Theorem brstruct 16849

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 16848	. 2 ⊢ Struct = {⟨𝑓, 𝑥⟩ ∣ (𝑥 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝑓 ∖ {∅}) ∧ dom 𝑓 ⊆ (...‘𝑥))}
2	1	relopabiv 5730	1 ⊢ Rel Struct

Syntax hints:   ∧ w3a 1086   ∈ wcel 2106   ∖ cdif 3884   ∩ cin 3886   ⊆ wss 3887  ∅c0 4256  {csn 4561   × cxp 5587  dom cdm 5589  Rel wrel 5594  Fun wfun 6427  ‘cfv 6433   ≤ cle 11010  ℕcn 11973  ...cfz 13239   Struct cstr 16847

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1542  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-v 3434  df-in 3894  df-ss 3904  df-opab 5137  df-xp 5595  df-rel 5596  df-struct 16848

This theorem is referenced by:  isstruct2  16850  structex  16851