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Theorem fnimadisj 5528
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 5507 . . . . 5  |-  ( F  Fn  A  ->  dom  F  =  A )
21ineq1d 3548 . . . 4  |-  ( F  Fn  A  ->  ( dom  F  i^i  C )  =  ( A  i^i  C ) )
32eqeq1d 2449 . . 3  |-  ( F  Fn  A  ->  (
( dom  F  i^i  C )  =  (/)  <->  ( A  i^i  C )  =  (/) ) )
43biimpar 482 . 2  |-  ( ( F  Fn  A  /\  ( A  i^i  C )  =  (/) )  ->  ( dom  F  i^i  C )  =  (/) )
5 imadisj 5185 . 2  |-  ( ( F " C )  =  (/)  <->  ( dom  F  i^i  C )  =  (/) )
64, 5sylibr 212 1  |-  ( ( F  Fn  A  /\  ( A  i^i  C )  =  (/) )  ->  ( F " C )  =  (/) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1364    i^i cin 3324   (/)c0 3634   dom cdm 4836   "cima 4839    Fn wfn 5410
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-sep 4410  ax-nul 4418  ax-pr 4528
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-rab 2722  df-v 2972  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-nul 3635  df-if 3789  df-sn 3875  df-pr 3877  df-op 3881  df-br 4290  df-opab 4348  df-xp 4842  df-cnv 4844  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-fn 5418
This theorem is referenced by: (None)
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