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

Theorem reldmsets 15867
 Description: The structure override operator is a proper operator. (Contributed by Stefan O'Rear, 29-Jan-2015.)
Assertion
Ref Expression
reldmsets Rel dom sSet

Proof of Theorem reldmsets
Dummy variables 𝑒 𝑠 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-sets 15845 . 2 sSet = (𝑠 ∈ V, 𝑒 ∈ V ↦ ((𝑠 ↾ (V ∖ dom {𝑒})) ∪ {𝑒}))
21reldmmpt2 6756 1 Rel dom sSet
 Colors of variables: wff setvar class Syntax hints:  Vcvv 3195   ∖ cdif 3564   ∪ cun 3565  {csn 4168  dom cdm 5104   ↾ cres 5106  Rel wrel 5109   sSet csts 15836 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pr 4897 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-rab 2918  df-v 3197  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-sn 4169  df-pr 4171  df-op 4175  df-br 4645  df-opab 4704  df-xp 5110  df-rel 5111  df-dm 5114  df-oprab 6639  df-mpt2 6640  df-sets 15845 This theorem is referenced by:  setsnid  15896  oduval  17111  oduleval  17112  oppgval  17758  oppgplusfval  17759  mgpval  18473  opprval  18605
 Copyright terms: Public domain W3C validator