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Theorem ndmima 4732
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 4384 . 2  |-  ( B
" { A }
)  =  ran  ( B  |`  { A }
)
2 dmres 4660 . . . . 5  |-  dom  ( B  |`  { A }
)  =  ( { A }  i^i  dom  B )
3 incom 3165 . . . . 5  |-  ( { A }  i^i  dom  B )  =  ( dom 
B  i^i  { A } )
42, 3eqtri 2102 . . . 4  |-  dom  ( B  |`  { A }
)  =  ( dom 
B  i^i  { A } )
5 disjsn 3462 . . . . 5  |-  ( ( dom  B  i^i  { A } )  =  (/)  <->  -.  A  e.  dom  B )
65biimpri 131 . . . 4  |-  ( -.  A  e.  dom  B  ->  ( dom  B  i^i  { A } )  =  (/) )
74, 6syl5eq 2126 . . 3  |-  ( -.  A  e.  dom  B  ->  dom  ( B  |`  { A } )  =  (/) )
8 dm0rn0 4580 . . 3  |-  ( dom  ( B  |`  { A } )  =  (/)  <->  ran  ( B  |`  { A } )  =  (/) )
97, 8sylib 120 . 2  |-  ( -.  A  e.  dom  B  ->  ran  ( B  |`  { A } )  =  (/) )
101, 9syl5eq 2126 1  |-  ( -.  A  e.  dom  B  ->  ( B " { A } )  =  (/) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1285    e. wcel 1434    i^i cin 2973   (/)c0 3258   {csn 3406   dom cdm 4371   ran crn 4372    |` cres 4373   "cima 4374
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3904  ax-pow 3956  ax-pr 3972
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-rex 2355  df-v 2604  df-dif 2976  df-un 2978  df-in 2980  df-ss 2987  df-nul 3259  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-br 3794  df-opab 3848  df-xp 4377  df-cnv 4379  df-dm 4381  df-rn 4382  df-res 4383  df-ima 4384
This theorem is referenced by:  fvun1  5271
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