Theorem brstruct 16845

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 16844	. 2 ⊢ Struct = {⟨𝑓, 𝑥⟩ ∣ (𝑥 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝑓 ∖ {∅}) ∧ dom 𝑓 ⊆ (...‘𝑥))}
2	1	relopabiv 5728	1 ⊢ Rel Struct

Syntax hints:   ∧ w3a 1086   ∈ wcel 2110   ∖ cdif 3889   ∩ cin 3891   ⊆ wss 3892  ∅c0 4262  {csn 4567   × cxp 5587  dom cdm 5589  Rel wrel 5594  Fun wfun 6425  ‘cfv 6431   ≤ cle 11009  ℕcn 11971  ...cfz 13236   Struct cstr 16843

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2015  ax-8 2112  ax-9 2120  ax-ext 2711

This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1545  df-ex 1787  df-sb 2072  df-clab 2718  df-cleq 2732  df-clel 2818  df-v 3433  df-in 3899  df-ss 3909  df-opab 5142  df-xp 5595  df-rel 5596  df-struct 16844

This theorem is referenced by:  isstruct2  16846  structex  16847