Theorem brstruct 17059

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 17058	. 2 ⊢ Struct = {⟨𝑓, 𝑥⟩ ∣ (𝑥 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝑓 ∖ {∅}) ∧ dom 𝑓 ⊆ (...‘𝑥))}
2	1	relopabiv 5759	1 ⊢ Rel Struct

Syntax hints:   ∧ w3a 1086   ∈ wcel 2111   ∖ cdif 3894   ∩ cin 3896   ⊆ wss 3897  ∅c0 4280  {csn 4573   × cxp 5612  dom cdm 5614  Rel wrel 5619  Fun wfun 6475  ‘cfv 6481   ≤ cle 11147  ℕcn 12125  ...cfz 13407   Struct cstr 17057

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1544  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-v 3438  df-ss 3914  df-opab 5152  df-xp 5620  df-rel 5621  df-struct 17058

This theorem is referenced by:  isstruct2  17060  structex  17061