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Theorem resimass 45183
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 6021 . 2 (𝐴𝐵) ⊆ 𝐴
2 imass1 6121 . 2 ((𝐴𝐵) ⊆ 𝐴 → ((𝐴𝐵) “ 𝐶) ⊆ (𝐴𝐶))
31, 2ax-mp 5 1 ((𝐴𝐵) “ 𝐶) ⊆ (𝐴𝐶)
Colors of variables: wff setvar class
Syntax hints:  wss 3962  cres 5690  cima 5691
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-ext 2705
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-sb 2062  df-clab 2712  df-cleq 2726  df-clel 2813  df-rab 3433  df-v 3479  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-br 5148  df-opab 5210  df-cnv 5696  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701
This theorem is referenced by:  limsupres  45660  limsupresxr  45721  liminfresxr  45722
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