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Theorem fnresdisj 5137
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 4754 . . 3  |-  Rel  ( F  |`  B )
2 reldm0 4667 . . 3  |-  ( Rel  ( F  |`  B )  ->  ( ( F  |`  B )  =  (/)  <->  dom  ( F  |`  B )  =  (/) ) )
31, 2ax-mp 7 . 2  |-  ( ( F  |`  B )  =  (/)  <->  dom  ( F  |`  B )  =  (/) )
4 dmres 4747 . . . . 5  |-  dom  ( F  |`  B )  =  ( B  i^i  dom  F )
5 incom 3193 . . . . 5  |-  ( B  i^i  dom  F )  =  ( dom  F  i^i  B )
64, 5eqtri 2109 . . . 4  |-  dom  ( F  |`  B )  =  ( dom  F  i^i  B )
7 fndm 5126 . . . . 5  |-  ( F  Fn  A  ->  dom  F  =  A )
87ineq1d 3201 . . . 4  |-  ( F  Fn  A  ->  ( dom  F  i^i  B )  =  ( A  i^i  B ) )
96, 8syl5eq 2133 . . 3  |-  ( F  Fn  A  ->  dom  ( F  |`  B )  =  ( A  i^i  B ) )
109eqeq1d 2097 . 2  |-  ( F  Fn  A  ->  ( dom  ( F  |`  B )  =  (/)  <->  ( A  i^i  B )  =  (/) ) )
113, 10syl5rbb 192 1  |-  ( F  Fn  A  ->  (
( A  i^i  B
)  =  (/)  <->  ( F  |`  B )  =  (/) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 104    = wceq 1290    i^i cin 2999   (/)c0 3287   dom cdm 4452    |` cres 4454   Rel wrel 4457    Fn wfn 5023
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 580  ax-in2 581  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-sep 3963  ax-pow 4015  ax-pr 4045
This theorem depends on definitions:  df-bi 116  df-3an 927  df-tru 1293  df-fal 1296  df-nf 1396  df-sb 1694  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ral 2365  df-rex 2366  df-v 2622  df-dif 3002  df-un 3004  df-in 3006  df-ss 3013  df-nul 3288  df-pw 3435  df-sn 3456  df-pr 3457  df-op 3459  df-br 3852  df-opab 3906  df-xp 4458  df-rel 4459  df-dm 4462  df-res 4464  df-fn 5031
This theorem is referenced by:  fvsnun2  5509  fseq1p1m1  9569
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