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Theorem dff14a 6163
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 6152 . 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 2664 . . . . . . 7  |-  ( x  =/=  y  <->  -.  x  =  y )
43bicomi 202 . . . . . 6  |-  ( -.  x  =  y  <->  x  =/=  y )
5 df-ne 2664 . . . . . . 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 2896 . . 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 1379    =/= wne 2662   A.wral 2814   -->wf 5582   -1-1->wf1 5583   ` cfv 5586
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fv 5594
This theorem is referenced by:  dff14b  6164
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