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Theorem dff14a 30149
Description: A one-to-one function in terms of different function values for different arguments. (Contributed by Alexander van der Vekens, 26-Jan-2018.)
Assertion
Ref Expression
dff14a  |-  ( F : A -1-1-> B  <->  ( F : A --> B  /\  A. x  e.  A  A. y  e.  A  (
x  =/=  y  -> 
( F `  x
)  =/=  ( F `
 y ) ) ) )
Distinct variable groups:    x, y, A    x, F, y
Allowed substitution hints:    B( x, y)

Proof of Theorem dff14a
StepHypRef Expression
1 dff13 5976 . 2  |-  ( F : A -1-1-> B  <->  ( F : A --> B  /\  A. x  e.  A  A. y  e.  A  (
( F `  x
)  =  ( F `
 y )  ->  x  =  y )
) )
2 con34b 292 . . . . 5  |-  ( ( ( F `  x
)  =  ( F `
 y )  ->  x  =  y )  <->  ( -.  x  =  y  ->  -.  ( F `  x )  =  ( F `  y ) ) )
3 df-ne 2613 . . . . . . 7  |-  ( x  =/=  y  <->  -.  x  =  y )
43bicomi 202 . . . . . 6  |-  ( -.  x  =  y  <->  x  =/=  y )
5 df-ne 2613 . . . . . . 7  |-  ( ( F `  x )  =/=  ( F `  y )  <->  -.  ( F `  x )  =  ( F `  y ) )
65bicomi 202 . . . . . 6  |-  ( -.  ( F `  x
)  =  ( F `
 y )  <->  ( F `  x )  =/=  ( F `  y )
)
74, 6imbi12i 326 . . . . 5  |-  ( ( -.  x  =  y  ->  -.  ( F `  x )  =  ( F `  y ) )  <->  ( x  =/=  y  ->  ( F `  x )  =/=  ( F `  y )
) )
82, 7bitri 249 . . . 4  |-  ( ( ( F `  x
)  =  ( F `
 y )  ->  x  =  y )  <->  ( x  =/=  y  -> 
( F `  x
)  =/=  ( F `
 y ) ) )
982ralbii 2746 . . 3  |-  ( A. x  e.  A  A. y  e.  A  (
( F `  x
)  =  ( F `
 y )  ->  x  =  y )  <->  A. x  e.  A  A. y  e.  A  (
x  =/=  y  -> 
( F `  x
)  =/=  ( F `
 y ) ) )
109anbi2i 694 . 2  |-  ( ( F : A --> 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  ( x  =/=  y  ->  ( F `
 x )  =/=  ( F `  y
) ) ) )
111, 10bitri 249 1  |-  ( F : A -1-1-> B  <->  ( F : A --> B  /\  A. x  e.  A  A. y  e.  A  (
x  =/=  y  -> 
( F `  x
)  =/=  ( F `
 y ) ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1369    =/= wne 2611   A.wral 2720   -->wf 5419   -1-1->wf1 5420   ` cfv 5423
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 4418  ax-nul 4426  ax-pr 4536
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 2573  df-ne 2613  df-ral 2725  df-rex 2726  df-rab 2729  df-v 2979  df-sbc 3192  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-nul 3643  df-if 3797  df-sn 3883  df-pr 3885  df-op 3889  df-uni 4097  df-br 4298  df-opab 4356  df-id 4641  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fv 5431
This theorem is referenced by:  dff14b  30150
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