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Theorem dfrel4v 5503
 Description: A relation can be expressed as the set of ordered pairs in it. An analogue of dffn5 6151 for relations. (Contributed by Mario Carneiro, 16-Aug-2015.)
Assertion
Ref Expression
dfrel4v (Rel 𝑅𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑅𝑦})
Distinct variable group:   𝑥,𝑦,𝑅

Proof of Theorem dfrel4v
StepHypRef Expression
1 dfrel2 5502 . 2 (Rel 𝑅𝑅 = 𝑅)
2 eqcom 2617 . 2 (𝑅 = 𝑅𝑅 = 𝑅)
3 cnvcnv3 5501 . . 3 𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑅𝑦}
43eqeq2i 2622 . 2 (𝑅 = 𝑅𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑅𝑦})
51, 2, 43bitri 285 1 (Rel 𝑅𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑅𝑦})
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 195   = wceq 1475   class class class wbr 4583  {copab 4642  ◡ccnv 5037  Rel wrel 5043 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-opab 4644  df-xp 5044  df-rel 5045  df-cnv 5046 This theorem is referenced by:  dfrel4  5504  dffn5  6151  fsplit  7169  pwsle  15975  tgphaus  21730  fneer  31518  dfafn5a  39889
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