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Theorem imarnf1pr 30147
Description: The image of the range of a function  F under a function  E if  F is a function of a pair into the domain of  E. (Contributed by Alexander van der Vekens, 2-Feb-2018.)
Assertion
Ref Expression
imarnf1pr  |-  ( ( X  e.  V  /\  Y  e.  W )  ->  ( ( ( F : { X ,  Y } --> dom  E  /\  E : dom  E --> R )  /\  ( ( E `
 ( F `  X ) )  =  A  /\  ( E `
 ( F `  Y ) )  =  B ) )  -> 
( E " ran  F )  =  { A ,  B } ) )

Proof of Theorem imarnf1pr
StepHypRef Expression
1 ffn 5557 . . . . . . . . 9  |-  ( E : dom  E --> R  ->  E  Fn  dom  E )
21adantl 466 . . . . . . . 8  |-  ( ( F : { X ,  Y } --> dom  E  /\  E : dom  E --> R )  ->  E  Fn  dom  E )
32adantr 465 . . . . . . 7  |-  ( ( ( F : { X ,  Y } --> dom  E  /\  E : dom  E --> R )  /\  ( X  e.  V  /\  Y  e.  W
) )  ->  E  Fn  dom  E )
4 simpll 753 . . . . . . . 8  |-  ( ( ( F : { X ,  Y } --> dom  E  /\  E : dom  E --> R )  /\  ( X  e.  V  /\  Y  e.  W
) )  ->  F : { X ,  Y }
--> dom  E )
5 prid1g 3979 . . . . . . . . . 10  |-  ( X  e.  V  ->  X  e.  { X ,  Y } )
65adantr 465 . . . . . . . . 9  |-  ( ( X  e.  V  /\  Y  e.  W )  ->  X  e.  { X ,  Y } )
76adantl 466 . . . . . . . 8  |-  ( ( ( F : { X ,  Y } --> dom  E  /\  E : dom  E --> R )  /\  ( X  e.  V  /\  Y  e.  W
) )  ->  X  e.  { X ,  Y } )
84, 7ffvelrnd 5842 . . . . . . 7  |-  ( ( ( F : { X ,  Y } --> dom  E  /\  E : dom  E --> R )  /\  ( X  e.  V  /\  Y  e.  W
) )  ->  ( F `  X )  e.  dom  E )
9 prid2g 3980 . . . . . . . . 9  |-  ( Y  e.  W  ->  Y  e.  { X ,  Y } )
109ad2antll 728 . . . . . . . 8  |-  ( ( ( F : { X ,  Y } --> dom  E  /\  E : dom  E --> R )  /\  ( X  e.  V  /\  Y  e.  W
) )  ->  Y  e.  { X ,  Y } )
114, 10ffvelrnd 5842 . . . . . . 7  |-  ( ( ( F : { X ,  Y } --> dom  E  /\  E : dom  E --> R )  /\  ( X  e.  V  /\  Y  e.  W
) )  ->  ( F `  Y )  e.  dom  E )
12 fnimapr 5753 . . . . . . 7  |-  ( ( E  Fn  dom  E  /\  ( F `  X
)  e.  dom  E  /\  ( F `  Y
)  e.  dom  E
)  ->  ( E " { ( F `  X ) ,  ( F `  Y ) } )  =  {
( E `  ( F `  X )
) ,  ( E `
 ( F `  Y ) ) } )
133, 8, 11, 12syl3anc 1218 . . . . . 6  |-  ( ( ( F : { X ,  Y } --> dom  E  /\  E : dom  E --> R )  /\  ( X  e.  V  /\  Y  e.  W
) )  ->  ( E " { ( F `
 X ) ,  ( F `  Y
) } )  =  { ( E `  ( F `  X ) ) ,  ( E `
 ( F `  Y ) ) } )
1413ex 434 . . . . 5  |-  ( ( F : { X ,  Y } --> dom  E  /\  E : dom  E --> R )  ->  (
( X  e.  V  /\  Y  e.  W
)  ->  ( E " { ( F `  X ) ,  ( F `  Y ) } )  =  {
( E `  ( F `  X )
) ,  ( E `
 ( F `  Y ) ) } ) )
1514adantr 465 . . . 4  |-  ( ( ( F : { X ,  Y } --> dom  E  /\  E : dom  E --> R )  /\  ( ( E `  ( F `  X ) )  =  A  /\  ( E `  ( F `
 Y ) )  =  B ) )  ->  ( ( X  e.  V  /\  Y  e.  W )  ->  ( E " { ( F `
 X ) ,  ( F `  Y
) } )  =  { ( E `  ( F `  X ) ) ,  ( E `
 ( F `  Y ) ) } ) )
1615impcom 430 . . 3  |-  ( ( ( X  e.  V  /\  Y  e.  W
)  /\  ( ( F : { X ,  Y } --> dom  E  /\  E : dom  E --> R )  /\  ( ( E `
 ( F `  X ) )  =  A  /\  ( E `
 ( F `  Y ) )  =  B ) ) )  ->  ( E " { ( F `  X ) ,  ( F `  Y ) } )  =  {
( E `  ( F `  X )
) ,  ( E `
 ( F `  Y ) ) } )
