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Theorem imadisj 5178
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 4838 . . 3  |-  ( A
" B )  =  ran  ( A  |`  B )
21eqeq1i 2411 . 2  |-  ( ( A " B )  =  (/)  <->  ran  ( A  |`  B )  =  (/) )
3 dm0rn0 5042 . 2  |-  ( dom  ( A  |`  B )  =  (/)  <->  ran  ( A  |`  B )  =  (/) )
4 dmres 5116 . . . 4  |-  dom  ( A  |`  B )  =  ( B  i^i  dom  A )
5 incom 3634 . . . 4  |-  ( B  i^i  dom  A )  =  ( dom  A  i^i  B )
64, 5eqtri 2433 . . 3  |-  dom  ( A  |`  B )  =  ( dom  A  i^i  B )
76eqeq1i 2411 . 2  |-  ( dom  ( A  |`  B )  =  (/)  <->  ( dom  A  i^i  B )  =  (/) )
82, 3, 73bitr2i 275 1  |-  ( ( A " B )  =  (/)  <->  ( dom  A  i^i  B )  =  (/) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 186    = wceq 1407    i^i cin 3415   (/)c0 3740   dom cdm 4825   ran crn 4826    |` cres 4827   "cima 4828
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1641  ax-4 1654  ax-5 1727  ax-6 1773  ax-7 1816  ax-9 1848  ax-10 1863  ax-11 1868  ax-12 1880  ax-13 2028  ax-ext 2382  ax-sep 4519  ax-nul 4527  ax-pr 4632
This theorem depends on definitions:  df-bi 187  df-or 370  df-an 371  df-3an 978  df-tru 1410  df-ex 1636  df-nf 1640  df-sb 1766  df-eu 2244  df-mo 2245  df-clab 2390  df-cleq 2396  df-clel 2399  df-nfc 2554  df-ne 2602  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3063  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3741  df-if 3888  df-sn 3975  df-pr 3977  df-op 3981  df-br 4398  df-opab 4456  df-xp 4831  df-cnv 4833  df-dm 4835  df-rn 4836  df-res 4837  df-ima 4838
This theorem is referenced by:  ndmima  5195  fnimadisj  5684  fnimaeq0  5685  fimacnvdisj  5748  acndom2  8469  isf34lem5  8792  isf34lem7  8793  isf34lem6  8794  limsupgre  13455  isercolllem3  13640  pf1rcl  18707  cnconn  20217  1stcfb  20240  xkohaus  20448  qtopeu  20511  fbasrn  20679  mbflimsup  22367  eulerpartlemt  28829  erdszelem5  29505  fnwe2lem2  35372  imadisjld  35996  imadisjlnd  35997  wnefimgd  35998
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