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Theorem fintm 5159
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 3687 . . . 4  |-  ( ran 
F  C_  |^| B  <->  A. x  e.  B  ran  F  C_  x )
21anbi2i 445 . . 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 3361 . . . 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 7 . . 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 185 . 2  |-  ( ( F  Fn  A  /\  ran  F  C_  |^| B )  <->  A. x  e.  B  ( F  Fn  A  /\  ran  F  C_  x
) )
7 df-f 4985 . 2  |-  ( F : A --> |^| B  <->  ( F  Fn  A  /\  ran  F  C_  |^| B ) )
8 df-f 4985 . . 3  |-  ( F : A --> x  <->  ( F  Fn  A  /\  ran  F  C_  x ) )
98ralbii 2380 . 2  |-  ( A. x  e.  B  F : A --> x  <->  A. x  e.  B  ( F  Fn  A  /\  ran  F  C_  x ) )
106, 7, 93bitr4i 210 1  |-  ( F : A --> |^| B  <->  A. x  e.  B  F : A --> x )
Colors of variables: wff set class
Syntax hints:    /\ wa 102    <-> wb 103   E.wex 1424    e. wcel 1436   A.wral 2355    C_ wss 2988   |^|cint 3671   ran crn 4412    Fn wfn 4976   -->wf 4977
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067
This theorem depends on definitions:  df-bi 115  df-tru 1290  df-nf 1393  df-sb 1690  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ral 2360  df-v 2617  df-in 2994  df-ss 3001  df-int 3672  df-f 4985
This theorem is referenced by: (None)
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