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Theorem dff1o6f 25092
Description: A one-to-one onto function in terms of function values. (Contributed by FL, 1-Jan-2011.) (Proof shortened by Mario Carneiro, 11-Dec-2016.)
Hypotheses
Ref Expression
dff1o6f.1  |-  F/_ x F
dff1o6f.2  |-  F/_ y F
Assertion
Ref Expression
dff1o6f  |-  ( 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 group:    x, A, y
Allowed substitution hints:    B( x, y)    F( x, y)

Proof of Theorem dff1o6f
StepHypRef Expression
1 df-fo 5261 . . 3  |-  ( F : A -onto-> B  <->  ( F  Fn  A  /\  ran  F  =  B ) )
21anbi1i 676 . 2  |-  ( ( F : A -onto-> 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 ) ) )
3 df-f1o 5262 . . 3  |-  ( F : A -1-1-onto-> B  <->  ( F : A -1-1-> B  /\  F : A -onto-> B ) )
4 ancom 437 . . 3  |-  ( ( F : A -1-1-> B  /\  F : A -onto-> B
)  <->  ( F : A -onto-> B  /\  F : A -1-1-> B ) )
5 fof 5451 . . . . 5  |-  ( F : A -onto-> B  ->  F : A --> B )
6 dff1o6f.1 . . . . . . 7  |-  F/_ x F
7 dff1o6f.2 . . . . . . 7  |-  F/_ y F
86, 7dff13f 5784 . . . . . 6  |-  ( F : A -1-1-> B  <->  ( F : A --> B  /\  A. x  e.  A  A. y  e.  A  (
( F `  x
)  =  ( F `
 y )  ->  x  =  y )
) )
98baib 871 . . . . 5  |-  ( F : A --> B  -> 
( F : A -1-1-> B  <->  A. x  e.  A  A. y  e.  A  ( ( F `  x )  =  ( F `  y )  ->  x  =  y ) ) )
105, 9syl 15 . . . 4  |-  ( F : A -onto-> B  -> 
( F : A -1-1-> B  <->  A. x  e.  A  A. y  e.  A  ( ( F `  x )  =  ( F `  y )  ->  x  =  y ) ) )
1110pm5.32i 618 . . 3  |-  ( ( F : A -onto-> B  /\  F : A -1-1-> B
)  <->  ( F : A -onto-> B  /\  A. x  e.  A  A. y  e.  A  ( ( F `  x )  =  ( F `  y )  ->  x  =  y ) ) )
123, 4, 113bitri 262 . 2  |-  ( F : A -1-1-onto-> B  <->  ( F : A -onto-> B  /\  A. x  e.  A  A. y  e.  A  ( ( F `  x )  =  ( F `  y )  ->  x  =  y ) ) )
13 df-3an 936 . 2  |-  ( ( 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 )
) )
142, 12, 133bitr4i 268 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    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623   F/_wnfc 2406   A.wral 2543   ran crn 4690    Fn wfn 5250   -->wf 5251   -1-1->wf1 5252   -onto->wfo 5253   -1-1-onto->wf1o 5254   ` cfv 5255
This theorem is referenced by:  trooo  25394  rltrooo  25415
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263
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