Theorem brstruct 17113

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 17112	. 2 ⊢ Struct = {⟨𝑓, 𝑥⟩ ∣ (𝑥 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝑓 ∖ {∅}) ∧ dom 𝑓 ⊆ (...‘𝑥))}
2	1	relopabiv 5766	1 ⊢ Rel Struct

Syntax hints:   ∧ w3a 1093   ∈ wcel 2121   ∖ cdif 3882   ∩ cin 3884   ⊆ wss 3885  ∅c0 4264  {csn 4558   × cxp 5619  dom cdm 5621  Rel wrel 5626  Fun wfun 6483  ‘cfv 6489   ≤ cle 11175  ℕcn 12169  ...cfz 13456   Struct cstr 17111

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-ext 2713

This theorem depends on definitions:  df-bi 209  df-an 398  df-tru 1551  df-ex 1788  df-sb 2075  df-clab 2720  df-cleq 2733  df-clel 2816  df-v 3435  df-ss 3902  df-opab 5138  df-xp 5627  df-rel 5628  df-struct 17112

This theorem is referenced by:  isstruct2  17114  structex  17115