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Theorem fnresdisj 5354
Description: A function restricted to a class disjoint with its domain is empty. (Contributed by NM, 23-Sep-2004.)
Assertion
Ref Expression
fnresdisj  |-  ( F  Fn  A  ->  (
( A  i^i  B
)  =  (/)  <->  ( F  |`  B )  =  (/) ) )

Proof of Theorem fnresdisj
StepHypRef Expression
1 relres 4983 . . 3  |-  Rel  ( F  |`  B )
2 reldm0 4896 . . 3  |-  ( Rel  ( F  |`  B )  ->  ( ( F  |`  B )  =  (/)  <->  dom  ( F  |`  B )  =  (/) ) )
31, 2ax-mp 8 . 2  |-  ( ( F  |`  B )  =  (/)  <->  dom  ( F  |`  B )  =  (/) )
4 dmres 4976 . . . . 5  |-  dom  ( F  |`  B )  =  ( B  i^i  dom  F )
5 incom 3361 . . . . 5  |-  ( B  i^i  dom  F )  =  ( dom  F  i^i  B )
64, 5eqtri 2303 . . . 4  |-  dom  ( F  |`  B )  =  ( dom  F  i^i  B )
7 fndm 5343 . . . . 5  |-  ( F  Fn  A  ->  dom  F  =  A )
87ineq1d 3369 . . . 4  |-  ( F  Fn  A  ->  ( dom  F  i^i  B )  =  ( A  i^i  B ) )
96, 8syl5eq 2327 . . 3  |-  ( F  Fn  A  ->  dom  ( F  |`  B )  =  ( A  i^i  B ) )
109eqeq1d 2291 . 2  |-  ( F  Fn  A  ->  ( dom  ( F  |`  B )  =  (/)  <->  ( A  i^i  B )  =  (/) ) )
113, 10syl5rbb 249 1  |-  ( F  Fn  A  ->  (
( A  i^i  B
)  =  (/)  <->  ( F  |`  B )  =  (/) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    = wceq 1623    i^i cin 3151   (/)c0 3455   dom cdm 4689    |` cres 4691   Rel wrel 4694    Fn wfn 5250
This theorem is referenced by:  funressn  5706  fvsnun2  5716  axdc3lem4  8079  fseq1p1m1  10857  hashgval  11340  hashinf  11342  mplmonmul  16208  pwssplit1  27188  pwssplit4  27191
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-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-rel 4696  df-dm 4699  df-res 4701  df-fn 5258
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