Theorem eqrel 5724

Description: Extensionality principle for relations. Theorem 3.2(ii) of [Monk1] p. 33. (Contributed by NM, 2-Aug-1994.)

Assertion

Ref	Expression
eqrel	⊢ ((Rel 𝐴 ∧ Rel 𝐵) → (𝐴 = 𝐵 ↔ ∀𝑥∀𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐵)))

Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦

Proof of Theorem eqrel

Step	Hyp	Ref	Expression
1		ssrel 5723	. . 3 ⊢ (Rel 𝐴 → (𝐴 ⊆ 𝐵 ↔ ∀𝑥∀𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 → ⟨𝑥, 𝑦⟩ ∈ 𝐵)))
2		ssrel 5723	. . 3 ⊢ (Rel 𝐵 → (𝐵 ⊆ 𝐴 ↔ ∀𝑥∀𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐵 → ⟨𝑥, 𝑦⟩ ∈ 𝐴)))
3	1, 2	bi2anan9 638	. 2 ⊢ ((Rel 𝐴 ∧ Rel 𝐵) → ((𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐴) ↔ (∀𝑥∀𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 → ⟨𝑥, 𝑦⟩ ∈ 𝐵) ∧ ∀𝑥∀𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐵 → ⟨𝑥, 𝑦⟩ ∈ 𝐴))))
4		eqss 3950	. 2 ⊢ (𝐴 = 𝐵 ↔ (𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐴))
5		2albiim 1891	. 2 ⊢ (∀𝑥∀𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐵) ↔ (∀𝑥∀𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 → ⟨𝑥, 𝑦⟩ ∈ 𝐵) ∧ ∀𝑥∀𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐵 → ⟨𝑥, 𝑦⟩ ∈ 𝐴)))
6	3, 4, 5	3bitr4g 314	1 ⊢ ((Rel 𝐴 ∧ Rel 𝐵) → (𝐴 = 𝐵 ↔ ∀𝑥∀𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐵)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395  ∀wal 1539   = wceq 1541   ∈ wcel 2111   ⊆ wss 3902  ⟨cop 4582  Rel wrel 5621

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1544  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-v 3438  df-ss 3919  df-opab 5154  df-xp 5622  df-rel 5623

This theorem is referenced by:  eqrelriv  5729  eqrelrdv  5732  eqbrrdv  5733  eqrelrdv2  5735  opabid2  5768  reldm0  5868  iss  5984  asymref  6063  funssres  6525  fsn  7068  eqrelf  38289  iss2  38371