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Theorem prtlem12 38371
Description: Lemma for prtex 38384 and prter3 38386. (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 relopabv 5827 . 2 Rel {⟨𝑥, 𝑦⟩ ∣ ∃𝑢𝐴 (𝑥𝑢𝑦𝑢)}
2 releq 5782 . 2 ( = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢𝐴 (𝑥𝑢𝑦𝑢)} → (Rel ↔ Rel {⟨𝑥, 𝑦⟩ ∣ ∃𝑢𝐴 (𝑥𝑢𝑦𝑢)}))
31, 2mpbiri 257 1 ( = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢𝐴 (𝑥𝑢𝑦𝑢)} → Rel )
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394   = wceq 1533  wrex 3067  {copab 5214  Rel wrel 5687
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2699
This theorem depends on definitions:  df-bi 206  df-an 395  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2706  df-cleq 2720  df-clel 2806  df-v 3475  df-in 3956  df-ss 3966  df-opab 5215  df-xp 5688  df-rel 5689
This theorem is referenced by: (None)
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