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Theorem f11o 5711
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 5710 . . 3  |-  ( F : A --> B  <->  E. x
( F : A -onto->
x  /\  x  C_  B
) )
32anbi1i 678 . 2  |-  ( ( F : A --> B  /\  Fun  `' F )  <->  ( E. x ( F : A -onto-> x  /\  x  C_  B )  /\  Fun  `' F ) )
4 df-f1 5462 . 2  |-  ( F : A -1-1-> B  <->  ( F : A --> B  /\  Fun  `' F ) )
5 dff1o3 5683 . . . . . 6  |-  ( F : A -1-1-onto-> x  <->  ( F : A -onto-> x  /\  Fun  `' F ) )
65anbi1i 678 . . . . 5  |-  ( ( F : A -1-1-onto-> x  /\  x  C_  B )  <->  ( ( F : A -onto-> x  /\  Fun  `' F )  /\  x  C_  B ) )
7 an32 775 . . . . 5  |-  ( ( ( F : A -onto->
x  /\  Fun  `' F
)  /\  x  C_  B
)  <->  ( ( F : A -onto-> x  /\  x  C_  B )  /\  Fun  `' F ) )
86, 7bitri 242 . . . 4  |-  ( ( F : A -1-1-onto-> x  /\  x  C_  B )  <->  ( ( F : A -onto-> x  /\  x  C_  B )  /\  Fun  `' F ) )
98exbii 1593 . . 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 1925 . . 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 242 . 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 270 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 178    /\ wa 360   E.wex 1551    e. wcel 1726   _Vcvv 2958    C_ wss 3322   `'ccnv 4880   Fun wfun 5451   -->wf 5453   -1-1->wf1 5454   -onto->wfo 5455   -1-1-onto->wf1o 5456
This theorem is referenced by:  domen  7124
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pr 4406  ax-un 4704
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-rex 2713  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4216  df-opab 4270  df-cnv 4889  df-dm 4891  df-rn 4892  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464
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