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Theorem fin 5437
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 3404 . . . 4  |-  ( ( ran  F  C_  B  /\  ran  F  C_  C
)  <->  ran  F  C_  ( B  i^i  C ) )
21anbi2i 675 . . 3  |-  ( ( F  Fn  A  /\  ( ran  F  C_  B  /\  ran  F  C_  C
) )  <->  ( F  Fn  A  /\  ran  F  C_  ( B  i^i  C
) ) )
3 anandi 801 . . 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 242 . 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 5275 . 2  |-  ( F : A --> ( B  i^i  C )  <->  ( F  Fn  A  /\  ran  F  C_  ( B  i^i  C
) ) )
6 df-f 5275 . . 3  |-  ( F : A --> B  <->  ( F  Fn  A  /\  ran  F  C_  B ) )
7 df-f 5275 . . 3  |-  ( F : A --> C  <->  ( F  Fn  A  /\  ran  F  C_  C ) )
86, 7anbi12i 678 . 2  |-  ( ( F : A --> B  /\  F : A --> C )  <-> 
( ( F  Fn  A  /\  ran  F  C_  B )  /\  ( F  Fn  A  /\  ran  F  C_  C )
) )
94, 5, 83bitr4i 268 1  |-  ( F : A --> ( B  i^i  C )  <->  ( F : A --> B  /\  F : A --> C ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358    i^i cin 3164    C_ wss 3165   ran crn 4706    Fn wfn 5266   -->wf 5267
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-v 2803  df-in 3172  df-ss 3179  df-f 5275
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