Theorem rnresss 6006

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 3899  ran crn 5652   ↾ 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

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-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-cnv 5659  df-dm 5661  df-rn 5662  df-res 5663

This theorem is used by:  imadifssranOLD  6201  imadifssranOLDOLD  6202  oniso  28650  gsumhashmul  33621  esplyind  34200  nelrnres  46171  limsupvaluz2  46717  supcnvlimsup  46719  limsupgtlem  46756  sge0split  47388