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Theorem dfrel4v 4699
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 4698 . 2 (Rel 𝑅𝑅 = 𝑅)
2 eqcom 2024 . 2 (𝑅 = 𝑅𝑅 = 𝑅)
3 cnvcnv3 4697 . . 3 𝑅 = {⟨x, y⟩ ∣ x𝑅y}
43eqeq2i 2032 . 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 1228   class class class wbr 3738  {copab 3791  ccnv 4271  Rel wrel 4277
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 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-sep 3849  ax-pow 3901  ax-pr 3918
This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-nf 1330  df-sb 1628  df-eu 1885  df-mo 1886  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2289  df-rex 2290  df-v 2537  df-un 2899  df-in 2901  df-ss 2908  df-pw 3336  df-sn 3356  df-pr 3357  df-op 3359  df-br 3739  df-opab 3793  df-xp 4278  df-rel 4279  df-cnv 4280
This theorem is referenced by:  dffn5im  5144
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