Theorem rnssi 5924

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 5923	. 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 5656

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 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2732

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 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-rab 3413  df-v 3452  df-dif 3902  df-un 3904  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-br 5104  df-opab 5168  df-cnv 5663  df-dm 5665  df-rn 5666

This theorem is used by:  rnresss  6010  rnin  6137  ssrnres  6171  imadifssran  6197  fssres  6742  smores  8342  rnttrcl  9704  brdom4  10536  smobeth  10598  nqerf  10942  catcoppccl  18209  lern  18682  gsumzres  20039  gsumzaddlem  20051  gsumzadd  20052  dprdfadd  20152  txkgen  23881  dvlog  26891  perpln2  29068  pfxrn2  33389  fixssrn  36487  cnvrcl0  44468