Theorem rnssi 5934

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 5933	. 2 ⊢ (𝐴 ⊆ 𝐵 → ran 𝐴 ⊆ ran 𝐵)
3	1, 2	ax-mp 5	1 ⊢ ran 𝐴 ⊆ ran 𝐵

This proof depends on syntax axioms:   ⊆ wss 3900  ran crn 5659

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735

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 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4841  df-br 5109  df-opab 5174  df-cnv 5666  df-dm 5668  df-rn 5669

This theorem is used by:  rnresss  6023  rnin  6155  ssrnres  6188  imadifssranOLD  6221  fssres  6778  rnttrcl  9760  brdom4  10648  smobeth  10710  nqerf  11054  catcoppccl  18342  lern  18815  gsumzres  20174  gsumzaddlem  20186  gsumzadd  20187  dprdfadd  20287  txkgen  24026  dvlog  27036  perpln2  29243  pfxrn2  33603  fixssrn  36884  cnvrcl0  44886