MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  refrel Structured version   Visualization version   GIF version

Theorem refrel 22882
Description: Refinement is a relation. (Contributed by Jeff Hankins, 18-Jan-2010.) (Revised by Thierry Arnoux, 3-Feb-2020.)
Assertion
Ref Expression
refrel Rel Ref

Proof of Theorem refrel
Dummy variables 𝑥 𝑤 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-ref 22879 . 2 Ref = {⟨𝑥, 𝑦⟩ ∣ ( 𝑦 = 𝑥 ∧ ∀𝑧𝑥𝑤𝑦 𝑧𝑤)}
21relopabiv 5780 1 Rel Ref
Colors of variables: wff setvar class
Syntax hints:  wa 397   = wceq 1542  wral 3061  wrex 3070  wss 3914   cuni 4869  Rel wrel 5642  Refcref 22876
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-v 3449  df-in 3921  df-ss 3931  df-opab 5172  df-xp 5643  df-rel 5644  df-ref 22879
This theorem is referenced by:  isref  22883  refbas  22884  refssex  22885  reftr  22888  refun0  22889  locfinref  32486  refssfne  34883
  Copyright terms: Public domain W3C validator