Theorem resdmss 6229

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 6003	. 2 ⊢ dom (𝐴 ↾ 𝐵) = (𝐵 ∩ dom 𝐴)
2		inss1 4182	. 2 ⊢ (𝐵 ∩ dom 𝐴) ⊆ 𝐵
3	1, 2	eqsstri 3977	1 ⊢ dom (𝐴 ↾ 𝐵) ⊆ 𝐵

This proof depends on syntax axioms:   ∩ cin 3898   ⊆ wss 3899  dom cdm 5651   ↾ cres 5653

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 2147  ax-9 2155  ax-ext 2733  ax-sep 5249  ax-pr 5391

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 2740  df-cleq 2753  df-clel 2836  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-br 5104  df-opab 5168  df-xp 5657  df-dm 5661  df-res 5663

This theorem is used by:  ttrclse  9712  noinfbnd2  28070  esplyind  34189  imadomfi  43020  cnvrcl0  44584  tposres3  49933