Theorem resdmss 6201

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 5979	. 2 ⊢ dom (𝐴 ↾ 𝐵) = (𝐵 ∩ dom 𝐴)
2		inss1 4191	. 2 ⊢ (𝐵 ∩ dom 𝐴) ⊆ 𝐵
3	1, 2	eqsstri 3982	1 ⊢ dom (𝐴 ↾ 𝐵) ⊆ 𝐵

Syntax hints:   ∩ cin 3902   ⊆ wss 3903  dom cdm 5632   ↾ cres 5634

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 2709  ax-sep 5243  ax-pr 5379

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 2716  df-cleq 2729  df-clel 2812  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-br 5101  df-opab 5163  df-xp 5638  df-dm 5642  df-res 5644

This theorem is referenced by:  ttrclse  9648  noinfbnd2  27711  esplyind  33751  imadomfi  42369  cnvrcl0  43978  tposres3  49237