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

Theorem mptss 5898
 Description: Sufficient condition for inclusion among two functions in maps-to notation. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
Assertion
Ref Expression
mptss (𝐴𝐵 → (𝑥𝐴𝐶) ⊆ (𝑥𝐵𝐶))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵
Allowed substitution hint:   𝐶(𝑥)

Proof of Theorem mptss
StepHypRef Expression
1 resmpt 5893 . 2 (𝐴𝐵 → ((𝑥𝐵𝐶) ↾ 𝐴) = (𝑥𝐴𝐶))
2 resss 5866 . 2 ((𝑥𝐵𝐶) ↾ 𝐴) ⊆ (𝑥𝐵𝐶)
31, 2eqsstrrdi 4009 1 (𝐴𝐵 → (𝑥𝐴𝐶) ⊆ (𝑥𝐵𝐶))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ⊆ wss 3920   ↦ cmpt 5133   ↾ cres 5545 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5190  ax-nul 5197  ax-pr 5318 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-rab 3142  df-v 3483  df-dif 3923  df-un 3925  df-in 3927  df-ss 3937  df-nul 4278  df-if 4452  df-sn 4552  df-pr 4554  df-op 4558  df-opab 5116  df-mpt 5134  df-xp 5549  df-rel 5550  df-res 5555 This theorem is referenced by:  carsgclctunlem2  31637  sge0less  42958
 Copyright terms: Public domain W3C validator