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Theorem imadisj 4711
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 4378 . . 3  |-  ( A
" B )  =  ran  ( A  |`  B )
21eqeq1i 2089 . 2  |-  ( ( A " B )  =  (/)  <->  ran  ( A  |`  B )  =  (/) )
3 dm0rn0 4574 . 2  |-  ( dom  ( A  |`  B )  =  (/)  <->  ran  ( A  |`  B )  =  (/) )
4 dmres 4654 . . . 4  |-  dom  ( A  |`  B )  =  ( B  i^i  dom  A )
5 incom 3159 . . . 4  |-  ( B  i^i  dom  A )  =  ( dom  A  i^i  B )
64, 5eqtri 2102 . . 3  |-  dom  ( A  |`  B )  =  ( dom  A  i^i  B )
76eqeq1i 2089 . 2  |-  ( dom  ( A  |`  B )  =  (/)  <->  ( dom  A  i^i  B )  =  (/) )
82, 3, 73bitr2i 206 1  |-  ( ( A " B )  =  (/)  <->  ( dom  A  i^i  B )  =  (/) )
Colors of variables: wff set class
Syntax hints:    <-> wb 103    = wceq 1285    i^i cin 2973   (/)c0 3252   dom cdm 4365   ran crn 4366    |` cres 4367   "cima 4368
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3898  ax-pow 3950  ax-pr 3966
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-rex 2355  df-v 2604  df-dif 2976  df-un 2978  df-in 2980  df-ss 2987  df-nul 3253  df-pw 3386  df-sn 3406  df-pr 3407  df-op 3409  df-br 3788  df-opab 3842  df-xp 4371  df-cnv 4373  df-dm 4375  df-rn 4376  df-res 4377  df-ima 4378
This theorem is referenced by:  fnimadisj  5044  fnimaeq0  5045  fimacnvdisj  5099
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