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Theorem imarnf1pr 32147
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 5721 . . . . . . . . 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 4121 . . . . . . . . . 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 6017 . . . . . . 7  |-  ( ( ( F : { X ,  Y } --> dom  E  /\  E : dom  E --> R )  /\  ( X  e.  V  /\  Y  e.  W
) )  ->  ( F `  X )  e.  dom  E )
9 prid2g 4122 . . . . . . . . 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 6017 . . . . . . 7  |-  ( ( ( F : { X ,  Y } --> dom  E  /\  E : dom  E --> R )  /\  ( X  e.  V  /\  Y  e.  W
) )  ->  ( F `  Y )  e.  dom  E )
12 fnimapr 5922 . . . . . . 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 1229 . . . . . 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 5721 . . . . . . . . 9  |-  ( F : { X ,  Y } --> dom  E  ->  F  Fn  { X ,  Y } )
18 rnfdmpr 32146 . . . . . . . . 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 2451 . . . 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 5327 . . 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 4096 . . . 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 2492 . 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 1383    e. wcel 1804   {cpr 4016   dom cdm 4989   ran crn 4990   "cima 4992    Fn wfn 5573   -->wf 5574   ` cfv 5578
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-sep 4558  ax-nul 4566  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rex 2799  df-rab 2802  df-v 3097  df-sbc 3314  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3771  df-if 3927  df-sn 4015  df-pr 4017  df-op 4021  df-uni 4235  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-id 4785  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-fv 5586
This theorem is referenced by: (None)
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