Theorem refrel 23516

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 23513	. 2 ⊢ Ref = {⟨𝑥, 𝑦⟩ ∣ (∪ 𝑦 = ∪ 𝑥 ∧ ∀𝑧 ∈ 𝑥 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤)}
2	1	relopabiv 5830	1 ⊢ Rel Ref

Syntax hints:   ∧ wa 395   = wceq 1540  ∀wral 3061  ∃wrex 3070   ⊆ wss 3951  ∪ cuni 4907  Rel wrel 5690  Refcref 23510

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-v 3482  df-ss 3968  df-opab 5206  df-xp 5691  df-rel 5692  df-ref 23513

This theorem is referenced by:  isref  23517  refbas  23518  refssex  23519  reftr  23522  refun0  23523  locfinref  33840  refssfne  36359