Theorem brstruct 17126

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 17125	. 2 ⊢ Struct = {⟨𝑓, 𝑥⟩ ∣ (𝑥 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝑓 ∖ {∅}) ∧ dom 𝑓 ⊆ (...‘𝑥))}
2	1	relopabiv 5826	1 ⊢ Rel Struct

Syntax hints:   ∧ w3a 1084   ∈ wcel 2098   ∖ cdif 3946   ∩ cin 3948   ⊆ wss 3949  ∅c0 4326  {csn 4632   × cxp 5680  dom cdm 5682  Rel wrel 5687  Fun wfun 6547  ‘cfv 6553   ≤ cle 11289  ℕcn 12252  ...cfz 13526   Struct cstr 17124

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2699

This theorem depends on definitions:  df-bi 206  df-an 395  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2706  df-cleq 2720  df-clel 2806  df-v 3475  df-in 3956  df-ss 3966  df-opab 5215  df-xp 5688  df-rel 5689  df-struct 17125

This theorem is referenced by:  isstruct2  17127  structex  17128