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Theorem cnvepres 37802
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 6050 . 2 ( E ↾ 𝐴) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑥 E 𝑦)}
2 brcnvep 37769 . . . . 5 (𝑥 ∈ V → (𝑥 E 𝑦𝑦𝑥))
32elv 3479 . . . 4 (𝑥 E 𝑦𝑦𝑥)
43anbi2i 621 . . 3 ((𝑥𝐴𝑥 E 𝑦) ↔ (𝑥𝐴𝑦𝑥))
54opabbii 5219 . 2 {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑥 E 𝑦)} = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦𝑥)}
61, 5eqtri 2756 1 ( E ↾ 𝐴) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦𝑥)}
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 394   = wceq 1533  wcel 2098  Vcvv 3473   class class class wbr 5152  {copab 5214   E cep 5585  ccnv 5681  cres 5684
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-ext 2699  ax-sep 5303  ax-nul 5310  ax-pr 5433
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-sb 2060  df-clab 2706  df-cleq 2720  df-clel 2806  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-br 5153  df-opab 5215  df-eprel 5586  df-xp 5688  df-rel 5689  df-cnv 5690  df-res 5694
This theorem is referenced by:  rncnvepres  37807  n0el2  37837  cnvepresex  37838
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