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Theorem fint 5420
Description: Function into an intersection. (Contributed by NM, 14-Oct-1999.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)
Hypothesis
Ref Expression
fint.1  |-  B  =/=  (/)
Assertion
Ref Expression
fint  |-  ( F : A --> |^| B  <->  A. x  e.  B  F : A --> x )
Distinct variable groups:    x, A    x, B    x, F

Proof of Theorem fint
StepHypRef Expression
1 ssint 3878 . . . 4  |-  ( ran 
F  C_  |^| B  <->  A. x  e.  B  ran  F  C_  x )
21anbi2i 675 . . 3  |-  ( ( F  Fn  A  /\  ran  F  C_  |^| B )  <-> 
( F  Fn  A  /\  A. x  e.  B  ran  F  C_  x )
)
3 fint.1 . . . 4  |-  B  =/=  (/)
4 r19.28zv 3549 . . . 4  |-  ( B  =/=  (/)  ->  ( A. x  e.  B  ( F  Fn  A  /\  ran  F  C_  x )  <->  ( F  Fn  A  /\  A. x  e.  B  ran  F 
C_  x ) ) )
53, 4ax-mp 8 . . 3  |-  ( A. x  e.  B  ( F  Fn  A  /\  ran  F  C_  x )  <->  ( F  Fn  A  /\  A. x  e.  B  ran  F 
C_  x ) )
62, 5bitr4i 243 . 2  |-  ( ( F  Fn  A  /\  ran  F  C_  |^| B )  <->  A. x  e.  B  ( F  Fn  A  /\  ran  F  C_  x
) )
7 df-f 5259 . 2  |-  ( F : A --> |^| B  <->  ( F  Fn  A  /\  ran  F  C_  |^| B ) )
8 df-f 5259 . . 3  |-  ( F : A --> x  <->  ( F  Fn  A  /\  ran  F  C_  x ) )
98ralbii 2567 . 2  |-  ( A. x  e.  B  F : A --> x  <->  A. x  e.  B  ( F  Fn  A  /\  ran  F  C_  x ) )
106, 7, 93bitr4i 268 1  |-  ( F : A --> |^| B  <->  A. x  e.  B  F : A --> x )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358    =/= wne 2446   A.wral 2543    C_ wss 3152   (/)c0 3455   |^|cint 3862   ran crn 4690    Fn wfn 5250   -->wf 5251
This theorem is referenced by:  chintcli  21910
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-v 2790  df-dif 3155  df-in 3159  df-ss 3166  df-nul 3456  df-int 3863  df-f 5259
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