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Theorem imadisj 5048
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 4718 . . 3  |-  ( A
" B )  =  ran  ( A  |`  B )
21eqeq1i 2303 . 2  |-  ( ( A " B )  =  (/)  <->  ran  ( A  |`  B )  =  (/) )
3 dm0rn0 4911 . 2  |-  ( dom  ( A  |`  B )  =  (/)  <->  ran  ( A  |`  B )  =  (/) )
4 dmres 4992 . . . 4  |-  dom  ( A  |`  B )  =  ( B  i^i  dom  A )
5 incom 3374 . . . 4  |-  ( B  i^i  dom  A )  =  ( dom  A  i^i  B )
64, 5eqtri 2316 . . 3  |-  dom  ( A  |`  B )  =  ( dom  A  i^i  B )
76eqeq1i 2303 . 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 1632    i^i cin 3164   (/)c0 3468   dom cdm 4705   ran crn 4706    |` cres 4707   "cima 4708
This theorem is referenced by:  fnimadisj  5380  fnimaeq0  5381  fimacnvdisj  5435  acndom2  7697  isf34lem5  8020  isf34lem7  8021  isf34lem6  8022  limsupgre  11971  isercolllem3  12156  cnconn  17164  1stcfb  17187  xkohaus  17363  qtopeu  17423  fbasrn  17595  mbflimsup  19037  pf1rcl  19448  erdszelem5  23741  fnwe2lem2  27251
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-br 4040  df-opab 4094  df-xp 4711  df-cnv 4713  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718
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