Theorem elrel 5822

Description: A member of a relation is an ordered pair. (Contributed by NM, 17-Sep-2006.)

Assertion

Ref	Expression
elrel	⊢ ((Rel 𝑅 ∧ 𝐴 ∈ 𝑅) → ∃𝑥∃𝑦 𝐴 = ⟨𝑥, 𝑦⟩)

Distinct variable group:   𝑥,𝑦,𝐴

Allowed substitution hints:   𝑅(𝑥,𝑦)

Proof of Theorem elrel

Step	Hyp	Ref	Expression
1		df-rel 5707	. . . 4 ⊢ (Rel 𝑅 ↔ 𝑅 ⊆ (V × V))
2	1	biimpi 216	. . 3 ⊢ (Rel 𝑅 → 𝑅 ⊆ (V × V))
3	2	sselda 4008	. 2 ⊢ ((Rel 𝑅 ∧ 𝐴 ∈ 𝑅) → 𝐴 ∈ (V × V))
4		elvv 5774	. 2 ⊢ (𝐴 ∈ (V × V) ↔ ∃𝑥∃𝑦 𝐴 = ⟨𝑥, 𝑦⟩)
5	3, 4	sylib 218	1 ⊢ ((Rel 𝑅 ∧ 𝐴 ∈ 𝑅) → ∃𝑥∃𝑦 𝐴 = ⟨𝑥, 𝑦⟩)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1537  ∃wex 1777   ∈ wcel 2108  Vcvv 3488   ⊆ wss 3976  ⟨cop 4654   × cxp 5698  Rel wrel 5705

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-v 3490  df-dif 3979  df-un 3981  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-opab 5229  df-xp 5706  df-rel 5707

This theorem is referenced by:  eliunxp  5862  elinxp  6048  unielrel  6305  dfpo2  6327  frxp  8167  frxp2  8185  rntpos  8280  gsum2d2lem  20015  funen1cnv  35064  fundmpss  35730  sscoid  35877  elfuns  35879  eliunxp2  48058