MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  dff1o6 Structured version   Unicode version

Theorem dff1o6 5982
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 5425 . 2  |-  ( F : A -1-1-onto-> B  <->  ( F : A -1-1-> B  /\  F : A -onto-> B ) )
2 dff13 5971 . . 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 5424 . . 3  |-  ( F : A -onto-> B  <->  ( F  Fn  A  /\  ran  F  =  B ) )
42, 3anbi12i 697 . 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 967 . . 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 3408 . . . . . . 7  |-  ( ran 
F  =  B  ->  ran  F  C_  B )
76anim2i 569 . . . . . 6  |-  ( ( F  Fn  A  /\  ran  F  =  B )  ->  ( F  Fn  A  /\  ran  F  C_  B ) )
8 df-f 5422 . . . . . 6  |-  ( F : A --> B  <->  ( F  Fn  A  /\  ran  F  C_  B ) )
97, 8sylibr 212 . . . . 5  |-  ( ( F  Fn  A  /\  ran  F  =  B )  ->  F : A --> B )
109pm4.71ri 633 . . . 4  |-  ( ( F  Fn  A  /\  ran  F  =  B )  <-> 
( F : A --> B  /\  ( F  Fn  A  /\  ran  F  =  B ) ) )
1110anbi1i 695 . . 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 796 . . 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 272 . 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 271 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 184    /\ wa 369    /\ w3a 965    = wceq 1369   A.wral 2715    C_ wss 3328   ran crn 4841    Fn wfn 5413   -->wf 5414   -1-1->wf1 5415   -onto->wfo 5416   -1-1-onto->wf1o 5417   ` cfv 5418
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4413  ax-nul 4421  ax-pr 4531
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2720  df-rex 2721  df-rab 2724  df-v 2974  df-sbc 3187  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-sn 3878  df-pr 3880  df-op 3884  df-uni 4092  df-br 4293  df-opab 4351  df-id 4636  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426
This theorem is referenced by:  soisores  6018  f1otrg  23117  f1otrge  23118  nbgraf1olem1  23350  nbgraf1olem5  23354  grpoinvf  23727  bra11  25512  rmxypairf1o  29252  diaf11N  34694  dibf11N  34806  lcfrlem9  35195  mapd1o  35293  hdmapf1oN  35513  hgmapf1oN  35551
  Copyright terms: Public domain W3C validator