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Theorem resimass 45246
Description: The image of a restriction is a subset of the original image. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
Assertion
Ref Expression
resimass ((𝐴𝐵) “ 𝐶) ⊆ (𝐴𝐶)

Proof of Theorem resimass
StepHypRef Expression
1 resss 6019 . 2 (𝐴𝐵) ⊆ 𝐴
2 imass1 6119 . 2 ((𝐴𝐵) ⊆ 𝐴 → ((𝐴𝐵) “ 𝐶) ⊆ (𝐴𝐶))
31, 2ax-mp 5 1 ((𝐴𝐵) “ 𝐶) ⊆ (𝐴𝐶)
Colors of variables: wff setvar class
Syntax hints:  wss 3951  cres 5687  cima 5688
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-cnv 5693  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698
This theorem is referenced by:  limsupres  45720  limsupresxr  45781  liminfresxr  45782
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