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Theorem mptss 5418
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 5413 . 2 (𝐴𝐵 → ((𝑥𝐵𝐶) ↾ 𝐴) = (𝑥𝐴𝐶))
2 resss 5386 . 2 ((𝑥𝐵𝐶) ↾ 𝐴) ⊆ (𝑥𝐵𝐶)
31, 2syl6eqssr 3640 1 (𝐴𝐵 → (𝑥𝐴𝐶) ⊆ (𝑥𝐵𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wss 3559  cmpt 4678  cres 5081
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4746  ax-nul 4754  ax-pr 4872
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-rab 2916  df-v 3191  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-op 4160  df-opab 4679  df-mpt 4680  df-xp 5085  df-rel 5086  df-res 5091
This theorem is referenced by:  carsgclctunlem2  30186  sge0less  39942
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