝐴𝑉 → (𝑥 ∈ ( E “ {𝐴}) ↔ 𝑥 E 𝐴))
3 epelg 5542 . . 3 (𝐴𝑉 → (𝑥 E 𝐴𝑥𝐴))
42, 3bitrd 279 . 2 (𝐴𝑉 → (𝑥 ∈ ( E “ {𝐴}) ↔ 𝑥𝐴))
54eqrdv 2731 1 (𝐴𝑉 → ( E “ {𝐴}) = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1542  wcel 2107  {csn 4590   class class class wbr 5109   E cep 5540  ccnv 5636  cima 5640
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5260  ax-nul 5267  ax-pr 5388
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-sn 4591  df-pr 4593  df-op 4597  df-br 5110  df-opab 5172  df-eprel 5541  df-xp 5643  df-cnv 5645  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650
This theorem is referenced by:  epini  6052
  Copyright terms: Public domain W3C validator