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Theorem ndmima 5241
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 4891 . 2  |-  ( B
" { A }
)  =  ran  ( B  |`  { A }
)
2 dmres 5167 . . . . 5  |-  dom  ( B  |`  { A }
)  =  ( { A }  i^i  dom  B )
3 incom 3533 . . . . 5  |-  ( { A }  i^i  dom  B )  =  ( dom 
B  i^i  { A } )
42, 3eqtri 2456 . . . 4  |-  dom  ( B  |`  { A }
)  =  ( dom 
B  i^i  { A } )
5 disjsn 3868 . . . . 5  |-  ( ( dom  B  i^i  { A } )  =  (/)  <->  -.  A  e.  dom  B )
65biimpri 198 . . . 4  |-  ( -.  A  e.  dom  B  ->  ( dom  B  i^i  { A } )  =  (/) )
74, 6syl5eq 2480 . . 3  |-  ( -.  A  e.  dom  B  ->  dom  ( B  |`  { A } )  =  (/) )
8 dm0rn0 5086 . . 3  |-  ( dom  ( B  |`  { A } )  =  (/)  <->  ran  ( B  |`  { A } )  =  (/) )
97, 8sylib 189 . 2  |-  ( -.  A  e.  dom  B  ->  ran  ( B  |`  { A } )  =  (/) )
101, 9syl5eq 2480 1  |-  ( -.  A  e.  dom  B  ->  ( B " { A } )  =  (/) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1652    e. wcel 1725    i^i cin 3319   (/)c0 3628   {csn 3814   dom cdm 4878   ran crn 4879    |` cres 4880   "cima 4881
This theorem is referenced by:  funfv  5790  dffv2  5796  fpwwe2lem13  8517
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pr 4403
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-br 4213  df-opab 4267  df-xp 4884  df-cnv 4886  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891
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