Theorem dmresss 5982

Description: The domain of a restriction is a subset of the original domain. (Contributed by Glauco Siliprandi, 23-Oct-2021.) Proof shortened and axiom usage reduced. (Proof shortened by AV, 15-May-2025.)

Assertion

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

Proof of Theorem dmresss

Step	Hyp	Ref	Expression
1		resss 5972	. 2 ⊢ (𝐴 ↾ 𝐵) ⊆ 𝐴
2		dmss 5866	. 2 ⊢ ((𝐴 ↾ 𝐵) ⊆ 𝐴 → dom (𝐴 ↾ 𝐵) ⊆ dom 𝐴)
3	1, 2	ax-mp 5	1 ⊢ dom (𝐴 ↾ 𝐵) ⊆ dom 𝐴

Syntax hints:   ⊆ wss 3914  dom cdm 5638   ↾ cres 5640

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

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-rab 3406  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-sn 4590  df-pr 4592  df-op 4596  df-br 5108  df-dm 5648  df-res 5650

This theorem is referenced by:  limsupresuz2  45707  liminfresuz2  45785  isubgruhgr  47868