Theorem brstruct 17124

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 17123	. 2 ⊢ Struct = {⟨𝑓, 𝑥⟩ ∣ (𝑥 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝑓 ∖ {∅}) ∧ dom 𝑓 ⊆ (...‘𝑥))}
2	1	relopabiv 5791	1 ⊢ Rel Struct

Syntax hints:   ∧ w3a 1086   ∈ wcel 2109   ∖ cdif 3919   ∩ cin 3921   ⊆ wss 3922  ∅c0 4304  {csn 4597   × cxp 5644  dom cdm 5646  Rel wrel 5651  Fun wfun 6513  ‘cfv 6519   ≤ cle 11227  ℕcn 12197  ...cfz 13481   Struct cstr 17122

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2702

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-v 3457  df-ss 3939  df-opab 5178  df-xp 5652  df-rel 5653  df-struct 17123

This theorem is referenced by:  isstruct2  17125  structex  17126