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Theorem fintm 5276
Description: Function into an intersection. (Contributed by Jim Kingdon, 28-Dec-2018.)
Hypothesis
Ref Expression
fintm.1  |-  E. x  x  e.  B
Assertion
Ref Expression
fintm  |-  ( F : A --> |^| B  <->  A. x  e.  B  F : A --> x )
Distinct variable groups:    x, A    x, B    x, F

Proof of Theorem fintm
StepHypRef Expression
1 ssint 3755 . . . 4  |-  ( ran 
F  C_  |^| B  <->  A. x  e.  B  ran  F  C_  x )
21anbi2i 450 . . 3  |-  ( ( F  Fn  A  /\  ran  F  C_  |^| B )  <-> 
( F  Fn  A  /\  A. x  e.  B  ran  F  C_  x )
)
3 fintm.1 . . . 4  |-  E. x  x  e.  B
4 r19.28mv 3423 . . . 4  |-  ( E. x  x  e.  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 5 . . 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 186 . 2  |-  ( ( F  Fn  A  /\  ran  F  C_  |^| B )  <->  A. x  e.  B  ( F  Fn  A  /\  ran  F  C_  x
) )
7 df-f 5095 . 2  |-  ( F : A --> |^| B  <->  ( F  Fn  A  /\  ran  F  C_  |^| B ) )
8 df-f 5095 . . 3  |-  ( F : A --> x  <->  ( F  Fn  A  /\  ran  F  C_  x ) )
98ralbii 2416 . 2  |-  ( A. x  e.  B  F : A --> x  <->  A. x  e.  B  ( F  Fn  A  /\  ran  F  C_  x ) )
106, 7, 93bitr4i 211 1  |-  ( F : A --> |^| B  <->  A. x  e.  B  F : A --> x )
Colors of variables: wff set class
Syntax hints:    /\ wa 103    <-> wb 104   E.wex 1451    e. wcel 1463   A.wral 2391    C_ wss 3039   |^|cint 3739   ran crn 4508    Fn wfn 5086   -->wf 5087
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ral 2396  df-v 2660  df-in 3045  df-ss 3052  df-int 3740  df-f 5095
This theorem is referenced by: (None)
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