Mathbox for Peter Mazsa < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  eqrel2 Structured version   Visualization version   GIF version

Theorem eqrel2 34618
 Description: Equality of relations. (Contributed by Peter Mazsa, 8-Mar-2019.)
Assertion
Ref Expression
eqrel2 ((Rel 𝐴 ∧ Rel 𝐵) → (𝐴 = 𝐵 ↔ ∀𝑥𝑦(𝑥𝐴𝑦𝑥𝐵𝑦)))
Distinct variable groups:   𝑥,𝐴,𝑦   𝑥,𝐵,𝑦

Proof of Theorem eqrel2
StepHypRef Expression
1 ssrel3 34617 . . 3 (Rel 𝐴 → (𝐴𝐵 ↔ ∀𝑥𝑦(𝑥𝐴𝑦𝑥𝐵𝑦)))
2 ssrel3 34617 . . 3 (Rel 𝐵 → (𝐵𝐴 ↔ ∀𝑥𝑦(𝑥𝐵𝑦𝑥𝐴𝑦)))
31, 2bi2anan9 629 . 2 ((Rel 𝐴 ∧ Rel 𝐵) → ((𝐴𝐵𝐵𝐴) ↔ (∀𝑥𝑦(𝑥𝐴𝑦𝑥𝐵𝑦) ∧ ∀𝑥𝑦(𝑥𝐵𝑦𝑥𝐴𝑦))))
4 eqss 3842 . 2 (𝐴 = 𝐵 ↔ (𝐴𝐵𝐵𝐴))
5 2albiim 1992 . 2 (∀𝑥𝑦(𝑥𝐴𝑦𝑥𝐵𝑦) ↔ (∀𝑥𝑦(𝑥𝐴𝑦𝑥𝐵𝑦) ∧ ∀𝑥𝑦(𝑥𝐵𝑦𝑥𝐴𝑦)))
63, 4, 53bitr4g 306 1 ((Rel 𝐴 ∧ Rel 𝐵) → (𝐴 = 𝐵 ↔ ∀𝑥𝑦(𝑥𝐴𝑦𝑥𝐵𝑦)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 198   ∧ wa 386  ∀wal 1654   = wceq 1656   ⊆ wss 3798   class class class wbr 4875  Rel wrel 5351 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-ext 2803 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-clab 2812  df-cleq 2818  df-clel 2821  df-in 3805  df-ss 3812  df-br 4876  df-opab 4938  df-xp 5352  df-rel 5353 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator