Theorem brstruct 17031

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 17030	. 2 ⊢ Struct = {⟨𝑓, 𝑥⟩ ∣ (𝑥 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝑓 ∖ {∅}) ∧ dom 𝑓 ⊆ (...‘𝑥))}
2	1	relopabiv 5781	1 ⊢ Rel Struct

Syntax hints:   ∧ w3a 1087   ∈ wcel 2106   ∖ cdif 3910   ∩ cin 3912   ⊆ wss 3913  ∅c0 4287  {csn 4591   × cxp 5636  dom cdm 5638  Rel wrel 5643  Fun wfun 6495  ‘cfv 6501   ≤ cle 11199  ℕcn 12162  ...cfz 13434   Struct cstr 17029

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2702

This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1544  df-ex 1782  df-sb 2068  df-clab 2709  df-cleq 2723  df-clel 2809  df-v 3448  df-in 3920  df-ss 3930  df-opab 5173  df-xp 5644  df-rel 5645  df-struct 17030

This theorem is referenced by:  isstruct2  17032  structex  17033