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Theorem ndmimaOLD 5218
Description: The image of a singleton outside the domain is empty. (Contributed by NM, 22-May-1998.) Obsolete version of ndmima 5217 as of 3-Jul-2020. (New usage is discouraged.) (Proof modification is discouraged.)
Assertion
Ref Expression
ndmimaOLD  |-  ( -.  A  e.  dom  B  ->  ( B " { A } )  =  (/) )

Proof of Theorem ndmimaOLD
StepHypRef Expression
1 df-ima 4859 . 2  |-  ( B
" { A }
)  =  ran  ( B  |`  { A }
)
2 dmres 5137 . . . . 5  |-  dom  ( B  |`  { A }
)  =  ( { A }  i^i  dom  B )
3 incom 3652 . . . . 5  |-  ( { A }  i^i  dom  B )  =  ( dom 
B  i^i  { A } )
42, 3eqtri 2449 . . . 4  |-  dom  ( B  |`  { A }
)  =  ( dom 
B  i^i  { A } )
5 disjsn 4054 . . . . 5  |-  ( ( dom  B  i^i  { A } )  =  (/)  <->  -.  A  e.  dom  B )
65biimpri 209 . . . 4  |-  ( -.  A  e.  dom  B  ->  ( dom  B  i^i  { A } )  =  (/) )
74, 6syl5eq 2473 . . 3  |-  ( -.  A  e.  dom  B  ->  dom  ( B  |`  { A } )  =  (/) )
8 dm0rn0 5063 . . 3  |-  ( dom  ( B  |`  { A } )  =  (/)  <->  ran  ( B  |`  { A } )  =  (/) )
97, 8sylib 199 . 2  |-  ( -.  A  e.  dom  B  ->  ran  ( B  |`  { A } )  =  (/) )
101, 9syl5eq 2473 1  |-  ( -.  A  e.  dom  B  ->  ( B " { A } )  =  (/) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1437    e. wcel 1867    i^i cin 3432   (/)c0 3758   {csn 3993   dom cdm 4846   ran crn 4847    |` cres 4848   "cima 4849
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 1748  ax-6 1794  ax-7 1838  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-sep 4540  ax-nul 4548  ax-pr 4653
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 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-ral 2778  df-rex 2779  df-rab 2782  df-v 3080  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-nul 3759  df-if 3907  df-sn 3994  df-pr 3996  df-op 4000  df-br 4418  df-opab 4477  df-xp 4852  df-cnv 4854  df-dm 4856  df-rn 4857  df-res 4858  df-ima 4859
This theorem is referenced by: (None)
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