Users' Mathboxes Mathbox for Peter Mazsa < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  eccnvep Structured version   Visualization version   GIF version

Theorem eccnvep 37758
Description: The converse epsilon coset of a set is the set. (Contributed by Peter Mazsa, 27-Jan-2019.)
Assertion
Ref Expression
eccnvep (𝐴𝑉 → [𝐴] E = 𝐴)

Proof of Theorem eccnvep
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 eleccnvep 37757 . 2 (𝐴𝑉 → (𝑥 ∈ [𝐴] E ↔ 𝑥𝐴))
21eqrdv 2725 1 (𝐴𝑉 → [𝐴] E = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1533  wcel 2098   E cep 5583  ccnv 5679  [cec 8727
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-10 2129  ax-11 2146  ax-12 2166  ax-ext 2698  ax-sep 5301  ax-nul 5308  ax-pr 5431
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-nf 1778  df-sb 2060  df-clab 2705  df-cleq 2719  df-clel 2805  df-ne 2937  df-ral 3058  df-rex 3067  df-rab 3429  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4325  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-br 5151  df-opab 5213  df-eprel 5584  df-xp 5686  df-rel 5687  df-cnv 5688  df-dm 5690  df-rn 5691  df-res 5692  df-ima 5693  df-ec 8731
This theorem is referenced by:  extep  37759  disjeccnvep  37760  eccnvepres2  37761  dfeldisj5  38197
  Copyright terms: Public domain W3C validator