Theorem brstruct 17090

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 17089	. 2 ⊢ Struct = {⟨𝑓, 𝑥⟩ ∣ (𝑥 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝑓 ∖ {∅}) ∧ dom 𝑓 ⊆ (...‘𝑥))}
2	1	relopabiv 5813	1 ⊢ Rel Struct

Syntax hints:   ∧ w3a 1084   ∈ wcel 2098   ∖ cdif 3940   ∩ cin 3942   ⊆ wss 3943  ∅c0 4317  {csn 4623   × cxp 5667  dom cdm 5669  Rel wrel 5674  Fun wfun 6531  ‘cfv 6537   ≤ cle 11253  ℕcn 12216  ...cfz 13490   Struct cstr 17088

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 2697

This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-v 3470  df-in 3950  df-ss 3960  df-opab 5204  df-xp 5675  df-rel 5676  df-struct 17089

This theorem is referenced by:  isstruct2  17091  structex  17092