Theorem eqrelriv 5782

Description: Inference from extensionality principle for relations. (Contributed by FL, 15-Oct-2012.)

Hypothesis

Ref	Expression
eqrelriv.1	⊢ (⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐵)

Assertion

Ref	Expression
eqrelriv	⊢ ((Rel 𝐴 ∧ Rel 𝐵) → 𝐴 = 𝐵)

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

Proof of Theorem eqrelriv

Step	Hyp	Ref	Expression
1		eqrelriv.1	. . 3 ⊢ (⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐵)
2	1	gen2 1790	. 2 ⊢ ∀𝑥∀𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐵)
3		eqrel 5777	. 2 ⊢ ((Rel 𝐴 ∧ Rel 𝐵) → (𝐴 = 𝐵 ↔ ∀𝑥∀𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐵)))
4	2, 3	mpbiri 258	1 ⊢ ((Rel 𝐴 ∧ Rel 𝐵) → 𝐴 = 𝐵)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395  ∀wal 1531   = wceq 1533   ∈ wcel 2098  ⟨cop 4629  Rel wrel 5674

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2697

This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-v 3470  df-in 3950  df-ss 3960  df-opab 5204  df-xp 5675  df-rel 5676

This theorem is referenced by:  eqrelriiv  5783  dfrel2  6181  coi1  6254  cnviin  6278