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Theorem cnvepres 38299
Description: Restricted converse epsilon relation as a class of ordered pairs. (Contributed by Peter Mazsa, 10-Feb-2018.)
Assertion
Ref Expression
cnvepres ( E ↾ 𝐴) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦𝑥)}
Distinct variable group:   𝑥,𝐴,𝑦

Proof of Theorem cnvepres
StepHypRef Expression
1 dfres2 6059 . 2 ( E ↾ 𝐴) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑥 E 𝑦)}
2 brcnvep 38266 . . . . 5 (𝑥 ∈ V → (𝑥 E 𝑦𝑦𝑥))
32elv 3485 . . . 4 (𝑥 E 𝑦𝑦𝑥)
43anbi2i 623 . . 3 ((𝑥𝐴𝑥 E 𝑦) ↔ (𝑥𝐴𝑦𝑥))
54opabbii 5210 . 2 {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑥 E 𝑦)} = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦𝑥)}
61, 5eqtri 2765 1 ( E ↾ 𝐴) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦𝑥)}
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395   = wceq 1540  wcel 2108  Vcvv 3480   class class class wbr 5143  {copab 5205   E cep 5583  ccnv 5684  cres 5687
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-eprel 5584  df-xp 5691  df-rel 5692  df-cnv 5693  df-res 5697
This theorem is referenced by:  rncnvepres  38304  n0el2  38334  cnvepresex  38335
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