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Theorem resdmss 6191
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 5963 . 2 dom (𝐴𝐵) = (𝐵 ∩ dom 𝐴)
2 inss1 4192 . 2 (𝐵 ∩ dom 𝐴) ⊆ 𝐵
31, 2eqsstri 3982 1 dom (𝐴𝐵) ⊆ 𝐵
Colors of variables: wff setvar class
Syntax hints:  cin 3913  wss 3914  dom cdm 5637  cres 5639
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5260  ax-nul 5267  ax-pr 5388
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-sn 4591  df-pr 4593  df-op 4597  df-br 5110  df-opab 5172  df-xp 5643  df-dm 5647  df-res 5649
This theorem is referenced by:  ttrclse  9671  noinfbnd2  27102  cnvrcl0  41989
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