Theorem rnssi 5905

Description: Subclass inference for range. (Contributed by Peter Mazsa, 24-Sep-2022.)

Hypothesis

Ref	Expression
rnssi.1	⊢ 𝐴 ⊆ 𝐵

Assertion

Ref	Expression
rnssi	⊢ ran 𝐴 ⊆ ran 𝐵

Proof of Theorem rnssi

Step	Hyp	Ref	Expression
1		rnssi.1	. 2 ⊢ 𝐴 ⊆ 𝐵
2		rnss 5904	. 2 ⊢ (𝐴 ⊆ 𝐵 → ran 𝐴 ⊆ ran 𝐵)
3	1, 2	ax-mp 5	1 ⊢ ran 𝐴 ⊆ ran 𝐵

Syntax hints:   ⊆ wss 3895  ran crn 5637

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1805  ax-4 1819  ax-5 1920  ax-6 1977  ax-7 2018  ax-8 2134  ax-9 2142  ax-ext 2724

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 857  df-3an 1097  df-tru 1553  df-fal 1563  df-ex 1790  df-sb 2081  df-clab 2731  df-cleq 2744  df-clel 2827  df-rab 3405  df-v 3446  df-dif 3898  df-un 3900  df-ss 3912  df-nul 4277  df-if 4471  df-sn 4573  df-pr 4575  df-op 4579  df-br 5091  df-opab 5153  df-cnv 5644  df-dm 5646  df-rn 5647

This theorem is referenced by:  rnresss  5992  rnin  6116  ssrnres  6149  fssres  6715  smores  8307  rnttrcl  9663  brdom4  10473  smobeth  10530  nqerf  10874  catcoppccl  18122  lern  18595  gsumzres  19921  gsumzaddlem  19933  gsumzadd  19934  dprdfadd  20034  txkgen  23681  dvlog  26682  perpln2  28846  pfxrn2  33068  fixssrn  36193  cnvrcl0  44139