Theorem prtlem12 39451

Description: Lemma for prtex 39464 and prter3 39466. (Contributed by Rodolfo Medina, 13-Oct-2010.)

Assertion

Ref	Expression
prtlem12	⊢ ( ∼ = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ 𝐴 (𝑥 ∈ 𝑢 ∧ 𝑦 ∈ 𝑢)} → Rel ∼ )

Distinct variable group:   𝑥,𝑦

Allowed substitution hints:   𝐴(𝑥,𝑦,𝑢)   ∼ (𝑥,𝑦,𝑢)

Proof of Theorem prtlem12

Step	Hyp	Ref	Expression
1		relopabv 5790	. 2 ⊢ Rel {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ 𝐴 (𝑥 ∈ 𝑢 ∧ 𝑦 ∈ 𝑢)}
2		releq 5745	. 2 ⊢ ( ∼ = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ 𝐴 (𝑥 ∈ 𝑢 ∧ 𝑦 ∈ 𝑢)} → (Rel ∼ ↔ Rel {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ 𝐴 (𝑥 ∈ 𝑢 ∧ 𝑦 ∈ 𝑢)}))
3	1, 2	mpbiri 260	1 ⊢ ( ∼ = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ 𝐴 (𝑥 ∈ 𝑢 ∧ 𝑦 ∈ 𝑢)} → Rel ∼ )

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1559  ∃wrex 3085  {copab 5159  Rel wrel 5648

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-ext 2733

This theorem depends on definitions:  df-bi 209  df-an 400  df-tru 1562  df-ex 1799  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-v 3455  df-ss 3919  df-opab 5160  df-xp 5649  df-rel 5650

This theorem is referenced by: (None)