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Theorem fnimadisj 5120
Description: A class that is disjoint with the domain of a function has an empty image under the function. (Contributed by FL, 24-Jan-2007.)
Assertion
Ref Expression
fnimadisj  |-  ( ( F  Fn  A  /\  ( A  i^i  C )  =  (/) )  ->  ( F " C )  =  (/) )

Proof of Theorem fnimadisj
StepHypRef Expression
1 fndm 5099 . . . . 5  |-  ( F  Fn  A  ->  dom  F  =  A )
21ineq1d 3198 . . . 4  |-  ( F  Fn  A  ->  ( dom  F  i^i  C )  =  ( A  i^i  C ) )
32eqeq1d 2096 . . 3  |-  ( F  Fn  A  ->  (
( dom  F  i^i  C )  =  (/)  <->  ( A  i^i  C )  =  (/) ) )
43biimpar 291 . 2  |-  ( ( F  Fn  A  /\  ( A  i^i  C )  =  (/) )  ->  ( dom  F  i^i  C )  =  (/) )
5 imadisj 4781 . 2  |-  ( ( F " C )  =  (/)  <->  ( dom  F  i^i  C )  =  (/) )
64, 5sylibr 132 1  |-  ( ( F  Fn  A  /\  ( A  i^i  C )  =  (/) )  ->  ( F " C )  =  (/) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    = wceq 1289    i^i cin 2996   (/)c0 3284   dom cdm 4428   "cima 4431    Fn wfn 4997
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 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-pow 4001  ax-pr 4027
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-dif 2999  df-un 3001  df-in 3003  df-ss 3010  df-nul 3285  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-br 3838  df-opab 3892  df-xp 4434  df-cnv 4436  df-dm 4438  df-rn 4439  df-res 4440  df-ima 4441  df-fn 5005
This theorem is referenced by: (None)
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