Theorem brstruct 17089

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 17088	. 2 ⊢ Struct = {⟨𝑓, 𝑥⟩ ∣ (𝑥 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝑓 ∖ {∅}) ∧ dom 𝑓 ⊆ (...‘𝑥))}
2	1	relopabiv 5779	1 ⊢ Rel Struct

Syntax hints:   ∧ w3a 1087   ∈ wcel 2114   ∖ cdif 3900   ∩ cin 3902   ⊆ wss 3903  ∅c0 4287  {csn 4582   × cxp 5632  dom cdm 5634  Rel wrel 5639  Fun wfun 6496  ‘cfv 6502   ≤ cle 11181  ℕcn 12159  ...cfz 13437   Struct cstr 17087

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 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1545  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-v 3444  df-ss 3920  df-opab 5163  df-xp 5640  df-rel 5641  df-struct 17088

This theorem is referenced by:  isstruct2  17090  structex  17091