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Theorem refrel 23532
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 23529 . 2 Ref = {⟨𝑥, 𝑦⟩ ∣ ( 𝑦 = 𝑥 ∧ ∀𝑧𝑥𝑤𝑦 𝑧𝑤)}
21relopabiv 5833 1 Rel Ref
Colors of variables: wff setvar class
Syntax hints:  wa 395   = wceq 1537  wral 3059  wrex 3068  wss 3963   cuni 4912  Rel wrel 5694  Refcref 23526
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-v 3480  df-ss 3980  df-opab 5211  df-xp 5695  df-rel 5696  df-ref 23529
This theorem is referenced by:  isref  23533  refbas  23534  refssex  23535  reftr  23538  refun0  23539  locfinref  33802  refssfne  36341
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