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Theorem resimass 39967
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 5563 . 2 (𝐴𝐵) ⊆ 𝐴
2 imass1 5641 . 2 ((𝐴𝐵) ⊆ 𝐴 → ((𝐴𝐵) “ 𝐶) ⊆ (𝐴𝐶))
31, 2ax-mp 5 1 ((𝐴𝐵) “ 𝐶) ⊆ (𝐴𝐶)
Colors of variables: wff setvar class
Syntax hints:  wss 3723  cres 5251  cima 5252
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-rab 3070  df-v 3353  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-br 4787  df-opab 4847  df-cnv 5257  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262
This theorem is referenced by:  limsupres  40455  limsupresxr  40516  liminfresxr  40517
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