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Theorem fnimadisj 5351
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 5330 . . . . 5  |-  ( F  Fn  A  ->  dom  F  =  A )
21ineq1d 3350 . . . 4  |-  ( F  Fn  A  ->  ( dom  F  i^i  C )  =  ( A  i^i  C ) )
32eqeq1d 2198 . . 3  |-  ( F  Fn  A  ->  (
( dom  F  i^i  C )  =  (/)  <->  ( A  i^i  C )  =  (/) ) )
43biimpar 297 . 2  |-  ( ( F  Fn  A  /\  ( A  i^i  C )  =  (/) )  ->  ( dom  F  i^i  C )  =  (/) )
5 imadisj 5005 . 2  |-  ( ( F " C )  =  (/)  <->  ( dom  F  i^i  C )  =  (/) )
64, 5sylibr 134 1  |-  ( ( F  Fn  A  /\  ( A  i^i  C )  =  (/) )  ->  ( F " C )  =  (/) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    = wceq 1364    i^i cin 3143   (/)c0 3437   dom cdm 4641   "cima 4644    Fn wfn 5226
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4189  ax-pr 4224
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-br 4019  df-opab 4080  df-xp 4647  df-cnv 4649  df-dm 4651  df-rn 4652  df-res 4653  df-ima 4654  df-fn 5234
This theorem is referenced by: (None)
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