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Theorem dff1o6 6128
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 5546 . 2  |-  ( F : A -1-1-onto-> B  <->  ( F : A -1-1-> B  /\  F : A -onto-> B ) )
2 dff13 6113 . . 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 5545 . . 3  |-  ( F : A -onto-> B  <->  ( F  Fn  A  /\  ran  F  =  B ) )
42, 3anbi12i 701 . 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 984 . . 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 3454 . . . . . . 7  |-  ( ran 
F  =  B  ->  ran  F  C_  B )
76anim2i 571 . . . . . 6  |-  ( ( F  Fn  A  /\  ran  F  =  B )  ->  ( F  Fn  A  /\  ran  F  C_  B ) )
8 df-f 5543 . . . . . 6  |-  ( F : A --> B  <->  ( F  Fn  A  /\  ran  F  C_  B ) )
97, 8sylibr 215 . . . . 5  |-  ( ( F  Fn  A  /\  ran  F  =  B )  ->  F : A --> B )
109pm4.71ri 637 . . . 4  |-  ( ( F  Fn  A  /\  ran  F  =  B )  <-> 
( F : A --> B  /\  ( F  Fn  A  /\  ran  F  =  B ) ) )
1110anbi1i 699 . . 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 805 . . 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 275 . 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 274 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 setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370    /\ w3a 982    = wceq 1437   A.wral 2709    C_ wss 3374   ran crn 4792    Fn wfn 5534   -->wf 5535   -1-1->wf1 5536   -onto->wfo 5537   -1-1-onto->wf1o 5538   ` cfv 5539
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2058  ax-ext 2403  ax-sep 4484  ax-nul 4493  ax-pr 4598
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2275  df-mo 2276  df-clab 2410  df-cleq 2416  df-clel 2419  df-nfc 2553  df-ne 2596  df-ral 2714  df-rex 2715  df-rab 2718  df-v 3019  df-sbc 3238  df-dif 3377  df-un 3379  df-in 3381  df-ss 3388  df-nul 3700  df-if 3850  df-sn 3937  df-pr 3939  df-op 3943  df-uni 4158  df-br 4362  df-opab 4421  df-id 4706  df-xp 4797  df-rel 4798  df-cnv 4799  df-co 4800  df-dm 4801  df-iota 5503  df-fun 5541  df-fn 5542  df-f 5543  df-f1 5544  df-fo 5545  df-f1o 5546  df-fv 5547
This theorem is referenced by:  soisores  6172  f1otrg  24838  f1otrge  24839  nbgraf1olem1  25106  nbgraf1olem5  25110  grpoinvf  25905  bra11  27698  diaf11N  34529  dibf11N  34641  lcfrlem9  35030  mapd1o  35128  hdmapf1oN  35348  hgmapf1oN  35386  rmxypairf1o  35672
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