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Theorem ndmima 5225
Description: The image of a singleton outside the domain is empty. (Contributed by NM, 22-May-1998.) (Proof shortened by OpenAI, 3-Jul-2020.)
Assertion
Ref Expression
ndmima  |-  ( -.  A  e.  dom  B  ->  ( B " { A } )  =  (/) )

Proof of Theorem ndmima
StepHypRef Expression
1 imadisj 5207 . . 3  |-  ( ( B " { A } )  =  (/)  <->  ( dom  B  i^i  { A } )  =  (/) )
2 disjsn 4063 . . 3  |-  ( ( dom  B  i^i  { A } )  =  (/)  <->  -.  A  e.  dom  B )
31, 2bitri 252 . 2  |-  ( ( B " { A } )  =  (/)  <->  -.  A  e.  dom  B )
43biimpri 209 1  |-  ( -.  A  e.  dom  B  ->  ( B " { A } )  =  (/) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1437    e. wcel 1870    i^i cin 3441   (/)c0 3767   {csn 4002   dom cdm 4854   "cima 4857
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pr 4661
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-rab 2791  df-v 3089  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-sn 4003  df-pr 4005  df-op 4009  df-br 4427  df-opab 4485  df-xp 4860  df-cnv 4862  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867
This theorem is referenced by:  funfv  5948  dffv2  5954  fpwwe2lem13  9066
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