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Theorem dfrel4v 4772
 Description: A relation can be expressed as the set of ordered pairs in it. (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 4771 . 2 (Rel 𝑅𝑅 = 𝑅)
2 eqcom 2042 . 2 (𝑅 = 𝑅𝑅 = 𝑅)
3 cnvcnv3 4770 . . 3 𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑅𝑦}
43eqeq2i 2050 . 2 (𝑅 = 𝑅𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑅𝑦})
51, 2, 43bitri 195 1 (Rel 𝑅𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑅𝑦})
 Colors of variables: wff set class Syntax hints:   ↔ wb 98   = wceq 1243   class class class wbr 3764  {copab 3817  ◡ccnv 4344  Rel wrel 4350 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944 This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-br 3765  df-opab 3819  df-xp 4351  df-rel 4352  df-cnv 4353 This theorem is referenced by:  dffn5im  5219
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