Theorem brstruct 17287

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 17286	. 2 ⊢ Struct = {⟨𝑓, 𝑥⟩ ∣ (𝑥 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝑓 ∖ {∅}) ∧ dom 𝑓 ⊆ (...‘𝑥))}
2	1	relopabiv 5794	1 ⊢ Rel Struct

This proof depends on syntax axioms:   ∧ w3a 1103   ∈ wcel 2145   ∖ cdif 3895   ∩ cin 3897   ⊆ wss 3898  ∅c0 4278  {csn 4583   × cxp 5645  dom cdm 5647  Rel wrel 5652  Fun wfun 6521  ‘cfv 6527   ≤ cle 11315  ℕcn 12304  ...cfz 13608   Struct cstr 17285

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-v 3452  df-ss 3915  df-opab 5167  df-xp 5653  df-rel 5654  df-struct 17286

This theorem is used by:  isstruct2  17288  structex  17289