Theorem eqbrriv 5690

Description: Inference from extensionality principle for relations. (Contributed by NM, 12-Dec-2006.)

Hypotheses

Ref	Expression
eqbrriv.1	⊢ Rel 𝐴
eqbrriv.2	⊢ Rel 𝐵
eqbrriv.3	⊢ (𝑥𝐴𝑦 ↔ 𝑥𝐵𝑦)

Assertion

Ref	Expression
eqbrriv	⊢ 𝐴 = 𝐵

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

Proof of Theorem eqbrriv

Step	Hyp	Ref	Expression
1		eqbrriv.1	. 2 ⊢ Rel 𝐴
2		eqbrriv.2	. 2 ⊢ Rel 𝐵
3		eqbrriv.3	. . 3 ⊢ (𝑥𝐴𝑦 ↔ 𝑥𝐵𝑦)
4		df-br 5071	. . 3 ⊢ (𝑥𝐴𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐴)
5		df-br 5071	. . 3 ⊢ (𝑥𝐵𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐵)
6	3, 4, 5	3bitr3i 300	. 2 ⊢ (⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐵)
7	1, 2, 6	eqrelriiv 5689	1 ⊢ 𝐴 = 𝐵

Syntax hints:   ↔ wb 205   = wceq 1539   ∈ wcel 2108  ⟨cop 4564   class class class wbr 5070  Rel wrel 5585

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-12 2173  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1542  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-v 3424  df-in 3890  df-ss 3900  df-br 5071  df-opab 5133  df-xp 5586  df-rel 5587

This theorem is referenced by:  resco  6143  tpostpos  8033  sbthcl  8835  dfle2  12810  dflt2  12811  xpab  33579  idsset  34119  dfbigcup2  34128  imageval  34159  inxpxrn  36448