Theorem fnerel 36573

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 36572	. 2 ⊢ Fne = {⟨𝑥, 𝑦⟩ ∣ (∪ 𝑥 = ∪ 𝑦 ∧ ∀𝑧 ∈ 𝑥 𝑧 ⊆ ∪ (𝑦 ∩ 𝒫 𝑧))}
2	1	relopabiv 5770	1 ⊢ Rel Fne

Syntax hints:   ∧ wa 396   = wceq 1547  ∀wral 3054   ∩ cin 3889   ⊆ wss 3890  𝒫 cpw 4536  ∪ cuni 4845  Rel wrel 5630  Fnecfne 36571

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2712

This theorem depends on definitions:  df-bi 208  df-an 397  df-tru 1550  df-ex 1787  df-sb 2074  df-clab 2719  df-cleq 2732  df-clel 2815  df-v 3434  df-ss 3907  df-opab 5142  df-xp 5631  df-rel 5632  df-fne 36572

This theorem is referenced by:  isfne  36574  isfne4  36575  fnetr  36586  fneval  36587  fneer  36588  fnessref  36592