Theorem dmresss 5985

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 5975	. 2 ⊢ (𝐴 ↾ 𝐵) ⊆ 𝐴
2		dmss 5869	. 2 ⊢ ((𝐴 ↾ 𝐵) ⊆ 𝐴 → dom (𝐴 ↾ 𝐵) ⊆ dom 𝐴)
3	1, 2	ax-mp 5	1 ⊢ dom (𝐴 ↾ 𝐵) ⊆ dom 𝐴

Syntax hints:   ⊆ wss 3917  dom cdm 5641   ↾ cres 5643

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 2702

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 2709  df-cleq 2722  df-clel 2804  df-rab 3409  df-v 3452  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-sn 4593  df-pr 4595  df-op 4599  df-br 5111  df-dm 5651  df-res 5653

This theorem is referenced by:  limsupresuz2  45714  liminfresuz2  45792  isubgruhgr  47872