Theorem ima0 5945

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

Step	Hyp	Ref	Expression
1		df-ima 5568	. 2 ⊢ (𝐴 “ ∅) = ran (𝐴 ↾ ∅)
2		res0 5857	. . 3 ⊢ (𝐴 ↾ ∅) = ∅
3	2	rneqi 5807	. 2 ⊢ ran (𝐴 ↾ ∅) = ran ∅
4		rn0 5796	. 2 ⊢ ran ∅ = ∅
5	1, 3, 4	3eqtri 2848	1 ⊢ (𝐴 “ ∅) = ∅

Syntax hints:   = wceq 1537  ∅c0 4291  ran crn 5556   ↾ cres 5557   “ cima 5558

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-sep 5203  ax-nul 5210  ax-pr 5330

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-rab 3147  df-v 3496  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-sn 4568  df-pr 4570  df-op 4574  df-br 5067  df-opab 5129  df-xp 5561  df-cnv 5563  df-dm 5565  df-rn 5566  df-res 5567  df-ima 5568

This theorem is referenced by:  csbima12  5947  relimasn  5952  elimasni  5956  inisegn0  5961  dffv3  6666  suppco  7870  supp0cosupp0  7872  supp0cosupp0OLD  7873  imacosuppOLD  7875  ecexr  8294  domunfican  8791  fodomfi  8797  efgrelexlema  18875  dprdsn  19158  cnindis  21900  cnhaus  21962  cmpfi  22016  xkouni  22207  xkoccn  22227  mbfima  24231  ismbf2d  24241  limcnlp  24476  mdeg0  24664  pserulm  25010  spthispth  27507  pthdlem2  27549  0pth  27904  1pthdlem2  27915  eupth2lemb  28016  disjpreima  30334  imadifxp  30351  swrdrndisj  30631  dstrvprob  31729  opelco3  33018  funpartlem  33403  poimirlem1  34908  poimirlem2  34909  poimirlem3  34910  poimirlem4  34911  poimirlem5  34912  poimirlem6  34913  poimirlem7  34914  poimirlem10  34917  poimirlem11  34918  poimirlem12  34919  poimirlem13  34920  poimirlem16  34923  poimirlem17  34924  poimirlem19  34926  poimirlem20  34927  poimirlem22  34929  poimirlem23  34930  poimirlem24  34931  poimirlem25  34932  poimirlem28  34935  poimirlem29  34936  poimirlem31  34938  he0  40150  smfresal  43083