Theorem rnresss 6018

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 6002	. 2 ⊢ (𝐴 ↾ 𝐵) ⊆ 𝐴
2	1	rnssi 5932	1 ⊢ ran (𝐴 ↾ 𝐵) ⊆ ran 𝐴

This proof depends on syntax axioms:   ⊆ wss 3906  ran crn 5664   ↾ cres 5665

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 2148  ax-9 2156  ax-ext 2737

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 2744  df-cleq 2757  df-clel 2840  df-rab 3419  df-v 3459  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-br 5112  df-opab 5176  df-cnv 5671  df-dm 5673  df-rn 5674  df-res 5675

This theorem is used by:  imadifssran  6204  imadifssranOLD  6205  oniso  28515  gsumhashmul  33451  esplyind  34029  nelrnres  45963  limsupvaluz2  46510  supcnvlimsup  46512  limsupgtlem  46549  sge0split  47181