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Theorem fnresdisj 5557
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 5176 . . 3  |-  Rel  ( F  |`  B )
2 reldm0 5089 . . 3  |-  ( Rel  ( F  |`  B )  ->  ( ( F  |`  B )  =  (/)  <->  dom  ( F  |`  B )  =  (/) ) )
31, 2ax-mp 8 . 2  |-  ( ( F  |`  B )  =  (/)  <->  dom  ( F  |`  B )  =  (/) )
4 dmres 5169 . . . . 5  |-  dom  ( F  |`  B )  =  ( B  i^i  dom  F )
5 incom 3535 . . . . 5  |-  ( B  i^i  dom  F )  =  ( dom  F  i^i  B )
64, 5eqtri 2458 . . . 4  |-  dom  ( F  |`  B )  =  ( dom  F  i^i  B )
7 fndm 5546 . . . . 5  |-  ( F  Fn  A  ->  dom  F  =  A )
87ineq1d 3543 . . . 4  |-  ( F  Fn  A  ->  ( dom  F  i^i  B )  =  ( A  i^i  B ) )
96, 8syl5eq 2482 . . 3  |-  ( F  Fn  A  ->  dom  ( F  |`  B )  =  ( A  i^i  B ) )
109eqeq1d 2446 . 2  |-  ( F  Fn  A  ->  ( dom  ( F  |`  B )  =  (/)  <->  ( A  i^i  B )  =  (/) ) )
113, 10syl5rbb 251 1  |-  ( F  Fn  A  ->  (
( A  i^i  B
)  =  (/)  <->  ( F  |`  B )  =  (/) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    = wceq 1653    i^i cin 3321   (/)c0 3630   dom cdm 4880    |` cres 4882   Rel wrel 4885    Fn wfn 5451
This theorem is referenced by:  funressn  5921  fvsnun2  5931  axdc3lem4  8335  fseq1p1m1  11124  hashgval  11623  hashinf  11625  mplmonmul  16529  pwssplit1  27167  pwssplit4  27170
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pr 4405
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-br 4215  df-opab 4269  df-xp 4886  df-rel 4887  df-dm 4890  df-res 4892  df-fn 5459
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