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Theorem imadisj 5032
Description: A class whose image under another is empty is disjoint with the other's domain. (Contributed by FL, 24-Jan-2007.)
Assertion
Ref Expression
imadisj  |-  ( ( A " B )  =  (/)  <->  ( dom  A  i^i  B )  =  (/) )

Proof of Theorem imadisj
StepHypRef Expression
1 df-ima 4702 . . 3  |-  ( A
" B )  =  ran  ( A  |`  B )
21eqeq1i 2290 . 2  |-  ( ( A " B )  =  (/)  <->  ran  ( A  |`  B )  =  (/) )
3 dm0rn0 4895 . 2  |-  ( dom  ( A  |`  B )  =  (/)  <->  ran  ( A  |`  B )  =  (/) )
4 dmres 4976 . . . 4  |-  dom  ( A  |`  B )  =  ( B  i^i  dom  A )
5 incom 3361 . . . 4  |-  ( B  i^i  dom  A )  =  ( dom  A  i^i  B )
64, 5eqtri 2303 . . 3  |-  dom  ( A  |`  B )  =  ( dom  A  i^i  B )
76eqeq1i 2290 . 2  |-  ( dom  ( A  |`  B )  =  (/)  <->  ( dom  A  i^i  B )  =  (/) )
82, 3, 73bitr2i 264 1  |-  ( ( A " B )  =  (/)  <->  ( dom  A  i^i  B )  =  (/) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    = wceq 1623    i^i cin 3151   (/)c0 3455   dom cdm 4689   ran crn 4690    |` cres 4691   "cima 4692
This theorem is referenced by:  fnimadisj  5364  fnimaeq0  5365  fimacnvdisj  5419  acndom2  7681  isf34lem5  8004  isf34lem7  8005  isf34lem6  8006  limsupgre  11955  isercolllem3  12140  cnconn  17148  1stcfb  17171  xkohaus  17347  qtopeu  17407  fbasrn  17579  mbflimsup  19021  pf1rcl  19432  erdszelem5  23726  fnwe2lem2  27148
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-opab 4078  df-xp 4695  df-cnv 4697  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702
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