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Theorem fnimadisj 5364
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 5343 . . . . 5  |-  ( F  Fn  A  ->  dom  F  =  A )
21ineq1d 3369 . . . 4  |-  ( F  Fn  A  ->  ( dom  F  i^i  C )  =  ( A  i^i  C ) )
32eqeq1d 2291 . . 3  |-  ( F  Fn  A  ->  (
( dom  F  i^i  C )  =  (/)  <->  ( A  i^i  C )  =  (/) ) )
43biimpar 471 . 2  |-  ( ( F  Fn  A  /\  ( A  i^i  C )  =  (/) )  ->  ( dom  F  i^i  C )  =  (/) )
5 imadisj 5032 . 2  |-  ( ( F " C )  =  (/)  <->  ( dom  F  i^i  C )  =  (/) )
64, 5sylibr 203 1  |-  ( ( F  Fn  A  /\  ( A  i^i  C )  =  (/) )  ->  ( F " C )  =  (/) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    i^i cin 3151   (/)c0 3455   dom cdm 4689   "cima 4692    Fn wfn 5250
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-opab 4078  df-xp 4695  df-cnv 4697  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-fn 5258
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