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Theorem mptss 6046
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 6041 . 2 (𝐴𝐵 → ((𝑥𝐵𝐶) ↾ 𝐴) = (𝑥𝐴𝐶))
2 resss 6010 . 2 ((𝑥𝐵𝐶) ↾ 𝐴) ⊆ (𝑥𝐵𝐶)
31, 2eqsstrrdi 4035 1 (𝐴𝐵 → (𝑥𝐴𝐶) ⊆ (𝑥𝐵𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wss 3947  cmpt 5231  cres 5680
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-12 2167  ax-ext 2699  ax-sep 5299  ax-nul 5306  ax-pr 5429
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-rab 3430  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-opab 5211  df-mpt 5232  df-xp 5684  df-rel 5685  df-res 5690
This theorem is referenced by:  tdeglem4  25994  carsgclctunlem2  33939  mhphf  41830  sge0less  45780
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