Theorem fnerel 36405

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 36404	. 2 ⊢ Fne = {⟨𝑥, 𝑦⟩ ∣ (∪ 𝑥 = ∪ 𝑦 ∧ ∀𝑧 ∈ 𝑥 𝑧 ⊆ ∪ (𝑦 ∩ 𝒫 𝑧))}
2	1	relopabiv 5766	1 ⊢ Rel Fne

Syntax hints:   ∧ wa 395   = wceq 1541  ∀wral 3048   ∩ cin 3897   ⊆ wss 3898  𝒫 cpw 4551  ∪ cuni 4860  Rel wrel 5626  Fnecfne 36403

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2705

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1544  df-ex 1781  df-sb 2068  df-clab 2712  df-cleq 2725  df-clel 2808  df-v 3439  df-ss 3915  df-opab 5158  df-xp 5627  df-rel 5628  df-fne 36404

This theorem is referenced by:  isfne  36406  isfne4  36407  fnetr  36418  fneval  36419  fneer  36420  fnessref  36424