MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  elrel Structured version   Visualization version   GIF version

Theorem elrel 5557
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 5450 . . . 4 (Rel 𝑅𝑅 ⊆ (V × V))
21biimpi 217 . . 3 (Rel 𝑅𝑅 ⊆ (V × V))
32sselda 3889 . 2 ((Rel 𝑅𝐴𝑅) → 𝐴 ∈ (V × V))
4 elvv 5512 . 2 (𝐴 ∈ (V × V) ↔ ∃𝑥𝑦 𝐴 = ⟨𝑥, 𝑦⟩)
53, 4sylib 219 1 ((Rel 𝑅𝐴𝑅) → ∃𝑥𝑦 𝐴 = ⟨𝑥, 𝑦⟩)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1522  wex 1761  wcel 2081  Vcvv 3437  wss 3859  cop 4478   × cxp 5441  Rel wrel 5448
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-ext 2769  ax-sep 5094  ax-nul 5101  ax-pr 5221
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-v 3439  df-dif 3862  df-un 3864  df-in 3866  df-ss 3874  df-nul 4212  df-if 4382  df-sn 4473  df-pr 4475  df-op 4479  df-opab 5025  df-xp 5449  df-rel 5450
This theorem is referenced by:  eliunxp  5594  elinxp  5771  unielrel  6000  frxp  7673  rntpos  7756  gsum2d2lem  18813  funen1cnv  31971  dfpo2  32599  fundmpss  32617  sscoid  32983  elfuns  32985  eliunxp2  43860
  Copyright terms: Public domain W3C validator