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Theorem fnresdisj 5306
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 4917 . . 3  |-  Rel  ( F  |`  B )
2 reldm0 4827 . . 3  |-  ( Rel  ( F  |`  B )  ->  ( ( F  |`  B )  =  (/)  <->  dom  ( F  |`  B )  =  (/) ) )
31, 2ax-mp 5 . 2  |-  ( ( F  |`  B )  =  (/)  <->  dom  ( F  |`  B )  =  (/) )
4 dmres 4910 . . . . 5  |-  dom  ( F  |`  B )  =  ( B  i^i  dom  F )
5 incom 3319 . . . . 5  |-  ( B  i^i  dom  F )  =  ( dom  F  i^i  B )
64, 5eqtri 2191 . . . 4  |-  dom  ( F  |`  B )  =  ( dom  F  i^i  B )
7 fndm 5295 . . . . 5  |-  ( F  Fn  A  ->  dom  F  =  A )
87ineq1d 3327 . . . 4  |-  ( F  Fn  A  ->  ( dom  F  i^i  B )  =  ( A  i^i  B ) )
96, 8eqtrid 2215 . . 3  |-  ( F  Fn  A  ->  dom  ( F  |`  B )  =  ( A  i^i  B ) )
109eqeq1d 2179 . 2  |-  ( F  Fn  A  ->  ( dom  ( F  |`  B )  =  (/)  <->  ( A  i^i  B )  =  (/) ) )
113, 10bitr2id 192 1  |-  ( F  Fn  A  ->  (
( A  i^i  B
)  =  (/)  <->  ( F  |`  B )  =  (/) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 104    = wceq 1348    i^i cin 3120   (/)c0 3414   dom cdm 4609    |` cres 4611   Rel wrel 4614    Fn wfn 5191
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4105  ax-pow 4158  ax-pr 4192
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3566  df-sn 3587  df-pr 3588  df-op 3590  df-br 3988  df-opab 4049  df-xp 4615  df-rel 4616  df-dm 4619  df-res 4621  df-fn 5199
This theorem is referenced by:  fvsnun2  5691  fseq1p1m1  10037
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