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Theorem imadisj 5188
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 4853 . . 3  |-  ( A
" B )  =  ran  ( A  |`  B )
21eqeq1i 2450 . 2  |-  ( ( A " B )  =  (/)  <->  ran  ( A  |`  B )  =  (/) )
3 dm0rn0 5056 . 2  |-  ( dom  ( A  |`  B )  =  (/)  <->  ran  ( A  |`  B )  =  (/) )
4 dmres 5131 . . . 4  |-  dom  ( A  |`  B )  =  ( B  i^i  dom  A )
5 incom 3543 . . . 4  |-  ( B  i^i  dom  A )  =  ( dom  A  i^i  B )
64, 5eqtri 2463 . . 3  |-  dom  ( A  |`  B )  =  ( dom  A  i^i  B )
76eqeq1i 2450 . 2  |-  ( dom  ( A  |`  B )  =  (/)  <->  ( dom  A  i^i  B )  =  (/) )
82, 3, 73bitr2i 273 1  |-  ( ( A " B )  =  (/)  <->  ( dom  A  i^i  B )  =  (/) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    = wceq 1369    i^i cin 3327   (/)c0 3637   dom cdm 4840   ran crn 4841    |` cres 4842   "cima 4843
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4413  ax-nul 4421  ax-pr 4531
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2720  df-rex 2721  df-rab 2724  df-v 2974  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-sn 3878  df-pr 3880  df-op 3884  df-br 4293  df-opab 4351  df-xp 4846  df-cnv 4848  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853
This theorem is referenced by:  fnimadisj  5531  fnimaeq0  5532  fimacnvdisj  5589  acndom2  8224  isf34lem5  8547  isf34lem7  8548  isf34lem6  8549  limsupgre  12959  isercolllem3  13144  pf1rcl  17783  cnconn  19026  1stcfb  19049  xkohaus  19226  qtopeu  19289  fbasrn  19457  mbflimsup  21144  eulerpartlemt  26754  erdszelem5  27083  fnwe2lem2  29404
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