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Theorem mptss 6033
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 6028 . 2 (𝐴𝐵 → ((𝑥𝐵𝐶) ↾ 𝐴) = (𝑥𝐴𝐶))
2 resss 5997 . 2 ((𝑥𝐵𝐶) ↾ 𝐴) ⊆ (𝑥𝐵𝐶)
31, 2eqsstrrdi 4030 1 (𝐴𝐵 → (𝑥𝐴𝐶) ⊆ (𝑥𝐵𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wss 3941  cmpt 5222  cres 5669
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-12 2163  ax-ext 2695  ax-sep 5290  ax-nul 5297  ax-pr 5418
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-rab 3425  df-v 3468  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-sn 4622  df-pr 4624  df-op 4628  df-opab 5202  df-mpt 5223  df-xp 5673  df-rel 5674  df-res 5679
This theorem is referenced by:  tdeglem4  25939  carsgclctunlem2  33837  mhphf  41698  sge0less  45653
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