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Theorem 0ima 5441
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 5436 . . 3 (∅ “ 𝐴) ⊆ ran ∅
2 rn0 5337 . . 3 ran ∅ = ∅
31, 2sseqtri 3616 . 2 (∅ “ 𝐴) ⊆ ∅
4 0ss 3944 . 2 ∅ ⊆ (∅ “ 𝐴)
53, 4eqssi 3599 1 (∅ “ 𝐴) = ∅
Colors of variables: wff setvar class
Syntax hints:   = wceq 1480  c0 3891  ran crn 5075  cima 5077
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 4741  ax-nul 4749  ax-pr 4867
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-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-br 4614  df-opab 4674  df-xp 5080  df-cnv 5082  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087
This theorem is referenced by:  csbrn  5555  nghmfval  22436  isnghm  22437  mthmval  31177  0he  37555  limsup0  39327  0cnf  39390  mbf0  39477
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