MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  fint Structured version   Unicode version

Theorem fint 5622
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 4066 . . . 4  |-  ( ran 
F  C_  |^| B  <->  A. x  e.  B  ran  F  C_  x )
21anbi2i 676 . . 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 3723 . . . 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 244 . 2  |-  ( ( F  Fn  A  /\  ran  F  C_  |^| B )  <->  A. x  e.  B  ( F  Fn  A  /\  ran  F  C_  x
) )
7 df-f 5458 . 2  |-  ( F : A --> |^| B  <->  ( F  Fn  A  /\  ran  F  C_  |^| B ) )
8 df-f 5458 . . 3  |-  ( F : A --> x  <->  ( F  Fn  A  /\  ran  F  C_  x ) )
98ralbii 2729 . 2  |-  ( A. x  e.  B  F : A --> x  <->  A. x  e.  B  ( F  Fn  A  /\  ran  F  C_  x ) )
106, 7, 93bitr4i 269 1  |-  ( F : A --> |^| B  <->  A. x  e.  B  F : A --> x )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    /\ wa 359    =/= wne 2599   A.wral 2705    C_ wss 3320   (/)c0 3628   |^|cint 4050   ran crn 4879    Fn wfn 5449   -->wf 5450
This theorem is referenced by:  chintcli  22833
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-ne 2601  df-ral 2710  df-v 2958  df-dif 3323  df-in 3327  df-ss 3334  df-nul 3629  df-int 4051  df-f 5458
  Copyright terms: Public domain W3C validator