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Theorem dff1o6 5467
Description: A one-to-one onto function in terms of function values. (Contributed by NM, 29-Mar-2008.)
Assertion
Ref Expression
dff1o6  |-  ( F : A -1-1-onto-> B  <->  ( F  Fn  A  /\  ran  F  =  B  /\  A. x  e.  A  A. y  e.  A  ( ( F `  x )  =  ( F `  y )  ->  x  =  y ) ) )
Distinct variable groups:    x, y, A   
x, F, y
Allowed substitution hints:    B( x, y)

Proof of Theorem dff1o6
StepHypRef Expression
1 df-f1o 4959 . 2  |-  ( F : A -1-1-onto-> B  <->  ( F : A -1-1-> B  /\  F : A -onto-> B ) )
2 dff13 5459 . . 3  |-  ( F : A -1-1-> B  <->  ( F : A --> B  /\  A. x  e.  A  A. y  e.  A  (
( F `  x
)  =  ( F `
 y )  ->  x  =  y )
) )
3 df-fo 4958 . . 3  |-  ( F : A -onto-> B  <->  ( F  Fn  A  /\  ran  F  =  B ) )
42, 3anbi12i 448 . 2  |-  ( ( F : A -1-1-> B  /\  F : A -onto-> B
)  <->  ( ( F : A --> B  /\  A. x  e.  A  A. y  e.  A  (
( F `  x
)  =  ( F `
 y )  ->  x  =  y )
)  /\  ( F  Fn  A  /\  ran  F  =  B ) ) )
5 df-3an 922 . . 3  |-  ( ( F  Fn  A  /\  ran  F  =  B  /\  A. x  e.  A  A. y  e.  A  (
( F `  x
)  =  ( F `
 y )  ->  x  =  y )
)  <->  ( ( F  Fn  A  /\  ran  F  =  B )  /\  A. x  e.  A  A. y  e.  A  (
( F `  x
)  =  ( F `
 y )  ->  x  =  y )
) )
6 eqimss 3060 . . . . . . 7  |-  ( ran 
F  =  B  ->  ran  F  C_  B )
76anim2i 334 . . . . . 6  |-  ( ( F  Fn  A  /\  ran  F  =  B )  ->  ( F  Fn  A  /\  ran  F  C_  B ) )
8 df-f 4956 . . . . . 6  |-  ( F : A --> B  <->  ( F  Fn  A  /\  ran  F  C_  B ) )
97, 8sylibr 132 . . . . 5  |-  ( ( F  Fn  A  /\  ran  F  =  B )  ->  F : A --> B )
109pm4.71ri 384 . . . 4  |-  ( ( F  Fn  A  /\  ran  F  =  B )  <-> 
( F : A --> B  /\  ( F  Fn  A  /\  ran  F  =  B ) ) )
1110anbi1i 446 . . 3  |-  ( ( ( F  Fn  A  /\  ran  F  =  B )  /\  A. x  e.  A  A. y  e.  A  ( ( F `  x )  =  ( F `  y )  ->  x  =  y ) )  <-> 
( ( F : A
--> B  /\  ( F  Fn  A  /\  ran  F  =  B ) )  /\  A. x  e.  A  A. y  e.  A  ( ( F `
 x )  =  ( F `  y
)  ->  x  =  y ) ) )
12 an32 527 . . 3  |-  ( ( ( F : A --> B  /\  ( F  Fn  A  /\  ran  F  =  B ) )  /\  A. x  e.  A  A. y  e.  A  (
( F `  x
)  =  ( F `
 y )  ->  x  =  y )
)  <->  ( ( F : A --> B  /\  A. x  e.  A  A. y  e.  A  (
( F `  x
)  =  ( F `
 y )  ->  x  =  y )
)  /\  ( F  Fn  A  /\  ran  F  =  B ) ) )
135, 11, 123bitrri 205 . 2  |-  ( ( ( F : A --> B  /\  A. x  e.  A  A. y  e.  A  ( ( F `
 x )  =  ( F `  y
)  ->  x  =  y ) )  /\  ( F  Fn  A  /\  ran  F  =  B ) )  <->  ( F  Fn  A  /\  ran  F  =  B  /\  A. x  e.  A  A. y  e.  A  ( ( F `  x )  =  ( F `  y )  ->  x  =  y ) ) )
141, 4, 133bitri 204 1  |-  ( F : A -1-1-onto-> B  <->  ( F  Fn  A  /\  ran  F  =  B  /\  A. x  e.  A  A. y  e.  A  ( ( F `  x )  =  ( F `  y )  ->  x  =  y ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    /\ w3a 920    = wceq 1285   A.wral 2353    C_ wss 2982   ran crn 4392    Fn wfn 4947   -->wf 4948   -1-1->wf1 4949   -onto->wfo 4950   -1-1-onto->wf1o 4951   ` cfv 4952
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3916  ax-pow 3968  ax-pr 3992
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-v 2612  df-sbc 2825  df-un 2986  df-in 2988  df-ss 2995  df-pw 3402  df-sn 3422  df-pr 3423  df-op 3425  df-uni 3622  df-br 3806  df-opab 3860  df-id 4076  df-xp 4397  df-rel 4398  df-cnv 4399  df-co 4400  df-dm 4401  df-iota 4917  df-fun 4954  df-fn 4955  df-f 4956  df-f1 4957  df-fo 4958  df-f1o 4959  df-fv 4960
This theorem is referenced by: (None)
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