Theorem brstruct 17073

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 17072	. 2 ⊢ Struct = {⟨𝑓, 𝑥⟩ ∣ (𝑥 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝑓 ∖ {∅}) ∧ dom 𝑓 ⊆ (...‘𝑥))}
2	1	relopabiv 5767	1 ⊢ Rel Struct

Syntax hints:   ∧ w3a 1086   ∈ wcel 2113   ∖ cdif 3896   ∩ cin 3898   ⊆ wss 3899  ∅c0 4283  {csn 4578   × cxp 5620  dom cdm 5622  Rel wrel 5627  Fun wfun 6484  ‘cfv 6490   ≤ cle 11165  ℕcn 12143  ...cfz 13421   Struct cstr 17071

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1544  df-ex 1781  df-sb 2068  df-clab 2713  df-cleq 2726  df-clel 2809  df-v 3440  df-ss 3916  df-opab 5159  df-xp 5628  df-rel 5629  df-struct 17072

This theorem is referenced by:  isstruct2  17074  structex  17075