Theorem eqrelrdv 5764

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 1948	. 2 ⊢ (𝜑 → ∀𝑥∀𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐵))
3		eqrelrdv.1	. . 3 ⊢ Rel 𝐴
4		eqrelrdv.2	. . 3 ⊢ Rel 𝐵
5		eqrel 5756	. . 3 ⊢ ((Rel 𝐴 ∧ Rel 𝐵) → (𝐴 = 𝐵 ↔ ∀𝑥∀𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐵)))
6	3, 4, 5	mp2an 702	. 2 ⊢ (𝐴 = 𝐵 ↔ ∀𝑥∀𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐵))
7	2, 6	sylibr 236	1 ⊢ (𝜑 → 𝐴 = 𝐵)

Syntax hints:   → wi 4   ↔ wb 208  ∀wal 1558   = wceq 1560   ∈ wcel 2142  ⟨cop 4588  Rel wrel 5652

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-ext 2734

This theorem depends on definitions:  df-bi 209  df-an 400  df-tru 1563  df-ex 1800  df-sb 2091  df-clab 2741  df-cleq 2754  df-clel 2837  df-v 3456  df-ss 3921  df-opab 5163  df-xp 5653  df-rel 5654

This theorem is referenced by:  eqbrrdiv  5766  fcnvres  6741  fmptco  7111  fpwwe2lem7  10595  fpwwe2lem11  10599  fsumcom2  15801  fprodcom2  16014  gsumcom2  20015  lgsquadlem1  27441  lgsquadlem2  27442  fmptcof2  32856  dfcnv2  32874  dih1dimatlem  41950