Theorem rnresss 6009

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 5993	. 2 ⊢ (𝐴 ↾ 𝐵) ⊆ 𝐴
2	1	rnssi 5925	1 ⊢ ran (𝐴 ↾ 𝐵) ⊆ ran 𝐴

Syntax hints:   ⊆ wss 3931  ran crn 5660   ↾ cres 5661

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2715  df-cleq 2728  df-clel 2810  df-rab 3421  df-v 3466  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-nul 4314  df-if 4506  df-sn 4607  df-pr 4609  df-op 4613  df-br 5125  df-opab 5187  df-cnv 5667  df-dm 5669  df-rn 5670  df-res 5671

This theorem is referenced by:  imadifssran  6145  onsiso  28226  gsumhashmul  33060  nelrnres  45191  limsupvaluz2  45747  supcnvlimsup  45749  limsupgtlem  45786  sge0split  46418