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Theorem dfrel4v 5799
Description: A relation can be expressed as the set of ordered pairs in it. An analogue of dffn5 6464 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 5798 . 2 (Rel 𝑅𝑅 = 𝑅)
2 eqcom 2804 . 2 (𝑅 = 𝑅𝑅 = 𝑅)
3 cnvcnv3 5797 . . 3 𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑅𝑦}
43eqeq2i 2809 . 2 (𝑅 = 𝑅𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑅𝑦})
51, 2, 43bitri 289 1 (Rel 𝑅𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑅𝑦})
Colors of variables: wff setvar class
Syntax hints:  wb 198   = wceq 1653   class class class wbr 4841  {copab 4903  ccnv 5309  Rel wrel 5315
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2354  ax-ext 2775  ax-sep 4973  ax-nul 4981  ax-pr 5095
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2590  df-eu 2607  df-clab 2784  df-cleq 2790  df-clel 2793  df-nfc 2928  df-rab 3096  df-v 3385  df-dif 3770  df-un 3772  df-in 3774  df-ss 3781  df-nul 4114  df-if 4276  df-sn 4367  df-pr 4369  df-op 4373  df-br 4842  df-opab 4904  df-xp 5316  df-rel 5317  df-cnv 5318
This theorem is referenced by:  dfrel4  5800  dffn5  6464  fsplit  7517  pwsle  16464  tgphaus  22245  fneer  32852  inxp2  34615  dfxrn2  34624  1cosscnvxrn  34711  dfafn5a  42002  sprsymrelfo  42534
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