Theorem brstruct 17085

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 17084	. 2 ⊢ Struct = {⟨𝑓, 𝑥⟩ ∣ (𝑥 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝑓 ∖ {∅}) ∧ dom 𝑓 ⊆ (...‘𝑥))}
2	1	relopabiv 5819	1 ⊢ Rel Struct

Syntax hints:   ∧ w3a 1085   ∈ wcel 2104   ∖ cdif 3944   ∩ cin 3946   ⊆ wss 3947  ∅c0 4321  {csn 4627   × cxp 5673  dom cdm 5675  Rel wrel 5680  Fun wfun 6536  ‘cfv 6542   ≤ cle 11253  ℕcn 12216  ...cfz 13488   Struct cstr 17083

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-ext 2701

This theorem depends on definitions:  df-bi 206  df-an 395  df-tru 1542  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2722  df-clel 2808  df-v 3474  df-in 3954  df-ss 3964  df-opab 5210  df-xp 5681  df-rel 5682  df-struct 17084

This theorem is referenced by:  isstruct2  17086  structex  17087