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Theorem rnfdmpr 30074
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 5735 . . . 4  |-  ( F  Fn  { X ,  Y }  ->  ran  F  =  { x  |  E. i  e.  { X ,  Y } x  =  ( F `  i
) } )
21adantl 463 . . 3  |-  ( ( ( X  e.  V  /\  Y  e.  W
)  /\  F  Fn  { X ,  Y }
)  ->  ran  F  =  { x  |  E. i  e.  { X ,  Y } x  =  ( F `  i
) } )
3 fveq2 5688 . . . . . . . 8  |-  ( i  =  X  ->  ( F `  i )  =  ( F `  X ) )
43eqeq2d 2452 . . . . . . 7  |-  ( i  =  X  ->  (
x  =  ( F `
 i )  <->  x  =  ( F `  X ) ) )
54abbidv 2555 . . . . . 6  |-  ( i  =  X  ->  { x  |  x  =  ( F `  i ) }  =  { x  |  x  =  ( F `  X ) } )
6 fveq2 5688 . . . . . . . 8  |-  ( i  =  Y  ->  ( F `  i )  =  ( F `  Y ) )
76eqeq2d 2452 . . . . . . 7  |-  ( i  =  Y  ->  (
x  =  ( F `
 i )  <->  x  =  ( F `  Y ) ) )
87abbidv 2555 . . . . . 6  |-  ( i  =  Y  ->  { x  |  x  =  ( F `  i ) }  =  { x  |  x  =  ( F `  Y ) } )
95, 8iunxprg 30059 . . . . 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 462 . . . 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 4213 . . . 4  |-  U_ i  e.  { X ,  Y }  { x  |  x  =  ( F `  i ) }  =  { x  |  E. i  e.  { X ,  Y } x  =  ( F `  i
) }
12 df-sn 3875 . . . . . . 7  |-  { ( F `  X ) }  =  { x  |  x  =  ( F `  X ) }
1312eqcomi 2445 . . . . . 6  |-  { x  |  x  =  ( F `  X ) }  =  { ( F `  X ) }
14 df-sn 3875 . . . . . . 7  |-  { ( F `  Y ) }  =  { x  |  x  =  ( F `  Y ) }
1514eqcomi 2445 . . . . . 6  |-  { x  |  x  =  ( F `  Y ) }  =  { ( F `  Y ) }
1613, 15uneq12i 3505 . . . . 5  |-  ( { x  |  x  =  ( F `  X
) }  u.  {
x  |  x  =  ( F `  Y
) } )  =  ( { ( F `
 X ) }  u.  { ( F `
 Y ) } )
17 df-pr 3877 . . . . 5  |-  { ( F `  X ) ,  ( F `  Y ) }  =  ( { ( F `  X ) }  u.  { ( F `  Y
) } )
1816, 17eqtr4i 2464 . . . 4  |-  ( { x  |  x  =  ( F `  X
) }  u.  {
x  |  x  =  ( F `  Y
) } )  =  { ( F `  X ) ,  ( F `  Y ) }
1910, 11, 183eqtr3g 2496 . . 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 2473 . 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 1364    e. wcel 1761   {cab 2427   E.wrex 2714    u. cun 3323   {csn 3874   {cpr 3876   U_ciun 4168   ran crn 4837    Fn wfn 5410   ` cfv 5415
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-sep 4410  ax-nul 4418  ax-pr 4528
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-rab 2722  df-v 2972  df-sbc 3184  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-nul 3635  df-if 3789  df-sn 3875  df-pr 3877  df-op 3881  df-uni 4089  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-id 4632  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-iota 5378  df-fun 5417  df-fn 5418  df-fv 5423
This theorem is referenced by:  imarnf1pr  30075
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