Theorem eqrelrdv 5779

Description: Deduce equality of relations from equivalence of membership. (Contributed by Rodolfo Medina, 10-Oct-2010.)

Hypotheses

Ref	Expression
eqrelrdv.1	⊢ Rel 𝐴
eqrelrdv.2	⊢ Rel 𝐵
eqrelrdv.3	⊢ (𝜑 → (⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐵))

Assertion

Ref	Expression
eqrelrdv	⊢ (𝜑 → 𝐴 = 𝐵)

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

Proof of Theorem eqrelrdv

Step	Hyp	Ref	Expression
1		eqrelrdv.3	. . 3 ⊢ (𝜑 → (⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐵))
2	1	alrimivv 1955	. 2 ⊢ (𝜑 → ∀𝑥∀𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐵))
3		eqrelrdv.1	. . 3 ⊢ Rel 𝐴
4		eqrelrdv.2	. . 3 ⊢ Rel 𝐵
5		eqrel 5771	. . 3 ⊢ ((Rel 𝐴 ∧ Rel 𝐵) → (𝐴 = 𝐵 ↔ ∀𝑥∀𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐵)))
6	3, 4, 5	mp2an 704	. 2 ⊢ (𝐴 = 𝐵 ↔ ∀𝑥∀𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐵))
7	2, 6	sylibr 237	1 ⊢ (𝜑 → 𝐴 = 𝐵)

Syntax hints:   → wi 4   ↔ wb 209  ∀wal 1565   = wceq 1567   ∈ wcel 2149  ⟨cop 4600  Rel wrel 5667

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741

This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1570  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-v 3465  df-ss 3930  df-opab 5178  df-xp 5668  df-rel 5669

This theorem is referenced by:  eqbrrdiv  5781  fcnvres  6756  fmptco  7126  fpwwe2lem7  10621  fpwwe2lem11  10625  fsumcom2  15824  fprodcom2  16037  gsumcom2  20044  lgsquadlem1  27509  lgsquadlem2  27510  fmptcof2  32942  dfcnv2  32960  dih1dimatlem  41992