Theorem refrel 23646

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.

Step	Hyp	Ref	Expression
1		df-ref 23643	. 2 ⊢ Ref = {⟨𝑥, 𝑦⟩ ∣ (∪ 𝑦 = ∪ 𝑥 ∧ ∀𝑧 ∈ 𝑥 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤)}
2	1	relopabiv 5809	1 ⊢ Rel Ref

Syntax hints:   ∧ wa 400   = wceq 1570  ∀wral 3079  ∃wrex 3089   ⊆ wss 3906  ∪ cuni 4873  Rel wrel 5668  Refcref 23640

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735

This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1573  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-v 3457  df-ss 3923  df-opab 5175  df-xp 5669  df-rel 5670  df-ref 23643

This theorem is referenced by:  isref  23647  refbas  23648  refssex  23649  reftr  23652  refun0  23653  locfinref  34212  refssfne  36850