Theorem rnresss 6037

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 6022	. 2 ⊢ (𝐴 ↾ 𝐵) ⊆ 𝐴
2	1	rnssi 5954	1 ⊢ ran (𝐴 ↾ 𝐵) ⊆ ran 𝐴

Syntax hints:   ⊆ wss 3963  ran crn 5690   ↾ cres 5691

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-opab 5211  df-cnv 5697  df-dm 5699  df-rn 5700  df-res 5701

This theorem is referenced by:  gsumhashmul  33047  nelrnres  45130  limsupvaluz2  45694  supcnvlimsup  45696  limsupgtlem  45733  sge0split  46365