Theorem brstruct 17186

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 17185	. 2 ⊢ Struct = {⟨𝑓, 𝑥⟩ ∣ (𝑥 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝑓 ∖ {∅}) ∧ dom 𝑓 ⊆ (...‘𝑥))}
2	1	relopabiv 5795	1 ⊢ Rel Struct

Syntax hints:   ∧ w3a 1099   ∈ wcel 2144   ∖ cdif 3903   ∩ cin 3905   ⊆ wss 3906  ∅c0 4287  {csn 4584   × cxp 5647  dom cdm 5649  Rel wrel 5654  Fun wfun 6517  ‘cfv 6523   ≤ cle 11219  ℕcn 12212  ...cfz 13514   Struct cstr 17184

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-ext 2736

This theorem depends on definitions:  df-bi 209  df-an 400  df-tru 1565  df-ex 1802  df-sb 2093  df-clab 2743  df-cleq 2756  df-clel 2839  df-v 3458  df-ss 3923  df-opab 5165  df-xp 5655  df-rel 5656  df-struct 17185

This theorem is referenced by:  isstruct2  17187  structex  17188