Theorem fnerel 37106

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 37105	. 2 ⊢ Fne = {⟨𝑥, 𝑦⟩ ∣ (∪ 𝑥 = ∪ 𝑦 ∧ ∀𝑧 ∈ 𝑥 𝑧 ⊆ ∪ (𝑦 ∩ 𝒫 𝑧))}
2	1	relopabiv 5798	1 ⊢ Rel Fne

This proof depends on syntax axioms:   ∧ wa 401   = wceq 1570  ∀wral 3077   ∩ cin 3898   ⊆ wss 3899  𝒫 cpw 4557  ∪ cuni 4867  Rel wrel 5656  Fnecfne 37104

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2733

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-v 3453  df-ss 3916  df-opab 5168  df-xp 5657  df-rel 5658  df-fne 37105

This theorem is used by:  isfne  37107  isfne4  37108  fnetr  37119  fneval  37120  fneer  37121  fnessref  37125