Theorem brstruct 17190

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 17189	. 2 ⊢ Struct = {⟨𝑓, 𝑥⟩ ∣ (𝑥 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝑓 ∖ {∅}) ∧ dom 𝑓 ⊆ (...‘𝑥))}
2	1	relopabiv 5843	1 ⊢ Rel Struct

Syntax hints:   ∧ w3a 1087   ∈ wcel 2103   ∖ cdif 3967   ∩ cin 3969   ⊆ wss 3970  ∅c0 4347  {csn 4648   × cxp 5697  dom cdm 5699  Rel wrel 5704  Fun wfun 6566  ‘cfv 6572   ≤ cle 11321  ℕcn 12289  ...cfz 13563   Struct cstr 17188

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2105  ax-9 2113  ax-ext 2705

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-ex 1778  df-sb 2065  df-clab 2712  df-cleq 2726  df-clel 2813  df-v 3484  df-ss 3987  df-opab 5232  df-xp 5705  df-rel 5706  df-struct 17189

This theorem is referenced by:  isstruct2  17191  structex  17192