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Theorem fin 5107
Description: Mapping into an intersection. (Contributed by NM, 14-Sep-1999.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)
Assertion
Ref Expression
fin  |-  ( F : A --> ( B  i^i  C )  <->  ( F : A --> B  /\  F : A --> C ) )

Proof of Theorem fin
StepHypRef Expression
1 ssin 3195 . . . 4  |-  ( ( ran  F  C_  B  /\  ran  F  C_  C
)  <->  ran  F  C_  ( B  i^i  C ) )
21anbi2i 445 . . 3  |-  ( ( F  Fn  A  /\  ( ran  F  C_  B  /\  ran  F  C_  C
) )  <->  ( F  Fn  A  /\  ran  F  C_  ( B  i^i  C
) ) )
3 anandi 555 . . 3  |-  ( ( F  Fn  A  /\  ( ran  F  C_  B  /\  ran  F  C_  C
) )  <->  ( ( F  Fn  A  /\  ran  F  C_  B )  /\  ( F  Fn  A  /\  ran  F  C_  C
) ) )
42, 3bitr3i 184 . 2  |-  ( ( F  Fn  A  /\  ran  F  C_  ( B  i^i  C ) )  <->  ( ( F  Fn  A  /\  ran  F  C_  B )  /\  ( F  Fn  A  /\  ran  F  C_  C
) ) )
5 df-f 4936 . 2  |-  ( F : A --> ( B  i^i  C )  <->  ( F  Fn  A  /\  ran  F  C_  ( B  i^i  C
) ) )
6 df-f 4936 . . 3  |-  ( F : A --> B  <->  ( F  Fn  A  /\  ran  F  C_  B ) )
7 df-f 4936 . . 3  |-  ( F : A --> C  <->  ( F  Fn  A  /\  ran  F  C_  C ) )
86, 7anbi12i 448 . 2  |-  ( ( F : A --> B  /\  F : A --> C )  <-> 
( ( F  Fn  A  /\  ran  F  C_  B )  /\  ( F  Fn  A  /\  ran  F  C_  C )
) )
94, 5, 83bitr4i 210 1  |-  ( F : A --> ( B  i^i  C )  <->  ( F : A --> B  /\  F : A --> C ) )
Colors of variables: wff set class
Syntax hints:    /\ wa 102    <-> wb 103    i^i cin 2973    C_ wss 2974   ran crn 4372    Fn wfn 4927   -->wf 4928
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-v 2604  df-in 2980  df-ss 2987  df-f 4936
This theorem is referenced by: (None)
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