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

Theorem resdmss 6227
Description: Subset relationship for the domain of a restriction. (Contributed by Scott Fenton, 9-Aug-2024.)
Assertion
Ref Expression
resdmss dom (𝐴𝐵) ⊆ 𝐵

Proof of Theorem resdmss
StepHypRef Expression
1 dmres 5996 . 2 dom (𝐴𝐵) = (𝐵 ∩ dom 𝐴)
2 inss1 4223 . 2 (𝐵 ∩ dom 𝐴) ⊆ 𝐵
31, 2eqsstri 4011 1 dom (𝐴𝐵) ⊆ 𝐵
Colors of variables: wff setvar class
Syntax hints:  cin 3942  wss 3943  dom cdm 5669  cres 5671
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-br 5142  df-opab 5204  df-xp 5675  df-dm 5679  df-res 5681
This theorem is referenced by:  ttrclse  9721  noinfbnd2  27615  cnvrcl0  42933
  Copyright terms: Public domain W3C validator