Theorem elrel 5789

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 5673	. . . 4 ⊢ (Rel 𝑅 ↔ 𝑅 ⊆ (V × V))
2	1	biimpi 219	. . 3 ⊢ (Rel 𝑅 → 𝑅 ⊆ (V × V))
3	2	sselda 3940	. 2 ⊢ ((Rel 𝑅 ∧ 𝐴 ∈ 𝑅) → 𝐴 ∈ (V × V))
4		elvv 5741	. 2 ⊢ (𝐴 ∈ (V × V) ↔ ∃𝑥∃𝑦 𝐴 = ⟨𝑥, 𝑦⟩)
5	3, 4	sylib 221	1 ⊢ ((Rel 𝑅 ∧ 𝐴 ∈ 𝑅) → ∃𝑥∃𝑦 𝐴 = ⟨𝑥, 𝑦⟩)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570  ∃wex 1812   ∈ wcel 2146  Vcvv 3458   ⊆ wss 3908  ⟨cop 4600   × cxp 5664  Rel wrel 5671

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2738  ax-sep 5262  ax-pr 5409

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2745  df-cleq 2758  df-clel 2841  df-rab 3420  df-v 3460  df-un 3913  df-in 3915  df-ss 3925  df-sn 4595  df-pr 4597  df-op 4601  df-opab 5179  df-xp 5672  df-rel 5673

This theorem is used by:  eliunxp  5828  elinxp  6023  unielrel  6281  dfpo2  6304  frxp  8131  frxp2  8149  rntpos  8244  funen1cnv  9035  gsum2d2lem  20074  fundmpss  36280  sscoid  36424  elfuns  36426  eliunxp2  49155