Theorem brstruct 17094

Description: The structure relation is a relation. (Contributed by Mario Carneiro, 29-Aug-2015.)

Assertion

Ref	Expression
brstruct	⊢ Rel Struct

Proof of Theorem brstruct

Dummy variables 𝑥 𝑓 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-struct 17093	. 2 ⊢ Struct = {⟨𝑓, 𝑥⟩ ∣ (𝑥 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝑓 ∖ {∅}) ∧ dom 𝑓 ⊆ (...‘𝑥))}
2	1	relopabiv 5774	1 ⊢ Rel Struct

Syntax hints:   ∧ w3a 1086   ∈ wcel 2109   ∖ cdif 3908   ∩ cin 3910   ⊆ wss 3911  ∅c0 4292  {csn 4585   × cxp 5629  dom cdm 5631  Rel wrel 5636  Fun wfun 6493  ‘cfv 6499   ≤ cle 11185  ℕcn 12162  ...cfz 13444   Struct cstr 17092

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 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-v 3446  df-ss 3928  df-opab 5165  df-xp 5637  df-rel 5638  df-struct 17093

This theorem is referenced by:  isstruct2  17095  structex  17096