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Theorem elrel 5803
Description: A member of a relation is an ordered pair. (Contributed by NM, 17-Sep-2006.)
Assertion
Ref Expression
elrel ((Rel 𝑅𝐴𝑅) → ∃𝑥𝑦 𝐴 = ⟨𝑥, 𝑦⟩)
Distinct variable group:   𝑥,𝑦,𝐴
Allowed substitution hints:   𝑅(𝑥,𝑦)

Proof of Theorem elrel
StepHypRef Expression
1 df-rel 5688 . . . 4 (Rel 𝑅𝑅 ⊆ (V × V))
21biimpi 215 . . 3 (Rel 𝑅𝑅 ⊆ (V × V))
32sselda 3978 . 2 ((Rel 𝑅𝐴𝑅) → 𝐴 ∈ (V × V))
4 elvv 5755 . 2 (𝐴 ∈ (V × V) ↔ ∃𝑥𝑦 𝐴 = ⟨𝑥, 𝑦⟩)
53, 4sylib 217 1 ((Rel 𝑅𝐴𝑅) → ∃𝑥𝑦 𝐴 = ⟨𝑥, 𝑦⟩)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394   = wceq 1533  wex 1773  wcel 2098  Vcvv 3461  wss 3946  cop 4638   × cxp 5679  Rel wrel 5686
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2696  ax-sep 5303  ax-nul 5310  ax-pr 5432
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-v 3463  df-dif 3949  df-un 3951  df-ss 3963  df-nul 4325  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-opab 5215  df-xp 5687  df-rel 5688
This theorem is referenced by:  eliunxp  5843  elinxp  6027  unielrel  6284  dfpo2  6306  frxp  8139  frxp2  8157  rntpos  8253  gsum2d2lem  19966  funen1cnv  34881  fundmpss  35538  sscoid  35685  elfuns  35687  eliunxp2  47649
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