Theorem dmresss 5976

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 5966	. 2 ⊢ (𝐴 ↾ 𝐵) ⊆ 𝐴
2		dmss 5857	. 2 ⊢ ((𝐴 ↾ 𝐵) ⊆ 𝐴 → dom (𝐴 ↾ 𝐵) ⊆ dom 𝐴)
3	1, 2	ax-mp 5	1 ⊢ dom (𝐴 ↾ 𝐵) ⊆ dom 𝐴

Syntax hints:   ⊆ wss 3889  dom cdm 5631   ↾ cres 5633

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2715  df-cleq 2728  df-clel 2811  df-rab 3390  df-v 3431  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-sn 4568  df-pr 4570  df-op 4574  df-br 5086  df-dm 5641  df-res 5643

This theorem is referenced by:  limsupresuz2  46137  liminfresuz2  46215  isubgruhgr  48344