Theorem refrel 22113

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 22110	. 2 ⊢ Ref = {⟨𝑥, 𝑦⟩ ∣ (∪ 𝑦 = ∪ 𝑥 ∧ ∀𝑧 ∈ 𝑥 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤)}
2	1	relopabi 5658	1 ⊢ Rel Ref

Syntax hints:   ∧ wa 399   = wceq 1538  ∀wral 3106  ∃wrex 3107   ⊆ wss 3881  ∪ cuni 4800  Rel wrel 5524  Refcref 22107

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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-11 2158  ax-12 2175  ax-ext 2770

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-v 3443  df-un 3886  df-in 3888  df-ss 3898  df-sn 4526  df-pr 4528  df-op 4532  df-opab 5093  df-xp 5525  df-rel 5526  df-ref 22110

This theorem is referenced by:  isref  22114  refbas  22115  refssex  22116  reftr  22119  refun0  22120  locfinref  31194  refssfne  33819