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Theorem refrel 23012
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 23009 . 2 Ref = {⟨𝑥, 𝑦⟩ ∣ ( 𝑦 = 𝑥 ∧ ∀𝑧𝑥𝑤𝑦 𝑧𝑤)}
21relopabiv 5821 1 Rel Ref
Colors of variables: wff setvar class
Syntax hints:  wa 397   = wceq 1542  wral 3062  wrex 3071  wss 3949   cuni 4909  Rel wrel 5682  Refcref 23006
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 3477  df-in 3956  df-ss 3966  df-opab 5212  df-xp 5683  df-rel 5684  df-ref 23009
This theorem is referenced by:  isref  23013  refbas  23014  refssex  23015  reftr  23018  refun0  23019  locfinref  32852  refssfne  35291
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