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Theorem fin 5763
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 3654 . . . 4  |-  ( ( ran  F  C_  B  /\  ran  F  C_  C
)  <->  ran  F  C_  ( B  i^i  C ) )
21anbi2i 700 . . 3  |-  ( ( F  Fn  A  /\  ( ran  F  C_  B  /\  ran  F  C_  C
) )  <->  ( F  Fn  A  /\  ran  F  C_  ( B  i^i  C
) ) )
3 anandi 837 . . 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 255 . 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 5586 . 2  |-  ( F : A --> ( B  i^i  C )  <->  ( F  Fn  A  /\  ran  F  C_  ( B  i^i  C
) ) )
6 df-f 5586 . . 3  |-  ( F : A --> B  <->  ( F  Fn  A  /\  ran  F  C_  B ) )
7 df-f 5586 . . 3  |-  ( F : A --> C  <->  ( F  Fn  A  /\  ran  F  C_  C ) )
86, 7anbi12i 703 . 2  |-  ( ( F : A --> B  /\  F : A --> C )  <-> 
( ( F  Fn  A  /\  ran  F  C_  B )  /\  ( F  Fn  A  /\  ran  F  C_  C )
) )
94, 5, 83bitr4i 281 1  |-  ( F : A --> ( B  i^i  C )  <->  ( F : A --> B  /\  F : A --> C ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 188    /\ wa 371    i^i cin 3403    C_ wss 3404   ran crn 4835    Fn wfn 5577   -->wf 5578
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431
This theorem depends on definitions:  df-bi 189  df-an 373  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-v 3047  df-in 3411  df-ss 3418  df-f 5586
This theorem is referenced by:  maprnin  28316
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