Theorem brstruct 17168

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 17167	. 2 ⊢ Struct = {⟨𝑓, 𝑥⟩ ∣ (𝑥 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝑓 ∖ {∅}) ∧ dom 𝑓 ⊆ (...‘𝑥))}
2	1	relopabiv 5812	1 ⊢ Rel Struct

Syntax hints:   ∧ w3a 1086   ∈ wcel 2107   ∖ cdif 3930   ∩ cin 3932   ⊆ wss 3933  ∅c0 4315  {csn 4608   × cxp 5665  dom cdm 5667  Rel wrel 5672  Fun wfun 6536  ‘cfv 6542   ≤ cle 11279  ℕcn 12249  ...cfz 13530   Struct cstr 17166

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1542  df-ex 1779  df-sb 2064  df-clab 2713  df-cleq 2726  df-clel 2808  df-v 3466  df-ss 3950  df-opab 5188  df-xp 5673  df-rel 5674  df-struct 17167

This theorem is referenced by:  isstruct2  17169  structex  17170