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Theorem dfrel4v 6041
Description: A relation can be expressed as the set of ordered pairs in it. An analogue of dffn5 6718 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 6040 . 2 (Rel 𝑅𝑅 = 𝑅)
2 eqcom 2828 . 2 (𝑅 = 𝑅𝑅 = 𝑅)
3 cnvcnv3 6039 . . 3 𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑅𝑦}
43eqeq2i 2834 . 2 (𝑅 = 𝑅𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑅𝑦})
51, 2, 43bitri 298 1 (Rel 𝑅𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑅𝑦})
Colors of variables: wff setvar class
Syntax hints:  wb 207   = wceq 1528   class class class wbr 5058  {copab 5120  ccnv 5548  Rel wrel 5554
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2793  ax-sep 5195  ax-nul 5202  ax-pr 5321
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-rab 3147  df-v 3497  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4466  df-sn 4560  df-pr 4562  df-op 4566  df-br 5059  df-opab 5121  df-xp 5555  df-rel 5556  df-cnv 5557
This theorem is referenced by:  dfrel4  6042  dffn5  6718  fsplit  7803  fsplitOLD  7804  pwsle  16755  tgphaus  22654  fneer  33599  inxp2  35501  dfxrn2  35510  1cosscnvxrn  35597  dfafn5a  43240  sprsymrelfo  43506
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