Theorem resdmss 6235

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 6009	. 2 ⊢ dom (𝐴 ↾ 𝐵) = (𝐵 ∩ dom 𝐴)
2		inss1 4185	. 2 ⊢ (𝐵 ∩ dom 𝐴) ⊆ 𝐵
3	1, 2	eqsstri 3980	1 ⊢ dom (𝐴 ↾ 𝐵) ⊆ 𝐵

This proof depends on syntax axioms:   ∩ cin 3901   ⊆ wss 3902  dom cdm 5659   ↾ cres 5661

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 2734  ax-sep 5255  ax-pr 5402

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 2741  df-cleq 2754  df-clel 2837  df-ral 3079  df-rex 3089  df-rab 3415  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-sn 4588  df-pr 4590  df-op 4594  df-br 5108  df-opab 5172  df-xp 5665  df-dm 5669  df-res 5671

This theorem is used by:  ttrclse  9710  noinfbnd2  27975  esplyind  34093  imadomfi  42876  cnvrcl0  44473  tposres3  49815