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Theorem imass1 5469
Description: Subset theorem for image. (Contributed by NM, 16-Mar-2004.)
Assertion
Ref Expression
imass1 (𝐴𝐵 → (𝐴𝐶) ⊆ (𝐵𝐶))

Proof of Theorem imass1
StepHypRef Expression
1 ssres 5393 . . 3 (𝐴𝐵 → (𝐴𝐶) ⊆ (𝐵𝐶))
2 rnss 5324 . . 3 ((𝐴𝐶) ⊆ (𝐵𝐶) → ran (𝐴𝐶) ⊆ ran (𝐵𝐶))
31, 2syl 17 . 2 (𝐴𝐵 → ran (𝐴𝐶) ⊆ ran (𝐵𝐶))
4 df-ima 5097 . 2 (𝐴𝐶) = ran (𝐴𝐶)
5 df-ima 5097 . 2 (𝐵𝐶) = ran (𝐵𝐶)
63, 4, 53sstr4g 3631 1 (𝐴𝐵 → (𝐴𝐶) ⊆ (𝐵𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wss 3560  ran crn 5085  cres 5086  cima 5087
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
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 2917  df-v 3192  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-sn 4156  df-pr 4158  df-op 4162  df-br 4624  df-opab 4684  df-cnv 5092  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097
This theorem is referenced by:  vdwnnlem1  15642  dprdres  18367  imasnopn  21433  imasncld  21434  imasncls  21435  utoptop  21978  restutop  21981  ustuqtop3  21987  utopreg  21996  metustbl  22311  imadifxp  29300  esum2d  29978  eulerpartlemmf  30260  brtrclfv2  37539  frege97d  37564  frege109d  37569  frege131d  37576  hess  37595  resimass  38959
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