Theorem fnerel 36734

Description: Fineness is a relation. (Contributed by Jeff Hankins, 28-Sep-2009.)

Assertion

Ref	Expression
fnerel	⊢ Rel Fne

Proof of Theorem fnerel

Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-fne 36733	. 2 ⊢ Fne = {⟨𝑥, 𝑦⟩ ∣ (∪ 𝑥 = ∪ 𝑦 ∧ ∀𝑧 ∈ 𝑥 𝑧 ⊆ ∪ (𝑦 ∩ 𝒫 𝑧))}
2	1	relopabiv 5805	1 ⊢ Rel Fne

Syntax hints:   ∧ wa 400   = wceq 1567  ∀wral 3085   ∩ cin 3912   ⊆ wss 3913  𝒫 cpw 4564  ∪ cuni 4873  Rel wrel 5664  Fnecfne 36732

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741

This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1570  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-v 3465  df-ss 3930  df-opab 5175  df-xp 5665  df-rel 5666  df-fne 36733

This theorem is referenced by:  isfne  36735  isfne4  36736  fnetr  36747  fneval  36748  fneer  36749  fnessref  36753