Theorem eqbrriv 5815

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 5167	. . 3 ⊢ (𝑥𝐴𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐴)
5		df-br 5167	. . 3 ⊢ (𝑥𝐵𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐵)
6	3, 4, 5	3bitr3i 301	. 2 ⊢ (⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐵)
7	1, 2, 6	eqrelriiv 5814	1 ⊢ 𝐴 = 𝐵

Syntax hints:   ↔ wb 206   = wceq 1537   ∈ wcel 2108  ⟨cop 4654   class class class wbr 5166  Rel wrel 5705

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-v 3490  df-ss 3993  df-br 5167  df-opab 5229  df-xp 5706  df-rel 5707

This theorem is referenced by:  resco  6281  tpostpos  8287  sbthcl  9161  dfle2  13209  dflt2  13210  xpab  35688  idsset  35854  dfbigcup2  35863  imageval  35894  inxpxrn  38351