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Theorem eccnvepres 38238
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 38223 . . . 4 (𝐵𝑉 → (𝐵 E 𝑥𝑥𝐵))
21anbi1cd 634 . . 3 (𝐵𝑉 → ((𝐵𝐴𝐵 E 𝑥) ↔ (𝑥𝐵𝐵𝐴)))
32abbidv 2811 . 2 (𝐵𝑉 → {𝑥 ∣ (𝐵𝐴𝐵 E 𝑥)} = {𝑥 ∣ (𝑥𝐵𝐵𝐴)})
4 ecres 38236 . 2 [𝐵]( E ↾ 𝐴) = {𝑥 ∣ (𝐵𝐴𝐵 E 𝑥)}
5 df-rab 3444 . 2 {𝑥𝐵𝐵𝐴} = {𝑥 ∣ (𝑥𝐵𝐵𝐴)}
63, 4, 53eqtr4g 2805 1 (𝐵𝑉 → [𝐵]( E ↾ 𝐴) = {𝑥𝐵𝐵𝐴})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wcel 2108  {cab 2717  {crab 3443   class class class wbr 5166   E cep 5598  ccnv 5699  cres 5702  [cec 8763
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 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447
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 2718  df-cleq 2732  df-clel 2819  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-br 5167  df-opab 5229  df-eprel 5599  df-xp 5706  df-rel 5707  df-cnv 5708  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-ec 8767
This theorem is referenced by: (None)
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