Theorem brstruct 17118

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 17117	. 2 ⊢ Struct = {⟨𝑓, 𝑥⟩ ∣ (𝑥 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝑓 ∖ {∅}) ∧ dom 𝑓 ⊆ (...‘𝑥))}
2	1	relopabiv 5776	1 ⊢ Rel Struct

Syntax hints:   ∧ w3a 1087   ∈ wcel 2114   ∖ cdif 3886   ∩ cin 3888   ⊆ wss 3889  ∅c0 4273  {csn 4567   × cxp 5629  dom cdm 5631  Rel wrel 5636  Fun wfun 6492  ‘cfv 6498   ≤ cle 11180  ℕcn 12174  ...cfz 13461   Struct cstr 17116

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1545  df-ex 1782  df-sb 2069  df-clab 2715  df-cleq 2728  df-clel 2811  df-v 3431  df-ss 3906  df-opab 5148  df-xp 5637  df-rel 5638  df-struct 17117

This theorem is referenced by:  isstruct2  17119  structex  17120