MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  rnssi Structured version   Visualization version   GIF version

Theorem rnssi 5896
Description: Subclass inference for range. (Contributed by Peter Mazsa, 24-Sep-2022.)
Hypothesis
Ref Expression
rnssi.1 𝐴𝐵
Assertion
Ref Expression
rnssi ran 𝐴 ⊆ ran 𝐵

Proof of Theorem rnssi
StepHypRef Expression
1 rnssi.1 . 2 𝐴𝐵
2 rnss 5895 . 2 (𝐴𝐵 → ran 𝐴 ⊆ ran 𝐵)
31, 2ax-mp 5 1 ran 𝐴 ⊆ ran 𝐵
Colors of variables: wff setvar class
Syntax hints:  wss 3911  ran crn 5635
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-rab 3407  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-br 5107  df-opab 5169  df-cnv 5642  df-dm 5644  df-rn 5645
This theorem is referenced by:  rnresss  5974  ssrnres  6131  fssres  6709  smores  8299  rnttrcl  9663  brdom4  10471  smobeth  10527  nqerf  10871  catcoppccl  18008  catcoppcclOLD  18009  lern  18485  gsumzres  19691  gsumzaddlem  19703  gsumzadd  19704  dprdfadd  19804  txkgen  23019  dvlog  26022  perpln2  27695  pfxrn2  31845  fixssrn  34538  cnvrcl0  41985
  Copyright terms: Public domain W3C validator