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Theorem imass1 5641
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 5565 . . 3 (𝐴𝐵 → (𝐴𝐶) ⊆ (𝐵𝐶))
2 rnss 5492 . . 3 ((𝐴𝐶) ⊆ (𝐵𝐶) → ran (𝐴𝐶) ⊆ ran (𝐵𝐶))
31, 2syl 17 . 2 (𝐴𝐵 → ran (𝐴𝐶) ⊆ ran (𝐵𝐶))
4 df-ima 5262 . 2 (𝐴𝐶) = ran (𝐴𝐶)
5 df-ima 5262 . 2 (𝐵𝐶) = ran (𝐵𝐶)
63, 4, 53sstr4g 3793 1 (𝐴𝐵 → (𝐴𝐶) ⊆ (𝐵𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wss 3721  ran crn 5250  cres 5251  cima 5252
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-rab 3069  df-v 3351  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-sn 4315  df-pr 4317  df-op 4321  df-br 4785  df-opab 4845  df-cnv 5257  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262
This theorem is referenced by:  vdwnnlem1  15905  dprdres  18634  imasnopn  21713  imasncld  21714  imasncls  21715  utoptop  22257  restutop  22260  ustuqtop3  22266  utopreg  22275  metustbl  22590  imadifxp  29746  esum2d  30489  eulerpartlemmf  30771  brtrclfv2  38538  frege97d  38563  frege109d  38568  frege131d  38575  hess  38593  resimass  39961  setrecsss  42965
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