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Theorem ima0 5985
Description: Image of the empty set. Theorem 3.16(ii) of [Monk1] p. 38. (Contributed by NM, 20-May-1998.)
Assertion
Ref Expression
ima0 (𝐴 “ ∅) = ∅

Proof of Theorem ima0
StepHypRef Expression
1 df-ima 5602 . 2 (𝐴 “ ∅) = ran (𝐴 ↾ ∅)
2 res0 5895 . . 3 (𝐴 ↾ ∅) = ∅
32rneqi 5846 . 2 ran (𝐴 ↾ ∅) = ran ∅
4 rn0 5835 . 2 ran ∅ = ∅
51, 3, 43eqtri 2770 1 (𝐴 “ ∅) = ∅
Colors of variables: wff setvar class
Syntax hints:   = wceq 1539  c0 4256  ran crn 5590  cres 5591  cima 5592
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-br 5075  df-opab 5137  df-xp 5595  df-cnv 5597  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602
This theorem is referenced by:  csbima12  5987  relimasn  5992  elimasni  5999  inisegn0  6006  predprc  6241  dffv3  6770  suppco  8022  supp0cosupp0  8024  ecexr  8503  imafi  8958  domunfican  9087  fodomfi  9092  efgrelexlema  19355  dprdsn  19639  cnindis  22443  cnhaus  22505  cmpfi  22559  xkouni  22750  xkoccn  22770  mbfima  24794  ismbf2d  24804  limcnlp  25042  mdeg0  25235  pserulm  25581  spthispth  28094  pthdlem2  28136  0pth  28489  1pthdlem2  28500  eupth2lemb  28601  disjpreima  30923  imadifxp  30940  2ndimaxp  30984  swrdrndisj  31229  gsumpart  31315  zarclsint  31822  dstrvprob  32438  opelco3  33749  old0  34043  made0  34057  negs0s  34124  funpartlem  34244  poimirlem1  35778  poimirlem2  35779  poimirlem3  35780  poimirlem4  35781  poimirlem5  35782  poimirlem6  35783  poimirlem7  35784  poimirlem10  35787  poimirlem11  35788  poimirlem12  35789  poimirlem13  35790  poimirlem16  35793  poimirlem17  35794  poimirlem19  35796  poimirlem20  35797  poimirlem22  35799  poimirlem23  35800  poimirlem24  35801  poimirlem25  35802  poimirlem28  35805  poimirlem29  35806  poimirlem31  35808  he0  41392  smfresal  44322  predisj  46156
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