Theorem rnresss 5927

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 5916	. 2 ⊢ (𝐴 ↾ 𝐵) ⊆ 𝐴
2	1	rnssi 5849	1 ⊢ ran (𝐴 ↾ 𝐵) ⊆ ran 𝐴

Syntax hints:   ⊆ wss 3887  ran crn 5590   ↾ cres 5591

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-br 5075  df-opab 5137  df-cnv 5597  df-dm 5599  df-rn 5600  df-res 5601

This theorem is referenced by:  gsumhashmul  31316  nelrnres  42725  limsupvaluz2  43279  supcnvlimsup  43281  limsupgtlem  43318  sge0split  43947