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Theorem fnimadisj 5019
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 4998 . . . . 5  |-  ( F  Fn  A  ->  dom  F  =  A )
21ineq1d 3137 . . . 4  |-  ( F  Fn  A  ->  ( dom  F  i^i  C )  =  ( A  i^i  C ) )
32eqeq1d 2048 . . 3  |-  ( F  Fn  A  ->  (
( dom  F  i^i  C )  =  (/)  <->  ( A  i^i  C )  =  (/) ) )
43biimpar 281 . 2  |-  ( ( F  Fn  A  /\  ( A  i^i  C )  =  (/) )  ->  ( dom  F  i^i  C )  =  (/) )
5 imadisj 4687 . 2  |-  ( ( F " C )  =  (/)  <->  ( dom  F  i^i  C )  =  (/) )
64, 5sylibr 137 1  |-  ( ( F  Fn  A  /\  ( A  i^i  C )  =  (/) )  ->  ( F " C )  =  (/) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 97    = wceq 1243    i^i cin 2916   (/)c0 3224   dom cdm 4345   "cima 4348    Fn wfn 4897
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944
This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-br 3765  df-opab 3819  df-xp 4351  df-cnv 4353  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-fn 4905
This theorem is referenced by: (None)
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