Theorem elrel 5748

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 5632	. . . 4 ⊢ (Rel 𝑅 ↔ 𝑅 ⊆ (V × V))
2	1	biimpi 217	. . 3 ⊢ (Rel 𝑅 → 𝑅 ⊆ (V × V))
3	2	sselda 3922	. 2 ⊢ ((Rel 𝑅 ∧ 𝐴 ∈ 𝑅) → 𝐴 ∈ (V × V))
4		elvv 5700	. 2 ⊢ (𝐴 ∈ (V × V) ↔ ∃𝑥∃𝑦 𝐴 = ⟨𝑥, 𝑦⟩)
5	3, 4	sylib 219	1 ⊢ ((Rel 𝑅 ∧ 𝐴 ∈ 𝑅) → ∃𝑥∃𝑦 𝐴 = ⟨𝑥, 𝑦⟩)

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1547  ∃wex 1786   ∈ wcel 2119  Vcvv 3432   ⊆ wss 3890  ⟨cop 4568   × cxp 5623  Rel wrel 5630

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2712  ax-sep 5225  ax-pr 5369

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-ex 1787  df-sb 2074  df-clab 2719  df-cleq 2732  df-clel 2815  df-rab 3393  df-v 3434  df-un 3895  df-in 3897  df-ss 3907  df-sn 4563  df-pr 4565  df-op 4569  df-opab 5142  df-xp 5631  df-rel 5632

This theorem is referenced by:  eliunxp  5786  elinxp  5978  unielrel  6232  dfpo2  6254  frxp  8073  frxp2  8091  rntpos  8186  gsum2d2lem  19946  funen1cnv  35276  fundmpss  36002  sscoid  36146  elfuns  36148  eliunxp2  48832