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Theorem refrel 22437
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 22434 . 2 Ref = {⟨𝑥, 𝑦⟩ ∣ ( 𝑦 = 𝑥 ∧ ∀𝑧𝑥𝑤𝑦 𝑧𝑤)}
21relopabiv 5708 1 Rel Ref
Colors of variables: wff setvar class
Syntax hints:  wa 399   = wceq 1543  wral 3064  wrex 3065  wss 3883   cuni 4836  Rel wrel 5574  Refcref 22431
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2114  ax-9 2122  ax-ext 2710
This theorem depends on definitions:  df-bi 210  df-an 400  df-tru 1546  df-ex 1788  df-sb 2073  df-clab 2717  df-cleq 2731  df-clel 2818  df-v 3425  df-in 3890  df-ss 3900  df-opab 5133  df-xp 5575  df-rel 5576  df-ref 22434
This theorem is referenced by:  isref  22438  refbas  22439  refssex  22440  reftr  22443  refun0  22444  locfinref  31537  refssfne  34318
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