Theorem eqrelrdv 5804

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 1925	. 2 ⊢ (𝜑 → ∀𝑥∀𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐵))
3		eqrelrdv.1	. . 3 ⊢ Rel 𝐴
4		eqrelrdv.2	. . 3 ⊢ Rel 𝐵
5		eqrel 5796	. . 3 ⊢ ((Rel 𝐴 ∧ Rel 𝐵) → (𝐴 = 𝐵 ↔ ∀𝑥∀𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐵)))
6	3, 4, 5	mp2an 692	. 2 ⊢ (𝐴 = 𝐵 ↔ ∀𝑥∀𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐵))
7	2, 6	sylibr 234	1 ⊢ (𝜑 → 𝐴 = 𝐵)

Syntax hints:   → wi 4   ↔ wb 206  ∀wal 1534   = wceq 1536   ∈ wcel 2105  ⟨cop 4636  Rel wrel 5693

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-ext 2705

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1539  df-ex 1776  df-sb 2062  df-clab 2712  df-cleq 2726  df-clel 2813  df-v 3479  df-ss 3979  df-opab 5210  df-xp 5694  df-rel 5695

This theorem is referenced by:  eqbrrdiv  5806  fcnvres  6785  fmptco  7148  fpwwe2lem7  10674  fpwwe2lem11  10678  fsumcom2  15806  fprodcom2  16016  gsumcom2  20007  lgsquadlem1  27438  lgsquadlem2  27439  fmptcof2  32673  dfcnv2  32692  dih1dimatlem  41311