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Theorem f11o 5187
Description: Relationship between one-to-one and one-to-one onto function. (Contributed by NM, 4-Apr-1998.)
Hypothesis
Ref Expression
f11o.1  |-  F  e. 
_V
Assertion
Ref Expression
f11o  |-  ( F : A -1-1-> B  <->  E. x
( F : A -1-1-onto-> x  /\  x  C_  B ) )
Distinct variable groups:    x, F    x, A    x, B

Proof of Theorem f11o
StepHypRef Expression
1 f11o.1 . . . 4  |-  F  e. 
_V
21ffoss 5186 . . 3  |-  ( F : A --> B  <->  E. x
( F : A -onto->
x  /\  x  C_  B
) )
32anbi1i 439 . 2  |-  ( ( F : A --> B  /\  Fun  `' F )  <->  ( E. x ( F : A -onto-> x  /\  x  C_  B )  /\  Fun  `' F ) )
4 df-f1 4935 . 2  |-  ( F : A -1-1-> B  <->  ( F : A --> B  /\  Fun  `' F ) )
5 dff1o3 5160 . . . . . 6  |-  ( F : A -1-1-onto-> x  <->  ( F : A -onto-> x  /\  Fun  `' F ) )
65anbi1i 439 . . . . 5  |-  ( ( F : A -1-1-onto-> x  /\  x  C_  B )  <->  ( ( F : A -onto-> x  /\  Fun  `' F )  /\  x  C_  B ) )
7 an32 504 . . . . 5  |-  ( ( ( F : A -onto->
x  /\  Fun  `' F
)  /\  x  C_  B
)  <->  ( ( F : A -onto-> x  /\  x  C_  B )  /\  Fun  `' F ) )
86, 7bitri 177 . . . 4  |-  ( ( F : A -1-1-onto-> x  /\  x  C_  B )  <->  ( ( F : A -onto-> x  /\  x  C_  B )  /\  Fun  `' F ) )
98exbii 1512 . . 3  |-  ( E. x ( F : A
-1-1-onto-> x  /\  x  C_  B
)  <->  E. x ( ( F : A -onto-> x  /\  x  C_  B )  /\  Fun  `' F
) )
10 19.41v 1798 . . 3  |-  ( E. x ( ( F : A -onto-> x  /\  x  C_  B )  /\  Fun  `' F )  <->  ( E. x ( F : A -onto-> x  /\  x  C_  B )  /\  Fun  `' F ) )
119, 10bitri 177 . 2  |-  ( E. x ( F : A
-1-1-onto-> x  /\  x  C_  B
)  <->  ( E. x
( F : A -onto->
x  /\  x  C_  B
)  /\  Fun  `' F
) )
123, 4, 113bitr4i 205 1  |-  ( F : A -1-1-> B  <->  E. x
( F : A -1-1-onto-> x  /\  x  C_  B ) )
Colors of variables: wff set class
Syntax hints:    /\ wa 101    <-> wb 102   E.wex 1397    e. wcel 1409   _Vcvv 2574    C_ wss 2945   `'ccnv 4372   Fun wfun 4924   -->wf 4926   -1-1->wf1 4927   -onto->wfo 4928   -1-1-onto->wf1o 4929
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903  ax-pow 3955  ax-pr 3972  ax-un 4198
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-rex 2329  df-v 2576  df-un 2950  df-in 2952  df-ss 2959  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-uni 3609  df-br 3793  df-opab 3847  df-cnv 4381  df-dm 4383  df-rn 4384  df-f 4934  df-f1 4935  df-fo 4936  df-f1o 4937
This theorem is referenced by:  domen  6263
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