Theorem rnssi 5887

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 5886	. 2 ⊢ (𝐴 ⊆ 𝐵 → ran 𝐴 ⊆ ran 𝐵)
3	1, 2	ax-mp 5	1 ⊢ ran 𝐴 ⊆ ran 𝐵

Syntax hints:   ⊆ wss 3899  ran crn 5623

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2713  df-cleq 2726  df-clel 2809  df-rab 3398  df-v 3440  df-dif 3902  df-un 3904  df-ss 3916  df-nul 4284  df-if 4478  df-sn 4579  df-pr 4581  df-op 4585  df-br 5097  df-opab 5159  df-cnv 5630  df-dm 5632  df-rn 5633

This theorem is referenced by:  rnresss  5974  ssrnres  6134  fssres  6698  smores  8282  rnttrcl  9629  brdom4  10438  smobeth  10495  nqerf  10839  catcoppccl  18039  lern  18512  gsumzres  19836  gsumzaddlem  19848  gsumzadd  19849  dprdfadd  19949  txkgen  23594  dvlog  26614  perpln2  28732  pfxrn2  32971  fixssrn  36048  cnvrcl0  43808