Theorem elrel 5745

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 5630	. . . 4 ⊢ (Rel 𝑅 ↔ 𝑅 ⊆ (V × V))
2	1	biimpi 216	. . 3 ⊢ (Rel 𝑅 → 𝑅 ⊆ (V × V))
3	2	sselda 3937	. 2 ⊢ ((Rel 𝑅 ∧ 𝐴 ∈ 𝑅) → 𝐴 ∈ (V × V))
4		elvv 5698	. 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 1540  ∃wex 1779   ∈ wcel 2109  Vcvv 3438   ⊆ wss 3905  ⟨cop 4585   × cxp 5621  Rel wrel 5628

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701  ax-sep 5238  ax-nul 5248  ax-pr 5374

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-v 3440  df-dif 3908  df-un 3910  df-ss 3922  df-nul 4287  df-if 4479  df-sn 4580  df-pr 4582  df-op 4586  df-opab 5158  df-xp 5629  df-rel 5630

This theorem is referenced by:  eliunxp  5784  elinxp  5974  unielrel  6226  dfpo2  6248  frxp  8066  frxp2  8084  rntpos  8179  gsum2d2lem  19870  funen1cnv  35054  fundmpss  35739  sscoid  35886  elfuns  35888  eliunxp2  48319