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Theorem dfrel4v 5486
Description: A relation can be expressed as the set of ordered pairs in it. An analogue of dffn5 6133 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 5485 . 2 (Rel 𝑅𝑅 = 𝑅)
2 eqcom 2613 . 2 (𝑅 = 𝑅𝑅 = 𝑅)
3 cnvcnv3 5484 . . 3 𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑅𝑦}
43eqeq2i 2618 . 2 (𝑅 = 𝑅𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑅𝑦})
51, 2, 43bitri 284 1 (Rel 𝑅𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑅𝑦})
Colors of variables: wff setvar class
Syntax hints:  wb 194   = wceq 1474   class class class wbr 4574  {copab 4633  ccnv 5024  Rel wrel 5030
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2032  ax-13 2229  ax-ext 2586  ax-sep 4700  ax-nul 4709  ax-pr 4825
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2458  df-mo 2459  df-clab 2593  df-cleq 2599  df-clel 2602  df-nfc 2736  df-ral 2897  df-rex 2898  df-rab 2901  df-v 3171  df-dif 3539  df-un 3541  df-in 3543  df-ss 3550  df-nul 3871  df-if 4033  df-sn 4122  df-pr 4124  df-op 4128  df-br 4575  df-opab 4635  df-xp 5031  df-rel 5032  df-cnv 5033
This theorem is referenced by:  dfrel4  5487  dffn5  6133  fsplit  7143  pwsle  15918  tgphaus  21669  fneer  31321  dfafn5a  39691
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