17 ffn 5557 . . . . . . . . 9  |-  ( F : { X ,  Y } --> dom  E  ->  F  Fn  { X ,  Y } )
18 rnfdmpr 30146 . . . . . . . . 9  |-  ( ( X  e.  V  /\  Y  e.  W )  ->  ( F  Fn  { X ,  Y }  ->  ran  F  =  {
( F `  X
) ,  ( F `
 Y ) } ) )
1917, 18syl5com 30 . . . . . . . 8  |-  ( F : { X ,  Y } --> dom  E  ->  ( ( X  e.  V  /\  Y  e.  W
)  ->  ran  F  =  { ( F `  X ) ,  ( F `  Y ) } ) )
2019adantr 465 . . . . . . 7  |-  ( ( F : { X ,  Y } --> dom  E  /\  E : dom  E --> R )  ->  (
( X  e.  V  /\  Y  e.  W
)  ->  ran  F  =  { ( F `  X ) ,  ( F `  Y ) } ) )
2120adantr 465 . . . . . 6  |-  ( ( ( F : { X ,  Y } --> dom  E  /\  E : dom  E --> R )  /\  ( ( E `  ( F `  X ) )  =  A  /\  ( E `  ( F `
 Y ) )  =  B ) )  ->  ( ( X  e.  V  /\  Y  e.  W )  ->  ran  F  =  { ( F `
 X ) ,  ( F `  Y
) } ) )
2221impcom 430 . . . . 5  |-  ( ( ( X  e.  V  /\  Y  e.  W
)  /\  ( ( F : { X ,  Y } --> dom  E  /\  E : dom  E --> R )  /\  ( ( E `
 ( F `  X ) )  =  A  /\  ( E `
 ( F `  Y ) )  =  B ) ) )  ->  ran  F  =  { ( F `  X ) ,  ( F `  Y ) } )
2322eqcomd 2446 . . . 4  |-  ( ( ( X  e.  V  /\  Y  e.  W
)  /\  ( ( F : { X ,  Y } --> dom  E  /\  E : dom  E --> R )  /\  ( ( E `
 ( F `  X ) )  =  A  /\  ( E `
 ( F `  Y ) )  =  B ) ) )  ->  { ( F `
 X ) ,  ( F `  Y
) }  =  ran  F )
2423imaeq2d 5167 . . 3  |-  ( ( ( X  e.  V  /\  Y  e.  W
)  /\  ( ( F : { X ,  Y } --> dom  E  /\  E : dom  E --> R )  /\  ( ( E `
 ( F `  X ) )  =  A  /\  ( E `
 ( F `  Y ) )  =  B ) ) )  ->  ( E " { ( F `  X ) ,  ( F `  Y ) } )  =  ( E " ran  F
) )
25 preq12 3954 . . . 4  |-  ( ( ( E `  ( F `  X )
)  =  A  /\  ( E `  ( F `
 Y ) )  =  B )  ->  { ( E `  ( F `  X ) ) ,  ( E `
 ( F `  Y ) ) }  =  { A ,  B } )
2625ad2antll 728 . . 3  |-  ( ( ( X  e.  V  /\  Y  e.  W
)  /\  ( ( F : { X ,  Y } --> dom  E  /\  E : dom  E --> R )  /\  ( ( E `
 ( F `  X ) )  =  A  /\  ( E `
 ( F `  Y ) )  =  B ) ) )  ->  { ( E `
 ( F `  X ) ) ,  ( E `  ( F `  Y )
) }  =  { A ,  B }
)
2716, 24, 263eqtr3d 2481 . 2  |-  ( ( ( X  e.  V  /\  Y  e.  W
)  /\  ( ( F : { X ,  Y } --> dom  E  /\  E : dom  E --> R )  /\  ( ( E `
 ( F `  X ) )  =  A  /\  ( E `
 ( F `  Y ) )  =  B ) ) )  ->  ( E " ran  F )  =  { A ,  B }
)
2827ex 434 1  |-  ( ( X  e.  V  /\  Y  e.  W )  ->  ( ( ( F : { X ,  Y } --> dom  E  /\  E : dom  E --> R )  /\  ( ( E `
 ( F `  X ) )  =  A  /\  ( E `
 ( F `  Y ) )  =  B ) )  -> 
( E " ran  F )  =  { A ,  B } ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756   {cpr 3877   dom cdm 4838   ran crn 4839   "cima 4841    Fn wfn 5411   -->wf 5412   ` cfv 5416
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 2422  ax-sep 4411  ax-nul 4419  ax-pr 4529
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 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 3185  df-dif 3329  df-un 3331  df-in 3333  df-ss 3340  df-nul 3636  df-if 3790  df-sn 3876  df-pr 3878  df-op 3882  df-uni 4090  df-iun 4171  df-br 4291  df-opab 4349  df-mpt 4350  df-id 4634  df-xp 4844  df-rel 4845  df-cnv 4846  df-co 4847  df-dm 4848  df-rn 4849  df-res 4850  df-ima 4851  df-iota 5379  df-fun 5418  df-fn 5419  df-f 5420  df-fv 5424
This theorem is referenced by: (None)
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