Theorem resdmss 6188

Description: Subset relationship for the domain of a restriction. (Contributed by Scott Fenton, 9-Aug-2024.)

Assertion

Ref	Expression
resdmss	⊢ dom (𝐴 ↾ 𝐵) ⊆ 𝐵

Proof of Theorem resdmss

Step	Hyp	Ref	Expression
1		dmres 5967	. 2 ⊢ dom (𝐴 ↾ 𝐵) = (𝐵 ∩ dom 𝐴)
2		inss1 4190	. 2 ⊢ (𝐵 ∩ dom 𝐴) ⊆ 𝐵
3	1, 2	eqsstri 3984	1 ⊢ dom (𝐴 ↾ 𝐵) ⊆ 𝐵

Syntax hints:   ∩ cin 3904   ⊆ wss 3905  dom cdm 5623   ↾ cres 5625

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701  ax-sep 5238  ax-nul 5248  ax-pr 5374

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ral 3045  df-rex 3054  df-rab 3397  df-v 3440  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4479  df-sn 4580  df-pr 4582  df-op 4586  df-br 5096  df-opab 5158  df-xp 5629  df-dm 5633  df-res 5635

This theorem is referenced by:  ttrclse  9642  noinfbnd2  27659  imadomfi  41975  cnvrcl0  43598  tposres3  48866