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Theorem rnfdmpr 32551
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 5919 . . . 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 5872 . . . . . . . 8  |-  ( i  =  X  ->  ( F `  i )  =  ( F `  X ) )
43eqeq2d 2471 . . . . . . 7  |-  ( i  =  X  ->  (
x  =  ( F `
 i )  <->  x  =  ( F `  X ) ) )
54abbidv 2593 . . . . . 6  |-  ( i  =  X  ->  { x  |  x  =  ( F `  i ) }  =  { x  |  x  =  ( F `  X ) } )
6 fveq2 5872 . . . . . . . 8  |-  ( i  =  Y  ->  ( F `  i )  =  ( F `  Y ) )
76eqeq2d 2471 . . . . . . 7  |-  ( i  =  Y  ->  (
x  =  ( F `
 i )  <->  x  =  ( F `  Y ) ) )
87abbidv 2593 . . . . . 6  |-  ( i  =  Y  ->  { x  |  x  =  ( F `  i ) }  =  { x  |  x  =  ( F `  Y ) } )
95, 8iunxprg 32545 . . . . 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 4378 . . . 4  |-  U_ i  e.  { X ,  Y }  { x  |  x  =  ( F `  i ) }  =  { x  |  E. i  e.  { X ,  Y } x  =  ( F `  i
) }
12 df-sn 4033 . . . . . . 7  |-  { ( F `  X ) }  =  { x  |  x  =  ( F `  X ) }
1312eqcomi 2470 . . . . . 6  |-  { x  |  x  =  ( F `  X ) }  =  { ( F `  X ) }
14 df-sn 4033 . . . . . . 7  |-  { ( F `  Y ) }  =  { x  |  x  =  ( F `  Y ) }
1514eqcomi 2470 . . . . . 6  |-  { x  |  x  =  ( F `  Y ) }  =  { ( F `  Y ) }
1613, 15uneq12i 3652 . . . . 5  |-  ( { x  |  x  =  ( F `  X
) }  u.  {
x  |  x  =  ( F `  Y
) } )  =  ( { ( F `
 X ) }  u.  { ( F `
 Y ) } )
17 df-pr 4035 . . . . 5  |-  { ( F `  X ) ,  ( F `  Y ) }  =  ( { ( F `  X ) }  u.  { ( F `  Y
) } )
1816, 17eqtr4i 2489 . . . 4  |-  ( { x  |  x  =  ( F `  X
) }  u.  {
x  |  x  =  ( F `  Y
) } )  =  { ( F `  X ) ,  ( F `  Y ) }
1910, 11, 183eqtr3g 2521 . . 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 2498 . 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 1395    e. wcel 1819   {cab 2442   E.wrex 2808    u. cun 3469   {csn 4032   {cpr 4034   U_ciun 4332   ran crn 5009    Fn wfn 5589   ` cfv 5594
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pr 4695
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-iota 5557  df-fun 5596  df-fn 5597  df-fv 5602
This theorem is referenced by:  imarnf1pr  32552
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