Theorem resdmss 6241

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 6016	. 2 ⊢ dom (𝐴 ↾ 𝐵) = (𝐵 ∩ dom 𝐴)
2		inss1 4192	. 2 ⊢ (𝐵 ∩ dom 𝐴) ⊆ 𝐵
3	1, 2	eqsstri 3986	1 ⊢ dom (𝐴 ↾ 𝐵) ⊆ 𝐵

This proof depends on syntax axioms:   ∩ cin 3907   ⊆ wss 3908  dom cdm 5666   ↾ cres 5668

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2738  ax-sep 5262  ax-pr 5409

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2745  df-cleq 2758  df-clel 2841  df-ral 3083  df-rex 3093  df-rab 3420  df-v 3460  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4290  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-br 5115  df-opab 5179  df-xp 5672  df-dm 5676  df-res 5678

This theorem is used by:  ttrclse  9706  noinfbnd2  27932  esplyind  33996  imadomfi  42810  cnvrcl0  44392  tposres3  49700