Theorem ima0 5481

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 5127	. 2 ⊢ (𝐴 “ ∅) = ran (𝐴 ↾ ∅)
2		res0 5400	. . 3 ⊢ (𝐴 ↾ ∅) = ∅
3	2	rneqi 5352	. 2 ⊢ ran (𝐴 ↾ ∅) = ran ∅
4		rn0 5377	. 2 ⊢ ran ∅ = ∅
5	1, 3, 4	3eqtri 2648	1 ⊢ (𝐴 “ ∅) = ∅

Syntax hints:   = wceq 1483  ∅c0 3915  ran crn 5115   ↾ cres 5116   “ cima 5117

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-xp 5120  df-cnv 5122  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127

This theorem is referenced by:  csbima12  5483  relimasn  5488  elimasni  5492  inisegn0  5497  dffv3  6187  supp0cosupp0  7334  imacosupp  7335  ecexr  7747  domunfican  8233  fodomfi  8239  efgrelexlema  18162  dprdsn  18435  cnindis  21096  cnhaus  21158  cmpfi  21211  xkouni  21402  xkoccn  21422  mbfima  23399  ismbf2d  23408  limcnlp  23642  mdeg0  23830  pserulm  24176  spthispth  26622  pthdlem2  26664  0pth  26986  1pthdlem2  26996  eupth2lemb  27097  disjpreima  29397  imadifxp  29414  dstrvprob  30533  opelco3  31678  funpartlem  32049  poimirlem1  33410  poimirlem2  33411  poimirlem3  33412  poimirlem4  33413  poimirlem5  33414  poimirlem6  33415  poimirlem7  33416  poimirlem10  33419  poimirlem11  33420  poimirlem12  33421  poimirlem13  33422  poimirlem16  33425  poimirlem17  33426  poimirlem19  33428  poimirlem20  33429  poimirlem22  33431  poimirlem23  33432  poimirlem24  33433  poimirlem25  33434  poimirlem28  33437  poimirlem29  33438  poimirlem31  33440  he0  38078  smfresal  40995