Theorem fnerel 35161

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 35160	. 2 ⊢ Fne = {⟨𝑥, 𝑦⟩ ∣ (∪ 𝑥 = ∪ 𝑦 ∧ ∀𝑧 ∈ 𝑥 𝑧 ⊆ ∪ (𝑦 ∩ 𝒫 𝑧))}
2	1	relopabiv 5818	1 ⊢ Rel Fne

Syntax hints:   ∧ wa 397   = wceq 1542  ∀wral 3062   ∩ cin 3946   ⊆ wss 3947  𝒫 cpw 4601  ∪ cuni 4907  Rel wrel 5680  Fnecfne 35159

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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704

This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-v 3477  df-in 3954  df-ss 3964  df-opab 5210  df-xp 5681  df-rel 5682  df-fne 35160

This theorem is referenced by:  isfne  35162  isfne4  35163  fnetr  35174  fneval  35175  fneer  35176  fnessref  35180