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Theorem dff14b 6115
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
dff14b  |-  ( F : A -1-1-> B  <->  ( F : A --> B  /\  A. x  e.  A  A. y  e.  ( A  \  { x } ) ( F `  x
)  =/=  ( F `
 y ) ) )
Distinct variable groups:    x, y, A    x, F, y
Allowed substitution hints:    B( x, y)

Proof of Theorem dff14b
StepHypRef Expression
1 dff14a 6114 . 2  |-  ( F : A -1-1-> B  <->  ( F : A --> B  /\  A. x  e.  A  A. y  e.  A  (
x  =/=  y  -> 
( F `  x
)  =/=  ( F `
 y ) ) ) )
2 necom 2672 . . . . . . 7  |-  ( x  =/=  y  <->  y  =/=  x )
32imbi1i 323 . . . . . 6  |-  ( ( x  =/=  y  -> 
( F `  x
)  =/=  ( F `
 y ) )  <-> 
( y  =/=  x  ->  ( F `  x
)  =/=  ( F `
 y ) ) )
43ralbii 2834 . . . . 5  |-  ( A. y  e.  A  (
x  =/=  y  -> 
( F `  x
)  =/=  ( F `
 y ) )  <->  A. y  e.  A  ( y  =/=  x  ->  ( F `  x
)  =/=  ( F `
 y ) ) )
5 raldifsnb 4102 . . . . 5  |-  ( A. y  e.  A  (
y  =/=  x  -> 
( F `  x
)  =/=  ( F `
 y ) )  <->  A. y  e.  ( A  \  { x }
) ( F `  x )  =/=  ( F `  y )
)
64, 5bitri 249 . . . 4  |-  ( A. y  e.  A  (
x  =/=  y  -> 
( F `  x
)  =/=  ( F `
 y ) )  <->  A. y  e.  ( A  \  { x }
) ( F `  x )  =/=  ( F `  y )
)
76ralbii 2834 . . 3  |-  ( A. x  e.  A  A. y  e.  A  (
x  =/=  y  -> 
( F `  x
)  =/=  ( F `
 y ) )  <->  A. x  e.  A  A. y  e.  ( A  \  { x }
) ( F `  x )  =/=  ( F `  y )
)
87anbi2i 692 . 2  |-  ( ( F : A --> B  /\  A. x  e.  A  A. y  e.  A  (
x  =/=  y  -> 
( F `  x
)  =/=  ( F `
 y ) ) )  <->  ( F : A
--> B  /\  A. x  e.  A  A. y  e.  ( A  \  {
x } ) ( F `  x )  =/=  ( F `  y ) ) )
91, 8bitri 249 1  |-  ( F : A -1-1-> B  <->  ( F : A --> B  /\  A. x  e.  A  A. y  e.  ( A  \  { x } ) ( F `  x
)  =/=  ( F `
 y ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    =/= wne 2598   A.wral 2753    \ cdif 3410   {csn 3971   -->wf 5521   -1-1->wf1 5522   ` cfv 5525
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4516  ax-nul 4524  ax-pr 4629
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2758  df-rex 2759  df-rab 2762  df-v 3060  df-sbc 3277  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-sn 3972  df-pr 3974  df-op 3978  df-uni 4191  df-br 4395  df-opab 4453  df-id 4737  df-xp 4948  df-rel 4949  df-cnv 4950  df-co 4951  df-dm 4952  df-iota 5489  df-fun 5527  df-fn 5528  df-f 5529  df-f1 5530  df-fv 5533
This theorem is referenced by:  f12dfv  6116  f13dfv  6117
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