Theorem rnresss 6007

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 5992	. 2 ⊢ (𝐴 ↾ 𝐵) ⊆ 𝐴
2	1	rnssi 5922	1 ⊢ ran (𝐴 ↾ 𝐵) ⊆ ran 𝐴

This proof depends on syntax axioms:   ⊆ wss 3896  ran crn 5651   ↾ cres 5652

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-rab 3413  df-v 3452  df-ss 3897  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  df-if 4482  df-sn 4585  df-pr 4587  df-op 4835  df-br 5103  df-opab 5167  df-cnv 5658  df-dm 5660  df-rn 5661  df-res 5662

This theorem is used by:  imadifssranOLD  6204  imadifssranOLDOLD  6205  smores  8365  oniso  28664  gsumhashmul  33676  esplyind  34255  nelrnres  46309  limsupvaluz2  46855  supcnvlimsup  46857  limsupgtlem  46894  sge0split  47526