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Theorem resimass 41869
 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 5847 . 2 (𝐴𝐵) ⊆ 𝐴
2 imass1 5935 . 2 ((𝐴𝐵) ⊆ 𝐴 → ((𝐴𝐵) “ 𝐶) ⊆ (𝐴𝐶))
31, 2ax-mp 5 1 ((𝐴𝐵) “ 𝐶) ⊆ (𝐴𝐶)
 Colors of variables: wff setvar class Syntax hints:   ⊆ wss 3884   ↾ cres 5525   “ cima 5526 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-v 3446  df-un 3889  df-in 3891  df-ss 3901  df-sn 4529  df-pr 4531  df-op 4535  df-br 5034  df-opab 5096  df-cnv 5531  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536 This theorem is referenced by:  limsupres  42340  limsupresxr  42401  liminfresxr  42402
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