Theorem brstruct 17188

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 17187	. 2 ⊢ Struct = {⟨𝑓, 𝑥⟩ ∣ (𝑥 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝑓 ∖ {∅}) ∧ dom 𝑓 ⊆ (...‘𝑥))}
2	1	relopabiv 5834	1 ⊢ Rel Struct

Syntax hints:   ∧ w3a 1086   ∈ wcel 2107   ∖ cdif 3961   ∩ cin 3963   ⊆ wss 3964  ∅c0 4340  {csn 4632   × cxp 5688  dom cdm 5690  Rel wrel 5695  Fun wfun 6560  ‘cfv 6566   ≤ cle 11300  ℕcn 12270  ...cfz 13550   Struct cstr 17186

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 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1541  df-ex 1778  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-v 3481  df-ss 3981  df-opab 5212  df-xp 5696  df-rel 5697  df-struct 17187

This theorem is referenced by:  isstruct2  17189  structex  17190