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Theorem 0ima 5788
Description: Image under the empty relation. (Contributed by FL, 11-Jan-2007.)
Assertion
Ref Expression
0ima (∅ “ 𝐴) = ∅

Proof of Theorem 0ima
StepHypRef Expression
1 imassrn 5783 . . 3 (∅ “ 𝐴) ⊆ ran ∅
2 rn0 5677 . . 3 ran ∅ = ∅
31, 2sseqtri 3895 . 2 (∅ “ 𝐴) ⊆ ∅
4 0ss 4237 . 2 ∅ ⊆ (∅ “ 𝐴)
53, 4eqssi 3876 1 (∅ “ 𝐴) = ∅
Colors of variables: wff setvar class
Syntax hints:   = wceq 1507  c0 4180  ran crn 5409  cima 5411
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2750  ax-sep 5061  ax-nul 5068  ax-pr 5187
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2583  df-clab 2759  df-cleq 2771  df-clel 2846  df-nfc 2918  df-ral 3093  df-rex 3094  df-rab 3097  df-v 3417  df-dif 3834  df-un 3836  df-in 3838  df-ss 3845  df-nul 4181  df-if 4352  df-sn 4443  df-pr 4445  df-op 4449  df-br 4931  df-opab 4993  df-xp 5414  df-cnv 5416  df-dm 5418  df-rn 5419  df-res 5420  df-ima 5421
This theorem is referenced by:  csbrn  5901  nghmfval  23037  isnghm  23038  mthmval  32342  ec0  35066  0he  39491  limsup0  41407  0cnf  41591
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