Theorem rnssi 5926

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 5925	. 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 3899  ran crn 5655

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2101  df-clab 2742  df-cleq 2755  df-clel 2838  df-rab 3416  df-v 3455  df-ss 3900  df-dif 4049  df-un 4153  df-nul 4281  df-if 4485  df-sn 4588  df-pr 4590  df-op 4838  df-br 5106  df-opab 5170  df-cnv 5662  df-dm 5664  df-rn 5665

This theorem is used by:  rnresss  6011  rnin  6142  ssrnres  6175  imadifssranOLD  6208  fssres  6756  rnttrcl  9733  brdom4  10619  smobeth  10681  nqerf  11025  catcoppccl  18304  lern  18777  gsumzres  20135  gsumzaddlem  20147  gsumzadd  20148  dprdfadd  20248  txkgen  23983  dvlog  26991  perpln2  29198  pfxrn2  33560  fixssrn  36796  cnvrcl0  44765