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Theorem rnfdmpr 30149
Description: The range of a one-to-one function  F of an unordered pair into a set is the unordered pair of the function values. (Contributed by Alexander van der Vekens, 2-Feb-2018.)
Assertion
Ref Expression
rnfdmpr  |-  ( ( X  e.  V  /\  Y  e.  W )  ->  ( F  Fn  { X ,  Y }  ->  ran  F  =  {
( F `  X
) ,  ( F `
 Y ) } ) )

Proof of Theorem rnfdmpr
Dummy variables  x  i are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fnrnfv 5738 . . . 4  |-  ( F  Fn  { X ,  Y }  ->  ran  F  =  { x  |  E. i  e.  { X ,  Y } x  =  ( F `  i
) } )
21adantl 466 . . 3  |-  ( ( ( X  e.  V  /\  Y  e.  W
)  /\  F  Fn  { X ,  Y }
)  ->  ran  F  =  { x  |  E. i  e.  { X ,  Y } x  =  ( F `  i
) } )
3 fveq2 5691 . . . . . . . 8  |-  ( i  =  X  ->  ( F `  i )  =  ( F `  X ) )
43eqeq2d 2454 . . . . . . 7  |-  ( i  =  X  ->  (
x  =  ( F `
 i )  <->  x  =  ( F `  X ) ) )
54abbidv 2557 . . . . . 6  |-  ( i  =  X  ->  { x  |  x  =  ( F `  i ) }  =  { x  |  x  =  ( F `  X ) } )
6 fveq2 5691 . . . . . . . 8  |-  ( i  =  Y  ->  ( F `  i )  =  ( F `  Y ) )
76eqeq2d 2454 . . . . . . 7  |-  ( i  =  Y  ->  (
x  =  ( F `
 i )  <->  x  =  ( F `  Y ) ) )
87abbidv 2557 . . . . . 6  |-  ( i  =  Y  ->  { x  |  x  =  ( F `  i ) }  =  { x  |  x  =  ( F `  Y ) } )
95, 8iunxprg 30134 . . . . 5  |-  ( ( X  e.  V  /\  Y  e.  W )  ->  U_ i  e.  { X ,  Y }  { x  |  x  =  ( F `  i ) }  =  ( { x  |  x  =  ( F `  X ) }  u.  { x  |  x  =  ( F `  Y
) } ) )
109adantr 465 . . . 4  |-  ( ( ( X  e.  V  /\  Y  e.  W
)  /\  F  Fn  { X ,  Y }
)  ->  U_ i  e. 
{ X ,  Y }  { x  |  x  =  ( F `  i ) }  =  ( { x  |  x  =  ( F `  X ) }  u.  { x  |  x  =  ( F `  Y
) } ) )
11 iunab 4216 . . . 4  |-  U_ i  e.  { X ,  Y }  { x  |  x  =  ( F `  i ) }  =  { x  |  E. i  e.  { X ,  Y } x  =  ( F `  i
) }
12 df-sn 3878 . . . . . . 7  |-  { ( F `  X ) }  =  { x  |  x  =  ( F `  X ) }
1312eqcomi 2447 . . . . . 6  |-  { x  |  x  =  ( F `  X ) }  =  { ( F `  X ) }
14 df-sn 3878 . . . . . . 7  |-  { ( F `  Y ) }  =  { x  |  x  =  ( F `  Y ) }
1514eqcomi 2447 . . . . . 6  |-  { x  |  x  =  ( F `  Y ) }  =  { ( F `  Y ) }
1613, 15uneq12i 3508 . . . . 5  |-  ( { x  |  x  =  ( F `  X
) }  u.  {
x  |  x  =  ( F `  Y
) } )  =  ( { ( F `
 X ) }  u.  { ( F `
 Y ) } )
17 df-pr 3880 . . . . 5  |-  { ( F `  X ) ,  ( F `  Y ) }  =  ( { ( F `  X ) }  u.  { ( F `  Y
) } )
1816, 17eqtr4i 2466 . . . 4  |-  ( { x  |  x  =  ( F `  X
) }  u.  {
x  |  x  =  ( F `  Y
) } )  =  { ( F `  X ) ,  ( F `  Y ) }
1910, 11, 183eqtr3g 2498 . . 3  |-  ( ( ( X  e.  V  /\  Y  e.  W
)  /\  F  Fn  { X ,  Y }
)  ->  { x  |  E. i  e.  { X ,  Y }
x  =  ( F `
 i ) }  =  { ( F `
 X ) ,  ( F `  Y
) } )
202, 19eqtrd 2475 . 2  |-  ( ( ( X  e.  V  /\  Y  e.  W
)  /\  F  Fn  { X ,  Y }
)  ->  ran  F  =  { ( F `  X ) ,  ( F `  Y ) } )
2120ex 434 1  |-  ( ( X  e.  V  /\  Y  e.  W )  ->  ( F  Fn  { X ,  Y }  ->  ran  F  =  {
( F `  X
) ,  ( F `
 Y ) } ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756   {cab 2429   E.wrex 2716    u. cun 3326   {csn 3877   {cpr 3879   U_ciun 4171   ran crn 4841    Fn wfn 5413   ` cfv 5418
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 4413  ax-nul 4421  ax-pr 4531
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 2568  df-ne 2608  df-ral 2720  df-rex 2721  df-rab 2724  df-v 2974  df-sbc 3187  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-sn 3878  df-pr 3880  df-op 3884  df-uni 4092  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-id 4636  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-iota 5381  df-fun 5420  df-fn 5421  df-fv 5426
This theorem is referenced by:  imarnf1pr  30150
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