Theorem fnerel 36361

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 36360	. 2 ⊢ Fne = {⟨𝑥, 𝑦⟩ ∣ (∪ 𝑥 = ∪ 𝑦 ∧ ∀𝑧 ∈ 𝑥 𝑧 ⊆ ∪ (𝑦 ∩ 𝒫 𝑧))}
2	1	relopabiv 5804	1 ⊢ Rel Fne

Syntax hints:   ∧ wa 395   = wceq 1540  ∀wral 3052   ∩ cin 3930   ⊆ wss 3931  𝒫 cpw 4580  ∪ cuni 4888  Rel wrel 5664  Fnecfne 36359

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 2008  ax-8 2111  ax-9 2119  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2066  df-clab 2715  df-cleq 2728  df-clel 2810  df-v 3466  df-ss 3948  df-opab 5187  df-xp 5665  df-rel 5666  df-fne 36360

This theorem is referenced by:  isfne  36362  isfne4  36363  fnetr  36374  fneval  36375  fneer  36376  fnessref  36380