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Theorem ndmima 4886
Description: The image of a singleton outside the domain is empty. (Contributed by NM, 22-May-1998.)
Assertion
Ref Expression
ndmima  |-  ( -.  A  e.  dom  B  ->  ( B " { A } )  =  (/) )

Proof of Theorem ndmima
StepHypRef Expression
1 df-ima 4522 . 2  |-  ( B
" { A }
)  =  ran  ( B  |`  { A }
)
2 dmres 4810 . . . . 5  |-  dom  ( B  |`  { A }
)  =  ( { A }  i^i  dom  B )
3 incom 3238 . . . . 5  |-  ( { A }  i^i  dom  B )  =  ( dom 
B  i^i  { A } )
42, 3eqtri 2138 . . . 4  |-  dom  ( B  |`  { A }
)  =  ( dom 
B  i^i  { A } )
5 disjsn 3555 . . . . 5  |-  ( ( dom  B  i^i  { A } )  =  (/)  <->  -.  A  e.  dom  B )
65biimpri 132 . . . 4  |-  ( -.  A  e.  dom  B  ->  ( dom  B  i^i  { A } )  =  (/) )
74, 6syl5eq 2162 . . 3  |-  ( -.  A  e.  dom  B  ->  dom  ( B  |`  { A } )  =  (/) )
8 dm0rn0 4726 . . 3  |-  ( dom  ( B  |`  { A } )  =  (/)  <->  ran  ( B  |`  { A } )  =  (/) )
97, 8sylib 121 . 2  |-  ( -.  A  e.  dom  B  ->  ran  ( B  |`  { A } )  =  (/) )
101, 9syl5eq 2162 1  |-  ( -.  A  e.  dom  B  ->  ( B " { A } )  =  (/) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1316    e. wcel 1465    i^i cin 3040   (/)c0 3333   {csn 3497   dom cdm 4509   ran crn 4510    |` cres 4511   "cima 4512
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-fal 1322  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-v 2662  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-nul 3334  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-br 3900  df-opab 3960  df-xp 4515  df-cnv 4517  df-dm 4519  df-rn 4520  df-res 4521  df-ima 4522
This theorem is referenced by:  fvun1  5455
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