Theorem ima0 5338

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 4998	. 2 |- ( A " (/) ) = ran ( A |` (/) )
2		res0 5264	. . 3 |- ( A |` (/) ) = (/)
3	2	rneqi 5215	. 2 |- ran ( A |` (/) ) = ran (/)
4		rn0 5240	. 2 |- ran (/) = (/)
5	1, 3, 4	3eqtri 2474	1 |- ( A " (/) ) = (/)

Syntax hints:   = wceq 1381   (/)c0 3767   ran crn 4986   |` cres 4987   "cima 4988

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-sep 4554  ax-nul 4562  ax-pr 4672

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-rab 2800  df-v 3095  df-dif 3461  df-un 3463  df-in 3465  df-ss 3472  df-nul 3768  df-if 3923  df-sn 4011  df-pr 4013  df-op 4017  df-br 4434  df-opab 4492  df-xp 4991  df-cnv 4993  df-dm 4995  df-rn 4996  df-res 4997  df-ima 4998

This theorem is referenced by:  csbima12  5340  relimasn  5346  elimasni  5350  dffv3  5848  supp0cosupp0  6937  imacosupp  6938  ecexr  7314  domunfican  7791  fodomfi  7797  efgrelexlema  16636  gsumval3OLD  16777  dprdsn  16951  cnindis  19659  cnhaus  19721  cmpfi  19774  xkouni  19966  xkoccn  19986  mbfima  21905  ismbf2d  21914  limcnlp  22148  mdeg0  22336  pserulm  22682  0pth  24437  spthispth  24440  1pthonlem2  24457  eupath2  24845  disjpreima  27310  imadifxp  27323  dstrvprob  28276  opelco3  29176  funpartlem  29560  inisegn0  30957