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Theorem f11o 5612
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 5611 . . 3  |-  ( F : A --> B  <->  E. x
( F : A -onto->
x  /\  x  C_  B
) )
32anbi1i 676 . 2  |-  ( ( F : A --> B  /\  Fun  `' F )  <->  ( E. x ( F : A -onto-> x  /\  x  C_  B )  /\  Fun  `' F ) )
4 df-f1 5363 . 2  |-  ( F : A -1-1-> B  <->  ( F : A --> B  /\  Fun  `' F ) )
5 dff1o3 5584 . . . . . 6  |-  ( F : A -1-1-onto-> x  <->  ( F : A -onto-> x  /\  Fun  `' F ) )
65anbi1i 676 . . . . 5  |-  ( ( F : A -1-1-onto-> x  /\  x  C_  B )  <->  ( ( F : A -onto-> x  /\  Fun  `' F )  /\  x  C_  B ) )
7 an32 773 . . . . 5  |-  ( ( ( F : A -onto->
x  /\  Fun  `' F
)  /\  x  C_  B
)  <->  ( ( F : A -onto-> x  /\  x  C_  B )  /\  Fun  `' F ) )
86, 7bitri 240 . . . 4  |-  ( ( F : A -1-1-onto-> x  /\  x  C_  B )  <->  ( ( F : A -onto-> x  /\  x  C_  B )  /\  Fun  `' F ) )
98exbii 1587 . . 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 1911 . . 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 240 . 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 268 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:    <-> wb 176    /\ wa 358   E.wex 1546    e. wcel 1715   _Vcvv 2873    C_ wss 3238   `'ccnv 4791   Fun wfun 5352   -->wf 5354   -1-1->wf1 5355   -onto->wfo 5356   -1-1-onto->wf1o 5357
This theorem is referenced by:  domen  7018
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-13 1717  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-sep 4243  ax-nul 4251  ax-pr 4316  ax-un 4615
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-rex 2634  df-rab 2637  df-v 2875  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-nul 3544  df-if 3655  df-sn 3735  df-pr 3736  df-op 3738  df-uni 3930  df-br 4126  df-opab 4180  df-cnv 4800  df-dm 4802  df-rn 4803  df-f 5362  df-f1 5363  df-fo 5364  df-f1o 5365
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