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Theorem f11o 5671
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 5670 . . 3  |-  ( F : A --> B  <->  E. x
( F : A -onto->
x  /\  x  C_  B
) )
32anbi1i 677 . 2  |-  ( ( F : A --> B  /\  Fun  `' F )  <->  ( E. x ( F : A -onto-> x  /\  x  C_  B )  /\  Fun  `' F ) )
4 df-f1 5422 . 2  |-  ( F : A -1-1-> B  <->  ( F : A --> B  /\  Fun  `' F ) )
5 dff1o3 5643 . . . . . 6  |-  ( F : A -1-1-onto-> x  <->  ( F : A -onto-> x  /\  Fun  `' F ) )
65anbi1i 677 . . . . 5  |-  ( ( F : A -1-1-onto-> x  /\  x  C_  B )  <->  ( ( F : A -onto-> x  /\  Fun  `' F )  /\  x  C_  B ) )
7 an32 774 . . . . 5  |-  ( ( ( F : A -onto->
x  /\  Fun  `' F
)  /\  x  C_  B
)  <->  ( ( F : A -onto-> x  /\  x  C_  B )  /\  Fun  `' F ) )
86, 7bitri 241 . . . 4  |-  ( ( F : A -1-1-onto-> x  /\  x  C_  B )  <->  ( ( F : A -onto-> x  /\  x  C_  B )  /\  Fun  `' F ) )
98exbii 1589 . . 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 1920 . . 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 241 . 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 269 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 177    /\ wa 359   E.wex 1547    e. wcel 1721   _Vcvv 2920    C_ wss 3284   `'ccnv 4840   Fun wfun 5411   -->wf 5413   -1-1->wf1 5414   -onto->wfo 5415   -1-1-onto->wf1o 5416
This theorem is referenced by:  domen  7084
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389  ax-sep 4294  ax-nul 4302  ax-pr 4367  ax-un 4664
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-rex 2676  df-rab 2679  df-v 2922  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-nul 3593  df-if 3704  df-sn 3784  df-pr 3785  df-op 3787  df-uni 3980  df-br 4177  df-opab 4231  df-cnv 4849  df-dm 4851  df-rn 4852  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424
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