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Theorem fint 5436
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 3894 . . . 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 3562 . . . 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 5275 . 2  |-  ( F : A --> |^| B  <->  ( F  Fn  A  /\  ran  F  C_  |^| B ) )
8 df-f 5275 . . 3  |-  ( F : A --> x  <->  ( F  Fn  A  /\  ran  F  C_  x ) )
98ralbii 2580 . 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 2459   A.wral 2556    C_ wss 3165   (/)c0 3468   |^|cint 3878   ran crn 4706    Fn wfn 5266   -->wf 5267
This theorem is referenced by:  chintcli  21926
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-ne 2461  df-ral 2561  df-v 2803  df-dif 3168  df-in 3172  df-ss 3179  df-nul 3469  df-int 3879  df-f 5275
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