Theorem brstruct 17214

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 17213	. 2 ⊢ Struct = {⟨𝑓, 𝑥⟩ ∣ (𝑥 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝑓 ∖ {∅}) ∧ dom 𝑓 ⊆ (...‘𝑥))}
2	1	relopabiv 5806	1 ⊢ Rel Struct

This proof depends on syntax axioms:   ∧ w3a 1102   ∈ wcel 2142   ∖ cdif 3901   ∩ cin 3903   ⊆ wss 3904  ∅c0 4285  {csn 4588   × cxp 5658  dom cdm 5660  Rel wrel 5665  Fun wfun 6530  ‘cfv 6536   ≤ cle 11250  ℕcn 12239  ...cfz 13541   Struct cstr 17212

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 401  df-tru 1572  df-ex 1809  df-sb 2096  df-clab 2741  df-cleq 2754  df-clel 2837  df-v 3456  df-ss 3921  df-opab 5173  df-xp 5666  df-rel 5667  df-struct 17213

This theorem is used by:  isstruct2  17215  structex  17216