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Theorem fin 5623
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 3563 . . . 4  |-  ( ( ran  F  C_  B  /\  ran  F  C_  C
)  <->  ran  F  C_  ( B  i^i  C ) )
21anbi2i 676 . . 3  |-  ( ( F  Fn  A  /\  ( ran  F  C_  B  /\  ran  F  C_  C
) )  <->  ( F  Fn  A  /\  ran  F  C_  ( B  i^i  C
) ) )
3 anandi 802 . . 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 243 . 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 5458 . 2  |-  ( F : A --> ( B  i^i  C )  <->  ( F  Fn  A  /\  ran  F  C_  ( B  i^i  C
) ) )
6 df-f 5458 . . 3  |-  ( F : A --> B  <->  ( F  Fn  A  /\  ran  F  C_  B ) )
7 df-f 5458 . . 3  |-  ( F : A --> C  <->  ( F  Fn  A  /\  ran  F  C_  C ) )
86, 7anbi12i 679 . 2  |-  ( ( F : A --> B  /\  F : A --> C )  <-> 
( ( F  Fn  A  /\  ran  F  C_  B )  /\  ( F  Fn  A  /\  ran  F  C_  C )
) )
94, 5, 83bitr4i 269 1  |-  ( F : A --> ( B  i^i  C )  <->  ( F : A --> B  /\  F : A --> C ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    /\ wa 359    i^i cin 3319    C_ wss 3320   ran crn 4879    Fn wfn 5449   -->wf 5450
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-v 2958  df-in 3327  df-ss 3334  df-f 5458
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