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Theorem mptss 5373
 Description: Sufficient condition for inclusion in map-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 5369 . 2 (𝐴𝐵 → ((𝑥𝐵𝐶) ↾ 𝐴) = (𝑥𝐴𝐶))
2 resss 5342 . 2 ((𝑥𝐵𝐶) ↾ 𝐴) ⊆ (𝑥𝐵𝐶)
31, 2syl6eqssr 3619 1 (𝐴𝐵 → (𝑥𝐴𝐶) ⊆ (𝑥𝐵𝐶))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ⊆ wss 3540   ↦ cmpt 4643   ↾ cres 5040 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-opab 4644  df-mpt 4645  df-xp 5044  df-rel 5045  df-res 5050 This theorem is referenced by:  carsgclctunlem2  29708  sge0less  39285
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