Theorem dmresss 42312

Description: The domain of a restriction is a subset of the original domain. (Contributed by Glauco Siliprandi, 23-Oct-2021.)

Assertion

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

Proof of Theorem dmresss

Step	Hyp	Ref	Expression
1		dmres 5847	. 2 ⊢ dom (𝐴 ↾ 𝐵) = (𝐵 ∩ dom 𝐴)
2		inss2 4120	. 2 ⊢ (𝐵 ∩ dom 𝐴) ⊆ dom 𝐴
3	1, 2	eqsstri 3911	1 ⊢ dom (𝐴 ↾ 𝐵) ⊆ dom 𝐴

Syntax hints:   ∩ cin 3842   ⊆ wss 3843  dom cdm 5525   ↾ cres 5527

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-ext 2710  ax-sep 5167  ax-nul 5174  ax-pr 5296

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-sb 2075  df-clab 2717  df-cleq 2730  df-clel 2811  df-ral 3058  df-rex 3059  df-rab 3062  df-v 3400  df-dif 3846  df-un 3848  df-in 3850  df-ss 3860  df-nul 4212  df-if 4415  df-sn 4517  df-pr 4519  df-op 4523  df-br 5031  df-opab 5093  df-xp 5531  df-dm 5535  df-res 5537

This theorem is referenced by:  limsupresuz2  42792  liminfresuz2  42870