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Theorem fnresdisj 5632
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 5249 . . 3  |-  Rel  ( F  |`  B )
2 reldm0 5168 . . 3  |-  ( Rel  ( F  |`  B )  ->  ( ( F  |`  B )  =  (/)  <->  dom  ( F  |`  B )  =  (/) ) )
31, 2ax-mp 5 . 2  |-  ( ( F  |`  B )  =  (/)  <->  dom  ( F  |`  B )  =  (/) )
4 dmres 5242 . . . . 5  |-  dom  ( F  |`  B )  =  ( B  i^i  dom  F )
5 incom 3654 . . . . 5  |-  ( B  i^i  dom  F )  =  ( dom  F  i^i  B )
64, 5eqtri 2483 . . . 4  |-  dom  ( F  |`  B )  =  ( dom  F  i^i  B )
7 fndm 5621 . . . . 5  |-  ( F  Fn  A  ->  dom  F  =  A )
87ineq1d 3662 . . . 4  |-  ( F  Fn  A  ->  ( dom  F  i^i  B )  =  ( A  i^i  B ) )
96, 8syl5eq 2507 . . 3  |-  ( F  Fn  A  ->  dom  ( F  |`  B )  =  ( A  i^i  B ) )
109eqeq1d 2456 . 2  |-  ( F  Fn  A  ->  ( dom  ( F  |`  B )  =  (/)  <->  ( A  i^i  B )  =  (/) ) )
113, 10syl5rbb 258 1  |-  ( F  Fn  A  ->  (
( A  i^i  B
)  =  (/)  <->  ( F  |`  B )  =  (/) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    = wceq 1370    i^i cin 3438   (/)c0 3748   dom cdm 4951    |` cres 4953   Rel wrel 4956    Fn wfn 5524
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4524  ax-nul 4532  ax-pr 4642
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-sn 3989  df-pr 3991  df-op 3995  df-br 4404  df-opab 4462  df-xp 4957  df-rel 4958  df-dm 4961  df-res 4963  df-fn 5532
This theorem is referenced by:  funressn  6007  fvsnun2  6026  axdc3lem4  8737  fseq1p1m1  11655  hashgval  12227  hashinf  12229  swrdn0  12446  pwssplit1  17273  mplmonmul  17677  eulerpartlemt  26921  pwssplit4  29613  wwlkm1edg  30538
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