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Theorem ndmima 4916
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 4552 . 2  |-  ( B
" { A }
)  =  ran  ( B  |`  { A }
)
2 dmres 4840 . . . . 5  |-  dom  ( B  |`  { A }
)  =  ( { A }  i^i  dom  B )
3 incom 3268 . . . . 5  |-  ( { A }  i^i  dom  B )  =  ( dom 
B  i^i  { A } )
42, 3eqtri 2160 . . . 4  |-  dom  ( B  |`  { A }
)  =  ( dom 
B  i^i  { A } )
5 disjsn 3585 . . . . 5  |-  ( ( dom  B  i^i  { A } )  =  (/)  <->  -.  A  e.  dom  B )
65biimpri 132 . . . 4  |-  ( -.  A  e.  dom  B  ->  ( dom  B  i^i  { A } )  =  (/) )
74, 6syl5eq 2184 . . 3  |-  ( -.  A  e.  dom  B  ->  dom  ( B  |`  { A } )  =  (/) )
8 dm0rn0 4756 . . 3  |-  ( dom  ( B  |`  { A } )  =  (/)  <->  ran  ( B  |`  { A } )  =  (/) )
97, 8sylib 121 . 2  |-  ( -.  A  e.  dom  B  ->  ran  ( B  |`  { A } )  =  (/) )
101, 9syl5eq 2184 1  |-  ( -.  A  e.  dom  B  ->  ( B " { A } )  =  (/) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1331    e. wcel 1480    i^i cin 3070   (/)c0 3363   {csn 3527   dom cdm 4539   ran crn 4540    |` cres 4541   "cima 4542
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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-br 3930  df-opab 3990  df-xp 4545  df-cnv 4547  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552
This theorem is referenced by:  fvun1  5487
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