MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  brstruct Structured version   Visualization version   GIF version

Theorem brstruct 15913
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.
StepHypRef Expression
1 df-struct 15906 . 2 Struct = {⟨𝑓, 𝑥⟩ ∣ (𝑥 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝑓 ∖ {∅}) ∧ dom 𝑓 ⊆ (...‘𝑥))}
21relopabi 5278 1 Rel Struct
Colors of variables: wff setvar class
Syntax hints:  w3a 1054  wcel 2030  cdif 3604  cin 3606  wss 3607  c0 3948  {csn 4210   × cxp 5141  dom cdm 5143  Rel wrel 5148  Fun wfun 5920  cfv 5926  cle 10113  cn 11058  ...cfz 12364   Struct cstr 15900
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-rab 2950  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-opab 4746  df-xp 5149  df-rel 5150  df-struct 15906
This theorem is referenced by:  isstruct2  15914  structex  15915
  Copyright terms: Public domain W3C validator