Theorem prtlem12 40033

Description: Lemma for prtex 40046 and prter3 40048. (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 5800	. 2 ⊢ Rel {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ 𝐴 (𝑥 ∈ 𝑢 ∧ 𝑦 ∈ 𝑢)}
2		releq 5754	. 2 ⊢ ( ∼ = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ 𝐴 (𝑥 ∈ 𝑢 ∧ 𝑦 ∈ 𝑢)} → (Rel ∼ ↔ Rel {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ 𝐴 (𝑥 ∈ 𝑢 ∧ 𝑦 ∈ 𝑢)}))
3	1, 2	mpbiri 261	1 ⊢ ( ∼ = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ 𝐴 (𝑥 ∈ 𝑢 ∧ 𝑦 ∈ 𝑢)} → Rel ∼ )

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1569  ∃wrex 3086  {copab 5166  Rel wrel 5656

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1572  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-v 3452  df-ss 3897  df-opab 5167  df-xp 5657  df-rel 5658

This theorem is used by: (None)