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

Theorem eccnvepres 38184
Description: Restricted converse epsilon coset of 𝐵. (Contributed by Peter Mazsa, 11-Feb-2018.) (Revised by Peter Mazsa, 21-Oct-2021.)
Assertion
Ref Expression
eccnvepres (𝐵𝑉 → [𝐵]( E ↾ 𝐴) = {𝑥𝐵𝐵𝐴})
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝑉

Proof of Theorem eccnvepres
StepHypRef Expression
1 brcnvep 38169 . . . 4 (𝐵𝑉 → (𝐵 E 𝑥𝑥𝐵))
21anbi1cd 634 . . 3 (𝐵𝑉 → ((𝐵𝐴𝐵 E 𝑥) ↔ (𝑥𝐵𝐵𝐴)))
32abbidv 2805 . 2 (𝐵𝑉 → {𝑥 ∣ (𝐵𝐴𝐵 E 𝑥)} = {𝑥 ∣ (𝑥𝐵𝐵𝐴)})
4 ecres 38182 . 2 [𝐵]( E ↾ 𝐴) = {𝑥 ∣ (𝐵𝐴𝐵 E 𝑥)}
5 df-rab 3439 . 2 {𝑥𝐵𝐵𝐴} = {𝑥 ∣ (𝑥𝐵𝐵𝐴)}
63, 4, 53eqtr4g 2799 1 (𝐵𝑉 → [𝐵]( E ↾ 𝐴) = {𝑥𝐵𝐵𝐴})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wcel 2103  {cab 2711  {crab 3438   class class class wbr 5169   E cep 5602  ccnv 5698  cres 5701  [cec 8757
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2105  ax-9 2113  ax-10 2136  ax-11 2153  ax-12 2173  ax-ext 2705  ax-sep 5320  ax-nul 5327  ax-pr 5450
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-clab 2712  df-cleq 2726  df-clel 2813  df-ne 2943  df-ral 3064  df-rex 3073  df-rab 3439  df-v 3484  df-dif 3973  df-un 3975  df-in 3977  df-ss 3987  df-nul 4348  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-br 5170  df-opab 5232  df-eprel 5603  df-xp 5705  df-rel 5706  df-cnv 5707  df-dm 5709  df-rn 5710  df-res 5711  df-ima 5712  df-ec 8761
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator