Theorem rnresss 6017

Description: The range of a restriction is a subset of the whole range. (Contributed by Glauco Siliprandi, 17-Aug-2020.)

Assertion

Ref	Expression
rnresss	⊢ ran (𝐴 ↾ 𝐵) ⊆ ran 𝐴

Proof of Theorem rnresss

Step	Hyp	Ref	Expression
1		resss 6006	. 2 ⊢ (𝐴 ↾ 𝐵) ⊆ 𝐴
2	1	rnssi 5939	1 ⊢ ran (𝐴 ↾ 𝐵) ⊆ ran 𝐴

Syntax hints:   ⊆ wss 3948  ran crn 5677   ↾ cres 5678

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2702

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-rab 3432  df-v 3475  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-br 5149  df-opab 5211  df-cnv 5684  df-dm 5686  df-rn 5687  df-res 5688

This theorem is referenced by:  gsumhashmul  32646  nelrnres  44348  limsupvaluz2  44916  supcnvlimsup  44918  limsupgtlem  44955  sge0split  45587