ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  f11o Unicode version

Theorem f11o 5533
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 5532 . . 3  |-  ( F : A --> B  <->  E. x
( F : A -onto->
x  /\  x  C_  B
) )
32anbi1i 458 . 2  |-  ( ( F : A --> B  /\  Fun  `' F )  <->  ( E. x ( F : A -onto-> x  /\  x  C_  B )  /\  Fun  `' F ) )
4 df-f1 5259 . 2  |-  ( F : A -1-1-> B  <->  ( F : A --> B  /\  Fun  `' F ) )
5 dff1o3 5506 . . . . . 6  |-  ( F : A -1-1-onto-> x  <->  ( F : A -onto-> x  /\  Fun  `' F ) )
65anbi1i 458 . . . . 5  |-  ( ( F : A -1-1-onto-> x  /\  x  C_  B )  <->  ( ( F : A -onto-> x  /\  Fun  `' F )  /\  x  C_  B ) )
7 an32 562 . . . . 5  |-  ( ( ( F : A -onto->
x  /\  Fun  `' F
)  /\  x  C_  B
)  <->  ( ( F : A -onto-> x  /\  x  C_  B )  /\  Fun  `' F ) )
86, 7bitri 184 . . . 4  |-  ( ( F : A -1-1-onto-> x  /\  x  C_  B )  <->  ( ( F : A -onto-> x  /\  x  C_  B )  /\  Fun  `' F ) )
98exbii 1616 . . 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 1914 . . 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 184 . 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 212 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 104    <-> wb 105   E.wex 1503    e. wcel 2164   _Vcvv 2760    C_ wss 3153   `'ccnv 4658   Fun wfun 5248   -->wf 5250   -1-1->wf1 5251   -onto->wfo 5252   -1-1-onto->wf1o 5253
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238  ax-un 4464
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-rex 2478  df-v 2762  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-br 4030  df-opab 4091  df-cnv 4667  df-dm 4669  df-rn 4670  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261
This theorem is referenced by:  domen  6805
  Copyright terms: Public domain W3C validator