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Theorem dff1o6f 25195
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 5277 . . 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 5278 . . 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 5467 . . . . 5  |-  ( F : A -onto-> B  ->  F : A --> B )
6 dff1o6f.1 . . . . . . 7  |-  F/_ x F
7 dff1o6f.2 . . . . . . 7  |-  F/_ y F
86, 7dff13f 5800 . . . . . 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 1632   F/_wnfc 2419   A.wral 2556   ran crn 4706    Fn wfn 5266   -->wf 5267   -1-1->wf1 5268   -onto->wfo 5269   -1-1-onto->wf1o 5270   ` cfv 5271
This theorem is referenced by:  trooo  25497  rltrooo  25518
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279
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