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Theorem prtlem12 36443
 Description: Lemma for prtex 36456 and prter3 36458. (Contributed by Rodolfo Medina, 13-Oct-2010.)
Assertion
Ref Expression
prtlem12 ( = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢𝐴 (𝑥𝑢𝑦𝑢)} → Rel )
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝐴(𝑥,𝑦,𝑢)   (𝑥,𝑦,𝑢)

Proof of Theorem prtlem12
StepHypRef Expression
1 relopab 5665 . 2 Rel {⟨𝑥, 𝑦⟩ ∣ ∃𝑢𝐴 (𝑥𝑢𝑦𝑢)}
2 releq 5620 . 2 ( = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢𝐴 (𝑥𝑢𝑦𝑢)} → (Rel ↔ Rel {⟨𝑥, 𝑦⟩ ∣ ∃𝑢𝐴 (𝑥𝑢𝑦𝑢)}))
31, 2mpbiri 261 1 ( = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢𝐴 (𝑥𝑢𝑦𝑢)} → Rel )
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538  ∃wrex 3071  {copab 5094  Rel wrel 5529 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-11 2158  ax-12 2175  ax-ext 2729 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-sb 2070  df-clab 2736  df-cleq 2750  df-clel 2830  df-v 3411  df-un 3863  df-in 3865  df-ss 3875  df-sn 4523  df-pr 4525  df-op 4529  df-opab 5095  df-xp 5530  df-rel 5531 This theorem is referenced by: (None)
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