Theorem fnerel 36830

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 36829	. 2 ⊢ Fne = {⟨𝑥, 𝑦⟩ ∣ (∪ 𝑥 = ∪ 𝑦 ∧ ∀𝑧 ∈ 𝑥 𝑧 ⊆ ∪ (𝑦 ∩ 𝒫 𝑧))}
2	1	relopabiv 5809	1 ⊢ Rel Fne

Syntax hints:   ∧ wa 400   = wceq 1570  ∀wral 3079   ∩ cin 3905   ⊆ wss 3906  𝒫 cpw 4563  ∪ cuni 4873  Rel wrel 5668  Fnecfne 36828

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735

This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1573  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-v 3457  df-ss 3923  df-opab 5175  df-xp 5669  df-rel 5670  df-fne 36829

This theorem is referenced by:  isfne  36831  isfne4  36832  fnetr  36843  fneval  36844  fneer  36845  fnessref  36849