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Theorem ima0 5445
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 5092 . 2 (𝐴 “ ∅) = ran (𝐴 ↾ ∅)
2 res0 5365 . . 3 (𝐴 ↾ ∅) = ∅
32rneqi 5317 . 2 ran (𝐴 ↾ ∅) = ran ∅
4 rn0 5342 . 2 ran ∅ = ∅
51, 3, 43eqtri 2647 1 (𝐴 “ ∅) = ∅
Colors of variables: wff setvar class
Syntax hints:   = wceq 1480  c0 3896  ran crn 5080  cres 5081  cima 5082
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  ax-sep 4746  ax-nul 4754  ax-pr 4872
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-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-rab 2916  df-v 3191  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-op 4160  df-br 4619  df-opab 4679  df-xp 5085  df-cnv 5087  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092
This theorem is referenced by:  csbima12  5447  relimasn  5452  elimasni  5456  inisegn0  5461  dffv3  6149  supp0cosupp0  7286  imacosupp  7287  ecexr  7699  domunfican  8185  fodomfi  8191  efgrelexlema  18094  dprdsn  18367  cnindis  21019  cnhaus  21081  cmpfi  21134  xkouni  21325  xkoccn  21345  mbfima  23322  ismbf2d  23331  limcnlp  23565  mdeg0  23751  pserulm  24097  spthispth  26508  pthdlem2  26550  0pth  26869  1pthdlem2  26879  eupth2lemb  26980  disjpreima  29265  imadifxp  29282  dstrvprob  30338  opelco3  31415  funpartlem  31726  poimirlem1  33077  poimirlem2  33078  poimirlem3  33079  poimirlem4  33080  poimirlem5  33081  poimirlem6  33082  poimirlem7  33083  poimirlem10  33086  poimirlem11  33087  poimirlem12  33088  poimirlem13  33089  poimirlem16  33092  poimirlem17  33093  poimirlem19  33095  poimirlem20  33096  poimirlem22  33098  poimirlem23  33099  poimirlem24  33100  poimirlem25  33101  poimirlem28  33104  poimirlem29  33105  poimirlem31  33107  he0  37595  smfresal  40328
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