Theorem fnerel 34527

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 34526	. 2 ⊢ Fne = {⟨𝑥, 𝑦⟩ ∣ (∪ 𝑥 = ∪ 𝑦 ∧ ∀𝑧 ∈ 𝑥 𝑧 ⊆ ∪ (𝑦 ∩ 𝒫 𝑧))}
2	1	relopabiv 5730	1 ⊢ Rel Fne

Syntax hints:   ∧ wa 396   = wceq 1539  ∀wral 3064   ∩ cin 3886   ⊆ wss 3887  𝒫 cpw 4533  ∪ cuni 4839  Rel wrel 5594  Fnecfne 34525

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1542  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-v 3434  df-in 3894  df-ss 3904  df-opab 5137  df-xp 5595  df-rel 5596  df-fne 34526

This theorem is referenced by:  isfne  34528  isfne4  34529  fnetr  34540  fneval  34541  fneer  34542  fnessref  34546