Theorem brstruct 17208

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 17207	. 2 ⊢ Struct = {⟨𝑓, 𝑥⟩ ∣ (𝑥 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝑓 ∖ {∅}) ∧ dom 𝑓 ⊆ (...‘𝑥))}
2	1	relopabiv 5808	1 ⊢ Rel Struct

Syntax hints:   ∧ w3a 1101   ∈ wcel 2149   ∖ cdif 3908   ∩ cin 3910   ⊆ wss 3911  ∅c0 4292  {csn 4592   × cxp 5660  dom cdm 5662  Rel wrel 5667  Fun wfun 6531  ‘cfv 6537   ≤ cle 11244  ℕcn 12233  ...cfz 13535   Struct cstr 17206

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741

This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1570  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-v 3463  df-ss 3928  df-opab 5176  df-xp 5668  df-rel 5669  df-struct 17207

This theorem is referenced by:  isstruct2  17209  structex  17210