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Theorem dff1o6 5973
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 5417 . 2  |-  ( F : A -1-1-onto-> B  <->  ( F : A -1-1-> B  /\  F : A -onto-> B ) )
2 dff13 5962 . . 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 5416 . . 3  |-  ( F : A -onto-> B  <->  ( F  Fn  A  /\  ran  F  =  B ) )
42, 3anbi12i 692 . 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 962 . . 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 3400 . . . . . . 7  |-  ( ran 
F  =  B  ->  ran  F  C_  B )
76anim2i 566 . . . . . 6  |-  ( ( F  Fn  A  /\  ran  F  =  B )  ->  ( F  Fn  A  /\  ran  F  C_  B ) )
8 df-f 5414 . . . . . 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 628 . . . 4  |-  ( ( F  Fn  A  /\  ran  F  =  B )  <-> 
( F : A --> B  /\  ( F  Fn  A  /\  ran  F  =  B ) ) )
1110anbi1i 690 . . 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 791 . . 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 960    = wceq 1364   A.wral 2709    C_ wss 3320   ran crn 4832    Fn wfn 5405   -->wf 5406   -1-1->wf1 5407   -onto->wfo 5408   -1-1-onto->wf1o 5409   ` cfv 5410
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1671  ax-6 1709  ax-7 1729  ax-9 1761  ax-10 1776  ax-11 1781  ax-12 1793  ax-13 1944  ax-ext 2418  ax-sep 4405  ax-nul 4413  ax-pr 4523
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1702  df-eu 2260  df-mo 2261  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-ral 2714  df-rex 2715  df-rab 2718  df-v 2968  df-sbc 3180  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3630  df-if 3784  df-sn 3870  df-pr 3872  df-op 3876  df-uni 4084  df-br 4285  df-opab 4343  df-id 4627  df-xp 4837  df-rel 4838  df-cnv 4839  df-co 4840  df-dm 4841  df-iota 5373  df-fun 5412  df-fn 5413  df-f 5414  df-f1 5415  df-fo 5416  df-f1o 5417  df-fv 5418
This theorem is referenced by:  soisores  6009  f1otrg  22944  f1otrge  22945  nbgraf1olem1  23177  nbgraf1olem5  23181  grpoinvf  23554  bra11  25339  rmxypairf1o  29101  diaf11N  34330  dibf11N  34442  lcfrlem9  34831  mapd1o  34929  hdmapf1oN  35149  hgmapf1oN  35187
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