Theorem eqrel 5734

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 5733	. . 3 ⊢ (Rel 𝐴 → (𝐴 ⊆ 𝐵 ↔ ∀𝑥∀𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 → ⟨𝑥, 𝑦⟩ ∈ 𝐵)))
2		ssrel 5733	. . 3 ⊢ (Rel 𝐵 → (𝐵 ⊆ 𝐴 ↔ ∀𝑥∀𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐵 → ⟨𝑥, 𝑦⟩ ∈ 𝐴)))
3	1, 2	bi2anan9 644	. 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 3937	. 2 ⊢ (𝐴 = 𝐵 ↔ (𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐴))
5		2albiim 1897	. 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 315	1 ⊢ ((Rel 𝐴 ∧ Rel 𝐵) → (𝐴 = 𝐵 ↔ ∀𝑥∀𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐵)))

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396  ∀wal 1545   = wceq 1547   ∈ wcel 2119   ⊆ wss 3890  ⟨cop 4568  Rel wrel 5630

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2712

This theorem depends on definitions:  df-bi 208  df-an 397  df-tru 1550  df-ex 1787  df-sb 2074  df-clab 2719  df-cleq 2732  df-clel 2815  df-v 3434  df-ss 3907  df-opab 5142  df-xp 5631  df-rel 5632

This theorem is referenced by:  eqrelriv  5739  eqrelrdv  5742  eqbrrdv  5743  eqrelrdv2  5745  opabid2  5778  reldm0  5877  iss  5994  asymref  6073  funssres  6536  fsn  7084  eqrelf  38632  iss2  38718