Users' Mathboxes Mathbox for Alexander van der Vekens < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  dff14b Structured version   Unicode version

Theorem dff14b 30316
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 30315 . 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 2721 . . . . . . 7  |-  ( x  =/=  y  <->  y  =/=  x )
32imbi1i 325 . . . . . 6  |-  ( ( x  =/=  y  -> 
( F `  x
)  =/=  ( F `
 y ) )  <-> 
( y  =/=  x  ->  ( F `  x
)  =/=  ( F `
 y ) ) )
43ralbii 2839 . . . . 5  |-  ( A. y  e.  A  (
x  =/=  y  -> 
( F `  x
)  =/=  ( F `
 y ) )  <->  A. y  e.  A  ( y  =/=  x  ->  ( F `  x
)  =/=  ( F `
 y ) ) )
5 raldifsnb 30294 . . . . 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 2839 . . 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 694 . 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 369    =/= wne 2648   A.wral 2799    \ cdif 3436   {csn 3988   -->wf 5525   -1-1->wf1 5526   ` cfv 5529
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4524  ax-nul 4532  ax-pr 4642
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-sbc 3295  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-br 4404  df-opab 4462  df-id 4747  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fv 5537
This theorem is referenced by:  f12dfv  30317  f13dfv  30318
  Copyright terms: Public domain W3C validator