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Theorem ima0 5400
 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 5051 . 2 (𝐴 “ ∅) = ran (𝐴 ↾ ∅)
2 res0 5321 . . 3 (𝐴 ↾ ∅) = ∅
32rneqi 5273 . 2 ran (𝐴 ↾ ∅) = ran ∅
4 rn0 5298 . 2 ran ∅ = ∅
51, 3, 43eqtri 2636 1 (𝐴 “ ∅) = ∅
 Colors of variables: wff setvar class Syntax hints:   = wceq 1475  ∅c0 3874  ran crn 5039   ↾ cres 5040   “ cima 5041 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-opab 4644  df-xp 5044  df-cnv 5046  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051 This theorem is referenced by:  csbima12  5402  relimasn  5407  elimasni  5411  inisegn0  5416  dffv3  6099  supp0cosupp0  7221  imacosupp  7222  ecexr  7634  domunfican  8118  fodomfi  8124  efgrelexlema  17985  dprdsn  18258  cnindis  20906  cnhaus  20968  cmpfi  21021  xkouni  21212  xkoccn  21232  mbfima  23205  ismbf2d  23214  limcnlp  23448  mdeg0  23634  pserulm  23980  0pth  26100  spthispth  26103  1pthonlem2  26120  eupath2  26507  disjpreima  28779  imadifxp  28796  dstrvprob  29860  opelco3  30923  funpartlem  31219  poimirlem1  32580  poimirlem2  32581  poimirlem3  32582  poimirlem4  32583  poimirlem5  32584  poimirlem6  32585  poimirlem7  32586  poimirlem10  32589  poimirlem11  32590  poimirlem12  32591  poimirlem13  32592  poimirlem16  32595  poimirlem17  32596  poimirlem19  32598  poimirlem20  32599  poimirlem22  32601  poimirlem23  32602  poimirlem24  32603  poimirlem25  32604  poimirlem28  32607  poimirlem29  32608  poimirlem31  32610  he0  37098  smfresal  39673  sPthisPth  40932  pthdlem2  40974  0pth-av  41293  1pthdlem2  41303  eupth2lemb  41405
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