Theorem refrel 23473

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 23470	. 2 ⊢ Ref = {⟨𝑥, 𝑦⟩ ∣ (∪ 𝑦 = ∪ 𝑥 ∧ ∀𝑧 ∈ 𝑥 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤)}
2	1	relopabiv 5776	1 ⊢ Rel Ref

Syntax hints:   ∧ wa 395   = wceq 1542  ∀wral 3051  ∃wrex 3061   ⊆ wss 3889  ∪ cuni 4850  Rel wrel 5636  Refcref 23467

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1545  df-ex 1782  df-sb 2069  df-clab 2715  df-cleq 2728  df-clel 2811  df-v 3431  df-ss 3906  df-opab 5148  df-xp 5637  df-rel 5638  df-ref 23470

This theorem is referenced by:  isref  23474  refbas  23475  refssex  23476  reftr  23479  refun0  23480  locfinref  33985  refssfne  36540